# Extremes of reflecting Gaussian processes on discrete Grid

Krzysztof Dȩbicki  Krzysztof Dȩbicki, Mathematical Institute, University of
Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
<EMAIL_ADDRESS>and  Grigori Jasnovidov  Grigori
Jasnovidov, St. Petersburg Department of Steklov Mathematical Institute of
Russian Academy of Sciences, St. Petersburg, Russia; University of Lausanne,
Lausanne, Switzerland<EMAIL_ADDRESS>

Abstract: For $\\{X(t),t\in G_{\delta}\\}$ a centered Gaussian process with
stationary increments on a discrete grid
$G_{\delta}=\\{0,\delta,2\delta,...\\}$, where $\delta>0$, we investigate the
stationary reflected process

$Q_{\delta,X}(t)=\sup\limits_{s\in[t,\infty)\cap
G_{\delta}}\big{(}X(s)-X(t)-c(s-t)\big{)},\ t\in G_{\delta}$

with $c>0$. We derive the exact asymptotics of
$\mathbb{P}\left(\sup\limits_{t\in[0,T]\cap
G_{\delta}}Q_{\delta,X}(t)>u\right)$ and
$\mathbb{P}\left(\inf\limits_{t\in[0,T]\cap
G_{\delta}}Q_{\delta,X}(t)>u\right),$ as $u\to\infty$, with $T>0$. It appears
that $\varphi=\lim_{u\to\infty}\frac{\sigma^{2}(u)}{u}$ determines the
asymptotics, leading to three qualitatively different scenarios: $\varphi=0$,
$\varphi\in(0,\infty)$ and $\varphi=\infty$.

Key Words: Gaussian process; storage process; exact asymptotics; Piterbarg
property; fractional Brownian motion; discrete grid.

AMS Classification: Primary 60G15; secondary 60G70, 60K25

##  1\. Introduction

For $X(t),t\geq 0$ a centered Gaussian process with a.s. continuous sample
paths, stationary increments and variance function
$\sigma^{2}(t):=\operatorname{Var}\left(X(t)\right)$ such that
$\sigma^{2}(0)=0$, consider the reflected (at $0$) process

(1)
$\hat{Q}_{X}(t)=X(t)-ct+\max\left(\hat{Q}_{X}(0),-\inf_{s\in[0,t]}(X(s)-cs)\right),\
t\geq 0,$

where $c>0$.

Due to its relation with the solution of the Skorokhod problem that describes
the dynamics of the buffer content process in a fluid queue fed by $X$ and
emptied at rate $c$, properties of $\hat{Q}_{X}$ have been investigated for a
wide class of processes $X$, see e.g. [1, 2, 3, 4] and references therein.
Within this framework, the process $\hat{Q}_{X}(t)$ is called in the
literature as the storage process [3, 5].

Distributional properties of the unique stationary solution of (1), which has
the following representation

(2) $Q_{X}(t)=\sup_{t\leq s}\left(X(s)-X(t)-c(s-t)\right),$

were intensively analyzed. In particular, the exact asymptotics of
$\mathbb{P}\left(Q_{X}(0)>u\right)$, as $u\to\infty$, was derived for a wide
class of Gaussian processes $X$; see, e.g., [2, 6, 4, 7] and references
therein. An important direction of research on extremal behaviour of the
process $Q_{X}(t),t\geq 0$ was initiated by Piterbarg [3], where asymptotics
of $\sup_{t\in[0,T]}Q_{B_{\alpha}}(t)$ was derived, with $B_{\alpha}(t)$ a
fractional Brownian motion with Hurst parameter $\alpha\in(0,1)$. Notably, the
following asymptotic equivalence

$\mathbb{P}\left(Q_{B_{\alpha}}(0)>u\right)\sim\mathbb{P}\left(\sup_{t\in[0,T]}Q_{B_{\alpha}}(t)>u\right)$

for $\alpha>1/2$, called in [5] the Piterbarg property, took particular
interest and was further observed for more broad class of processes $X$ [5, 8,
4]. Complementary, in [4] it was shown that if $\alpha>1/2$, then the process
$Q_{B_{\alpha}}$ also possesses strong Piterbarg property, i.e. for all $T>0$

$\mathbb{P}\left(Q_{B_{\alpha}}(0)>u\right)\sim\mathbb{P}\left(\inf_{t\in[0,T]}Q_{B_{\alpha}}(t)>u\right)$

as $u\to\infty$.

From the point of view of the stochastic modelling or simulation techniques,
discrete-time models frequently appear to be more natural. However, despite of
its relevance in modelling of, e.g., queueing systems, much less is known on
distributional properties of the discrete counterpart of (2), i.e.,

(3) $\displaystyle Q_{\delta,X}(t)=\sup\limits_{s\in[t,\infty)\cap
G_{\delta}}\big{(}X(s)-X(t)-c(s-t)\big{)},\ t\in G_{\delta},$

where $G_{\delta}=\\{0,\delta,2\delta,\ldots,\\}$. A notable exceptions are
recent works [9, 10], where the exact asymptotics of
$\mathbb{P}\left(Q_{\delta,B_{\alpha}}(0)>u\right)$, as $u\to\infty$, was
derived.

In this contribution we extend the findings of [9, 11, 10] to a more general
class of Gaussian processes with stationary increments and derive the exact
asymptotics of

(4)
$\displaystyle\psi_{T,\delta}^{\sup}(u):=\mathbb{P}\left(\sup\limits_{t\in[0,T]\cap
G_{\delta}}Q_{\delta,X}(t)>u\right),\quad\psi^{\inf}_{T,\delta}(u):=\mathbb{P}\left(\inf\limits_{t\in[0,T]\cap
G_{\delta}}Q_{\delta,X}(t)>u\right),$

as $u\to\infty$, for $T>0$ and $\delta>0$, complementing results for
continuous time given in [3, 4].

It appears that the influence of the grid size $\delta$ on the asymptotics of
(4) strongly depends on the value of

(5)
$\displaystyle\varphi:=\lim_{u\rightarrow\infty}\frac{\sigma^{2}(u)}{u}\in[0,\infty],$

leading to three scenarios: $\varphi=0$, $\varphi\in(0,\infty)$ and
$\varphi=\infty$. The case $\varphi=\infty$ leads to the same asymptotics as
its continuous-time counterpart, which reflects the long-range dependance
property of $X$ when its variance $\sigma^{2}$ is superlinear. On the other
hand, the asymptotics for the case $\varphi=0$ strongly depends on the grid
size and is asymptotically negligible with respect to its continuous
counterpart. The third case $\varphi\in(0,\infty)$ needs particularly precise
analysis and leads to the asymptotics which differs with the continuous-time
case only by a constant (depending on $\delta$).

Outline of the paper. We introduce notation and assumptions on the process $X$
in Section 2. Then, in Theorem 2.1 we derive exact asymptotics of
$\mathbb{P}\left(Q_{\delta,X}(0)>u\right)$ as $u\to\infty$. Main results of
this contribution are presented in Section 3. In Section 4 we illustrate the
main findings of this paper by analysis of two important classes of Gaussian
processes, i.e. fractional Brownian motions and Gaussian integrated processes.
Section 5 consists of the proofs of the results derived in this paper.

##  2\. Notation and preliminary results

Let $X(t),t\in\mathbb{R}$ be a centered Gaussian process with stationary
increments, as introduced in Section 1. Suppose that

A: $\sigma^{2}$ is regularly varying at $\infty$ with index $2\alpha\in(0,2)$
and $\sigma^{2}(t)$ is twice continuously differentiable for any
$t\in(0,\infty)$. Further, the first and second derivatives of $\sigma^{2}$
are ultimately monotone.

We note that it follows straightforwardly from A that $\varphi=0$ if
$\alpha<1/2$ while for $\alpha>1/2$ we have $\varphi=\infty$. It appears that
case $\alpha=1/2$, i.e. when $\sigma^{2}$ is asymptotically close to a linear
function, needs particularly precise analysis, for which the following
condition is a tractable assumption:

B: If $\sigma^{2}$ satisfies A with $\alpha=\frac{1}{2}$, then $\varphi>0.$

Condition B excludes the cases when $\alpha=1/2$ but $\varphi=0$. For this
scenario we were able to give only partial results. We refer to Remark 3.2 for
the discussion of the extension of the results derived in Section 3 to the
case $\alpha=1/2,\varphi=0$ under an additional constrain on the variance
function $\sigma^{2}$ of $X$.

Conditions A and B are satisfied for a wide class of Gaussian processes with
stationary increments, including family of fractional Brownian motions and
integrated stationary Gaussian processes; see Section 4 for details. We note
that quantity $\varphi$ already appeared in [7], where it was observed that
the form of the asymptotic behavior, as $u\to\infty$, of
$\mathbb{P}\left(Q_{X}(0)>u\right)$ introduced in (2) is determined by the
value of $\varphi$. Let

(6) $\displaystyle\ \ \ \ \ \
\mathcal{H}_{\xi}(M)=\mathbb{E}\left(\sup\limits_{t\in
M}e^{\sqrt{2}\xi(t)-\operatorname{Var}(\xi(t))}\right)\in(0,\infty),\quad\mathcal{G}_{\xi}(M)=\mathbb{E}\left(\inf\limits_{t\in
M}e^{\sqrt{2}\xi(t)-\operatorname{Var}(\xi(t))}\right)\in(0,\infty),$

where $M$ is a compact subset of $\mathbb{R}$ and $\xi(t),\ t\in\mathbb{R}$ is
a Gaussian process with stationary increments and a.s. continuous sample
paths. Then, we define Pickands constant by

$\mathcal{H}_{\xi}^{\delta}=\lim_{S\to\infty}\frac{\mathcal{H}_{\xi}([0,S]\cap\delta\mathbb{Z})}{S},\
\ \ \ \ \ \ \delta\geq 0,$

where we set $\delta\mathbb{Z}=\mathbb{R}^{+}$ if $\delta=0$. We refer to [12]
for properties of $\mathcal{H}_{\xi}^{0}$. In Lemma 5.3 we prove that for
$\delta>0$ it is sufficient to suppose that $\xi$ satisfies A to claim that
$\mathcal{H}_{\xi}^{\delta}\in(0,\infty)$. Later on for $\delta=0$ we simply
write $\mathcal{H}_{\xi}$ instead of $\mathcal{H}_{\xi}^{0}$.

Next, let us recall the findings of [7][Proposition 2] (see also [8] [Theorems
3.1-3.3]) for the asymptotics of $\mathbb{P}\left(Q_{X}(0)>u\right)$, as
$u\to\infty$, which will be a useful benchmark for the results derived in the
next section. Let $\overleftarrow{\sigma}(t),t\geq 0$ stand for the asymptotic
inverse function of $\sigma$, i.e.,
$\overleftarrow{\sigma}(x)=\inf\\{y\in[0,\infty):f(y)>x\\}$ (for details and
properties of the asymptotic inverse functions see, e.g., [13]) and let

(7) $\displaystyle t_{*}=\frac{\alpha}{c(1-\alpha)},\ \ \ \
m(u)=\inf\limits_{t>0}\frac{u(1+ct)}{\sigma(ut)},\ \ \ \
\Delta(u)=\begin{cases}\overleftarrow{\sigma}\left(\frac{\sqrt{2}\sigma^{2}(ut_{*})}{u(1+ct_{*})}\right),&\varphi\notin(0,\infty)\\\
1,&\varphi\in(0,\infty).\end{cases}$

Let for $X$ such that $\varphi\in(0,\infty)$,

(8) $\displaystyle\eta(t)=\frac{c\sqrt{2}}{\varphi}X(t),\ \ \ \ \ t\geq 0.$

As shown in [7][Proposition 2], if $\sigma^{2}$ is regularly varying at $0$
with index $2\alpha_{0}\in(0,2]$ and A is satisfied, then for $\hat{Q}_{X}$
defined in (1), we have

(9) $\displaystyle\mathbb{P}\left(\hat{Q}_{X}(0)>u\right)\sim
f(u)\Psi(m(u))\times\begin{cases}\mathcal{H}_{B_{\alpha}},&\varphi=\infty\\\
\mathcal{H}_{\eta},&\varphi\in(0,\infty)\\\
\mathcal{H}_{B_{\alpha_{0}}},&\varphi=0\end{cases},\quad u\to\infty,$

where $\Psi$ is the survival function of a standard Gaussian random variable
and

(10) $\displaystyle f(u)=\sqrt{\frac{2\pi A}{B}}\frac{u}{m(u)\Delta(u)},\ \ \
\ A=\frac{1}{(1-\alpha)t_{*}^{\alpha}},\quad
B=\frac{\alpha}{t_{*}^{\alpha+2}}.$

The following result establishes the asymptotics of
$\mathbb{P}\left(Q_{\delta,X}(0)>u\right)$ for $\delta>0$ as $u\to\infty$. It
generalizes the findings of [9, 10], where the special case of $X$ being a
fractional Brownian motion was analyzed.

######  Theorem 2.1.

Let $X(t),t\geq 0$ be a centered Gaussian process with continuous trajectories
and stationary increments satisfying A, B. Then, for $\delta>0$, as
$u\to\infty$, it holds that

(11)
$\displaystyle\mathbb{P}\left(Q_{\delta,X}(0)>u\right)\sim\Psi(m(u))\times\begin{cases}\frac{\sqrt{2\pi\alpha}u}{\delta
c(1-\alpha)^{3/2}m(u)},&\varphi=0\\\
\mathcal{H}_{\eta}^{\delta}f(u),&\varphi\in(0,\infty)\\\
\mathcal{H}_{B_{\alpha}}f(u),&\varphi=\infty.\end{cases}$

######  Remark 2.2.

Providing that both the asymptotics (9) and (11) hold, we have that

$\displaystyle\lim_{u\to\infty}\mathbb{P}\left(Q_{\delta,X}(0)>u|Q_{X}(0)>u\right)=\begin{cases}0,&\varphi=0,\\\
\frac{\mathcal{H}_{\eta}^{\delta}}{\mathcal{H}_{\eta}},&\varphi\in(0,\infty),\\\
1,&\varphi=\infty.\end{cases}$

##  3\. Main Results

In this section we derive the exact asymptotics of

(12)
$\displaystyle\psi_{T,\delta}^{\sup}(u):=\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta}}Q_{\delta,X}(t)>u\right)\quad\text{and}\quad\psi^{\inf}_{T,\delta}(u):=\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}Q_{\delta,X}(t)>u\right),$

as $u\to\infty$, for $T>0$ and $\delta>0$, where for any real $a<b$ and
positive $\delta$

$[a,b]_{\delta}=[a,b]\cap\delta\mathbb{Z}.$

We begin with the asymptotics of $\psi_{T,\delta}^{\sup}(u)$ as $u\to\infty$.
Let in the following $[\cdot]$ stand for the integer part.

######  Theorem 3.1.

Let $X(t),t\geq 0$ be a centered Gaussian process with continuous trajectories
and stationary increments satisfying A-B. Then for $\delta>0$ as $u\to\infty$
it holds that

$\displaystyle\psi_{T,\delta}^{\sup}(u)\sim\Psi(m(u))\times\begin{cases}(1+[\frac{T}{\delta}])\frac{\sqrt{2\pi\alpha}u}{\delta
c(1-\alpha)^{3/2}m(u)},&\varphi=0\\\
\mathcal{H}_{\eta}([0,T]_{\delta})\mathcal{H}_{\eta}^{\delta}f(u),&\varphi\in(0,\infty)\\\
\mathcal{H}_{B_{\alpha}}f(u),&\varphi=\infty.\end{cases}$

######  Remark 3.2.

It follows straightforwardly from the proof of Theorem 2.1 and 3.1 that
condition B can be relaxed a bit. Namely, if $\varphi=0$ and
$\alpha=\frac{1}{2}$ and for
$\kappa=\sqrt{c\\!\\!\inf\limits_{t\in\\{\delta,2\delta,...\\}}\sigma(t)}-\varepsilon$,
with sufficiently small $\varepsilon>0$,

(13) $\displaystyle\sigma(u)\leq\kappa\frac{\sqrt{u}}{\ln^{1/4}u},\ \ \ \
u\to\infty$

then both Theorem 2.1 and 3.1 hold. If $\varphi=0$ and $\alpha=1/2$ in the
Theorem 3.1 and (13) does not hold, then it follows from its proof that (11)
reduces to the upper bound.

Next we analyze the asymptotical behaviour of $\psi^{\inf}_{T,\delta}(u)$, as
$u\to\infty$.

######  Theorem 3.3.

Let $X(t),t\geq 0$ be a centered Gaussian process with continuous trajectories
and stationary increments satisfying A-B. Then for $\delta>0$ as $u\to\infty$
it holds that

$\displaystyle\psi_{T,\delta}^{\inf}(u)\sim
f(u)\Psi(m(u))\times\begin{cases}\mathcal{G}_{\eta}([0,T]_{\delta})\mathcal{H}_{\eta}^{\delta},&\varphi\in(0,\infty)\\\
\mathcal{H}_{B_{\alpha}},&\varphi=\infty.\end{cases}$

If $\varphi=0$, then for the non-degenerated scenario (when set
$[0,T]_{\delta}$ consists of more than 1 element, i.e, for $T\geq\delta$) it
seems difficult to derive even logarithmic asymptotics of
$\psi_{T,\delta}^{\inf}(u)$. One can argue that $\psi_{T,\delta}^{\inf}(u)$ is
exponentially smaller than $\mathbb{P}\left(Q_{\delta,X}(0)>u\right)$ in this
case, as $u\to\infty$. We have the following proposition giving an upper bound
for $\psi_{T,\delta}^{\inf}(u)$.

######  Proposition 3.4.

If $\varphi=0$ and for some $\varepsilon>0$

(14) $\displaystyle\sigma(u)\leq\frac{\sqrt{u}}{\ln^{1/4+\varepsilon}u},\ \ \
\ \ \ \ u\to\infty,$

then for $T\geq\delta$ with any
$\widetilde{\mathbb{C}}<\frac{1+ct_{*}}{2t_{*}^{2\alpha}}\sup\limits_{t\in[0,T]_{\delta}}\sigma(t)$
it holds that

$\displaystyle\psi_{T,\delta}^{\inf}(u)\leq\Psi(m(u))\Psi\left(\widetilde{\mathbb{C}}\frac{u}{\sigma^{2}(u)}\right),\
\ \ \ u\to\infty.$

######  Remark 3.5.

Comparing the asymptotics in Theorems 2.1, 3.1 and 3.3 for case $\alpha>1/2$
we observe that the process $Q_{\delta,X}$ possesses the so-called strong
Piterbarg property, that is

$\displaystyle\psi_{T,\delta}^{\sup}(u)\sim\mathbb{P}\left(Q_{\delta,X}(0)>u\right)\sim\psi_{T,\delta}^{\inf}(u),\
\ \ \ \ \ \ T\geq 0,\ u\to\infty.$

The analogous property was observed for the continuous-times analog of
$Q_{\delta,X}$; see [4].

##  4\. Examples

We illustrate the findings of this contribution by application of Theorems 3.1
and 3.3 to the family of fractional Brownian motions and Gaussian integrated
processes.

###  4.1. Fractional Brownian motion.

Let

$C_{H}=\frac{c^{H}}{H^{H}(1-H)^{1-H}},\ \
D_{H}=\frac{\sqrt{2\pi}H^{H+1/2}}{c^{H+1}(1-H)^{H+1/2}},\ \
E_{H}=\frac{2^{\frac{1}{2}-\frac{1}{2H}}\sqrt{\pi}}{H^{1/2}(1-H)^{1/2}}.$

Applying Theorems 2.1, 3.1 and 3.3 for $X(t)=B_{H}(t)$ being a standard
fractional Brownian motion with Hurst parameter $H\in(0,1)$, we obtain the
following results.

######  Corollary 4.1.

As $u\to\infty$ it holds that

$\displaystyle\mathbb{P}\left(Q_{\delta,B_{H}}(0)>u\right)\sim\begin{cases}\frac{D_{H}u^{H}}{\delta}\Psi(C_{H}u^{1-H}),&H<1/2\\\
\mathcal{H}^{2c^{2}\delta}_{B_{1/2}}e^{-2cu},&H=1/2\\\
\mathcal{H}_{B_{H}}E_{H}(C_{H}u^{1-H})^{1/H-1}\Psi(C_{H}u^{1-H}),&H>1/2.\end{cases}$

######  Corollary 4.2.

For $T,\delta>0$ as $u\to\infty$ it holds that

$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta}}Q_{\delta,B_{H}}(t)>u\right)\sim\begin{cases}(1+[\frac{T}{\delta}])\frac{D_{H}u^{H}}{\delta}\Psi(C_{H}u^{1-H}),&H<1/2\\\
\mathcal{H}_{B_{1/2}}([0,2c^{2}T]_{2c^{2}\delta})\mathcal{H}^{2c^{2}\delta}_{B_{1/2}}e^{-2cu},&H=1/2\\\
\mathcal{H}_{B_{H}}E_{H}(C_{H}u^{1-H})^{1/H-1}\Psi(C_{H}u^{1-H}),&H>1/2.\end{cases}$

and

$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}Q_{\delta,B_{H}}(t)>u\right)\sim\begin{cases}\mathcal{G}_{B_{1/2}}([0,2c^{2}T]_{2c^{2}\delta})\mathcal{H}_{B_{1/2}}^{2c^{2}\delta}e^{-2cu},&H=1/2\\\
\mathcal{H}_{B_{H}}E_{H}(C_{H}u^{1-H})^{1/H-1}\Psi(C_{H}u^{1-H}),&H>1/2.\end{cases}$

Note that Corollary 4.1 intersects with the results in [10, 9, 11] while
Corollary 4.2 provides a discrete counterpart of Theorems 5-7 in [3] and
Theorem 1 in [4], respectively.

###  4.2. Gaussian integrated processes.

For a stationary centered Gaussian process with a.s. continuous sample paths
$\zeta(s),\ s\geq 0$ define the integrated process by

(15) $\displaystyle Z(t)=\int\limits_{0}^{t}\zeta(s)ds,\ \ \ \ \ t\geq 0.$

This process is also Gaussian, has a.s. continuous sample paths and stationary
increments. In what follows we consider two classes of processes $Z$, which
differ by property of the correlation function
$R(t):=\mathbb{E}\left(\zeta(0)\zeta(t)\right)$ of $\zeta$ as $t\to\infty$.

_SRD case._ Following, e.g., [14] (see also [15]), we impose the following
conditions on the correlation of $\zeta$:
S1: $R(t)\in C([0,\infty)),\ \lim\limits_{t\to\infty}tR(t)=0$;
S2: $\int\limits_{0}^{t}R(s)ds>0$ for all $t\in(0,\infty]$;
S3: $\int\limits_{0}^{\infty}t^{2}|R(t)|dt<\infty$.
The above assertions imply the existence of the first and second derivatives
of $\sigma^{2}_{Z}(t)=Var(Z(t))$ and establish the asymptotic behavior of
$\sigma^{2}_{Z}(t)$ at $\infty$ (see e.g., Remark 6.1 in [14]):

$\sigma^{2}_{Z}(t)=\frac{2}{G}t-2D+o(t^{-1}),\ \ \ \ \ \ \ t\to\infty,$

where $G=1/\int\limits_{0}^{\infty}R(t)dt$ and
$D=\int\limits_{0}^{\infty}tR(t)dt$. Thus, $\sigma^{2}_{Z}$ satisfies A-B with
$\alpha=1/2$ and $\varphi=\frac{2}{G}>0$. Hence applying Theorems 2.1, 3.1 and
3.3, the following corollary holds.

######  Corollary 4.3.

Suppose that $\zeta$ satisfies S1-S3. Then for $T\geq 0$ and $\delta>0$ as
$u\to\infty$

$\displaystyle\mathbb{P}\left(Q_{\delta,Z}(0)>u\right)$ $\displaystyle\sim$
$\displaystyle\mathcal{A}\mathcal{H}^{\delta}_{\xi}e^{-cGu},$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta}}Q_{\delta,Z}(t)>u\right)$
$\displaystyle\sim$
$\displaystyle\mathcal{A}\mathcal{H}_{\xi}([0,T]_{\delta})\mathcal{H}^{\delta}_{\xi}e^{-cGu},$
$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}Q_{\delta,Z}(t)>u\right)$
$\displaystyle\sim$
$\displaystyle\mathcal{A}\mathcal{G}_{\xi}([0,T]_{\delta})\mathcal{H}^{\delta}_{\xi}e^{-cGu},$

where $\mathcal{A}=\frac{1}{c^{2}Ge^{c^{2}G^{2}D}}$ and
$\xi(t)=cGZ(t)/\sqrt{2}$.

Note that the first asymptotics in Corollary 4.3 differs from its continuous-
time analog (Theorem 5.1 in [14]) only by the corresponding Pickands
constants.

_LRD case._ Following, e.g., [6, 7] we characterize LRD case by the following
assumptions on the covariance function $R(t)$ of the process $\zeta$:
L1: $R(t)$ is a continuous strictly positive function for $t\geq 0$;
L2: $R(t)$ is regularly varying at $\infty$ with index $2\alpha-2$,
$\alpha\in(1/2,1)$.

Under the above assumptions, by Karamata’s theorem, $\sigma_{Z}^{2}$ is
regularly varying at $\infty$ with index $2\alpha$. Since $2\alpha>1$ we are
in $\varphi=\infty$ scenario. Hence, applying Theorems 2.1, 3.1 and 3.3 we
obtain the following result, which shows that $Q_{\delta,Z}(t)$ possesses the
strong Piterbarg property.

######  Corollary 4.4.

Suppose that $\zeta$ satisfies L1-L2. Then as $u\to\infty$ it holds that

$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}Q_{\delta,Z}(t)>u\right)\sim\mathbb{P}\left(Q_{\delta,Z}(0)>u\right)$
$\displaystyle\sim$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta}}Q_{\delta,Z}(t)>u\right)$
$\displaystyle\sim$ $\displaystyle\mathcal{H}_{B_{\alpha}}f(u)\Psi(m(u)),$

where $m(u)$ and $f(u)$ are defined in (7) and (10), respectively.

##  5\. Proofs

In this section we give proofs of all the results presented in this
contribution. Hereafter, denote by $\mathbb{C}$, $\mathbb{C}_{i},\
i=1,2,3,\dots$ positive constants that may differ from line to line and
$\overline{X}:=\frac{X}{\sqrt{Var(X)}}$ for any nontrivial random variable
$X$. For any $u>0$ we have

$\mathbb{P}\left(Q_{\delta,X}(0)>u\right)=\mathbb{P}\left(\sup_{t\in
G_{\delta}}(X(t)-ct)>u\right)=\mathbb{P}\left(\sup_{t\in
G_{\delta/u}}X_{u}(t)>m(u)\right),$

where $m(u)$ is defined in (7) and

$X_{u}(t)=\frac{X(ut)}{u(1+ct)}m(u).$

Denote by $\sigma_{X_{u}}^{2}$ the variance function of $X_{u}(t),\ t\geq 0$.
In the next lemma we focus on asymptotic properties of the variance and
correlation functions of $X_{u}(t)$; we refer to, e.g., [8] for the proof.

######  Lemma 5.1.

Suppose that A is satisfied. For $u$ large enough the maximizer $t_{u}$ of
$\sigma_{X_{u}}$ is unique and $t_{u}\rightarrow
t_{*}=\frac{\alpha}{c(1-\alpha)}$ as $u\to\infty$. Moreover, for
$\delta_{u}>0$ satisfying $\lim\limits_{u\rightarrow\infty}\delta_{u}=0$ ($A$,
$B$ are defined in (10))

$\displaystyle\lim_{u\rightarrow\infty}\sup_{t\in(t_{u}-\delta_{u},t_{u}+\delta_{u})\setminus\\{t_{u}\\}}\left|\frac{1-\sigma_{X_{u}}(t)}{\frac{B}{2A}(t-t_{u})^{2}}-1\right|=0$

and (recall, $\sigma^{2}$ is the variance of $X$)

$\displaystyle\lim_{u\rightarrow\infty}\sup_{s\neq
t,s,t\in(t_{u}-\delta_{u},t_{u}+\delta_{u})}\left|\frac{1-Cor\left(X(us),X(ut)\right)}{\frac{\sigma^{2}(u|s-t|)}{2\sigma^{2}(ut_{*})}}-1\right|=0.$

By Lemma 5.1 we have that $t_{u}$ is the unique minimizer of
$\frac{u(1+ct)}{\sigma(ut)}$ for large $u$ and hence by Potter’s theorem
(Theorem 1.5.6 in [16]) we obtain useful in the following proofs asymptotics
of $m(u)$

(16) $\displaystyle
m(u)=\frac{u(1+ct_{u})}{\sigma(ut_{u})}\sim\frac{u(1+ct_{*})}{\sigma(ut_{*})}\cdot\frac{\sigma(ut_{*})}{\sigma(ut_{u})}\sim\frac{u(1+ct_{*})}{t_{*}^{\alpha}\sigma(u)},\quad
u\to\infty.$

Observe that

$\displaystyle\psi_{T,\delta}^{\sup}(u)=\mathbb{P}\left(\sup\limits_{t\in[0,T/u]_{\delta/u},t\leq
s\in G_{\delta/u}}Z_{u}(t,s)>m(u)\right),$

where

(17) $\displaystyle Z_{u}(t,s)=\frac{X(us)-X(ut)}{u(1+c(s-t))}m(u).$

Notice that for the variance $\sigma_{Z_{u}}^{2}$ of $Z_{u}$ it holds, that
$\sigma_{Z_{u}}^{2}(s,t)=\sigma_{X_{u}}^{2}(s-t)$ and for correlation
$r_{Z_{u}}$ we have for $\delta_{u}>0$ satisfying
$\lim_{u\rightarrow\infty}\delta_{u}=0$ (Lemma 5.4. in [8])

(18)
$\displaystyle\lim_{u\to\infty}\sup\limits_{|t-t_{1}|<\delta_{u},s-t,s_{1}-t_{1}\in(-\delta_{u}+t_{u},t_{u}+\delta_{u}),(s,t)\neq(s_{1},t_{1})}\Big{|}\frac{1-r_{Z_{u}}(s,t,s_{1},t_{1})}{\frac{\sigma^{2}(u|s-s_{1}|)+\sigma^{2}(u|t-t_{1}|)}{2\sigma^{2}(ut_{*})}}-1\Big{|}=0.$

To the rest of the paper we suppose that

$\displaystyle\delta_{u}=\begin{cases}u^{-1/2}\ln u,&\varphi<\infty\\\
u^{-1}\ln(u)\sigma(u),&\varphi=\infty\end{cases}$

and set

(19) $\displaystyle
I(t_{u})=G_{\frac{\delta}{u}}\bigcap(-\delta_{u}+t_{u},t_{u}+\delta_{u})$

for $u>0$. The following lemma allow us to extract the main area contributing
in the asymtotics of $\psi_{T,\delta}^{\sup}(u),\psi_{T,\delta}^{\inf}(u)$ and
$\mathbb{P}\left(Q_{\delta,X}(0)>u\right)$ as $u\to\infty$.

######  Lemma 5.2.

For any $T\geq 0$ it holds that, as $u\to\infty$,

$\displaystyle\psi_{T,\delta}^{\sup}(u)$ $\displaystyle\sim$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T/u]_{\delta/u},s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)$ $\displaystyle\psi_{T,\delta}^{\inf}(u)$
$\displaystyle\sim$
$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,T/u]_{\delta/u},s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right).$

In the next lemma we prove that the discrete Pickands constant appearing in
Theorems 2.1, 3.1 and 3.3 is well defined, positive and finite.

######  Lemma 5.3.

For any $\delta\geq 0$ and $\eta$ a centered Gaussian process with stationary
increments, a.s. continuous sample paths and variance satisfying A it holds,
that

(20)
$\displaystyle\lim_{S\to\infty}\frac{\mathcal{H}_{\eta}([0,S]_{\delta})}{S}=\mathcal{H}^{\delta}_{\eta}\in(0,\infty).$

The lemma below allows us to give upper bounds for the double-sum terms
appearing in case $\varphi=0$ in Theorems 2.1 and 3.3.

######  Lemma 5.4.

Assume that $\varphi=0$. Then uniformly for $t\neq s\in I(t_{u})$ and all
large $u$ with some $\varepsilon>0$ it holds that

$\displaystyle\mathbb{P}\left(X_{u}(t)>m(u),X_{u}(s)>m(u)\right)\leq
u^{-1/2-\varepsilon}\Psi(m(u)).$

The proofs of Lemmas 5.2, 5.3 and 5.4 are given in the Appendix.

###  5.1.  Proof of Theorem 2.1 .

Taking $T=0$ in Lemma 5.2 we obtain that

(21)
$\displaystyle\mathbb{P}\left(Q_{\delta_{X}}(0)>u\right)\sim\mathbb{P}\left(\sup\limits_{t\in
I(t_{u})}X_{u}(t)>m(u)\right),\ \ \ u\to\infty,$

where $I(t_{u})$ is defined in (19). Next we consider 3 cases: $\varphi=0$,
$\varphi\in(0,\infty)$ and $\varphi=\infty$.

_Case $\varphi=0$._ We have by Bonferroni inequality

(22) $\displaystyle\sum\limits_{t\in
I(t_{u})}\mathbb{P}\left(X_{u}(t)>m(u)\right)$ $\displaystyle\geq$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in I(t_{u})}X_{u}(t)>m(u)\right)$
$\displaystyle\geq$ $\displaystyle\sum\limits_{t\in
I(t_{u})}\mathbb{P}\left(X_{u}(t)>m(u)\right)-\sum\limits_{t\neq s\in
I(t_{u})}\mathbb{P}\left(X_{u}(t)>m(u),X_{u}(s)>m(u)\right).$

There are less then $\mathbb{C}u\ln^{2}u$ summands in the double-sum above,
hence by Lemma 5.4 we have

(23) $\displaystyle\sum\limits_{t\neq s\in
I(t_{u})}\mathbb{P}\left(X_{u}(t)>m(u),X_{u}(s)>m(u)\right)\leq\mathbb{C}\ln^{2}(u)u^{1/2-\varepsilon^{\prime}}\Psi(m(u)),\
\ \ \ u\to\infty.$

Next we focus on calculation of the single sum in (22). Since by Lemma 5.1
$\sup\limits_{t\in I(t_{u})}|\sigma_{X_{u}}(t)-1|\to 0$ as $u\to\infty$ the
following inequality (see, e.g., Lemma 2.1 in [17])

$(1-\frac{1}{x^{2}})\frac{1}{\sqrt{2\pi}x}e^{-x^{2}/2}\leq\Psi(x)\leq\frac{1}{\sqrt{2\pi}x}e^{-x^{2}/2},\
\ \ \ x>0,$

implies as $u\to\infty$

(24) $\displaystyle\sum\limits_{t\in
I(t_{u})}\mathbb{P}\left(X_{u}(t)>m(u)\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{t\in I(t_{u})}\Psi(\frac{m(u)}{\sigma_{X_{u}}(t)})$
$\displaystyle\sim$ $\displaystyle\sum\limits_{t\in
I(t_{u})}\frac{\sigma_{X_{u}}(t)}{\sqrt{2\pi}m(u)}e^{-\frac{m^{2}(u)}{2\sigma^{2}_{X_{u}}(t)}}$
$\displaystyle\sim$
$\displaystyle\frac{e^{-\frac{m^{2}(u)}{2}}}{\sqrt{2\pi}m(u)}\\!\sum\limits_{t\in
I(t_{u})}e^{-\frac{m^{2}(u)}{2\sigma_{X_{u}}^{2}(t)}+\frac{m^{2}(u)}{2}}$
$\displaystyle\sim$ $\displaystyle\Psi(m(u))\sum\limits_{t\in
I(t_{u})}e^{-\frac{m^{2}(u)}{2}\frac{(1-\sigma_{X_{u}}^{2}(t))}{\sigma^{2}_{X_{u}}(t)}}.$

By Lemma 5.1 we have that as $u\to\infty$ the last sum above is asymptotically
equivalent to

$\displaystyle\sum\limits_{t\in
I(t_{u})}e^{-m^{2}(u)\frac{B}{2A}(t-t_{u})^{2}}$ $\displaystyle=$
$\displaystyle\sum\limits_{t\in(-\frac{\ln u}{\sqrt{u}},\frac{\ln
u}{\sqrt{u}})_{\delta/u}}e^{-m^{2}(u)\frac{B}{2A}t^{2}}$ $\displaystyle=$
$\displaystyle\frac{u}{\delta m(u)}\Big{(}\frac{\delta
m(u)}{u}\sum\limits_{t\in(-\frac{m(u)\ln u}{\sqrt{u}},\frac{m(u)\ln
u}{\sqrt{u}})_{\delta m(u)/u}}e^{-\frac{B}{2A}t^{2}}\Big{)}$
$\displaystyle\sim$ $\displaystyle\frac{u}{\delta
m(u)}\int\limits_{\mathbb{R}}e^{-\frac{B}{2A}t^{2}}dt$ $\displaystyle=$
$\displaystyle\frac{u}{\delta m(u)}\sqrt{\frac{2\pi A}{B}}=\frac{u}{\delta
m(u)}\frac{\sqrt{2\pi\alpha}}{c(1-\alpha)^{3/2}},$

where the asymptotic equivalence in (5.1) holds, since by (16), $\frac{\delta
m(u)}{u}\to 0$ and $\frac{m(u)\ln u}{\sqrt{u}}\to\infty$ as $u\to\infty$.
Thus,

(26) $\displaystyle\sum\limits_{t\in
I(t_{u})}\mathbb{P}\left(X_{u}(t)>m(u)\right)\sim\frac{\sqrt{2\pi\alpha}u\Psi(m(u))}{\delta
c(1-\alpha)^{3/2}m(u)},\ \ \ u\to\infty$

and hence by (16), (22) and (23) we have that

$\mathbb{P}\left(\sup\limits_{t\in
I(t_{u})}X_{u}(t)>m(u)\right)\sim\frac{\sqrt{2\pi\alpha}u\Psi(m(u))}{\delta
c(1-\alpha)^{3/2}m(u)},\ \ \ u\to\infty$

and the claim follows by (21).

_Cases $\varphi\in(0,\infty)$ and $\varphi=\infty$._ For any fixed $u>0$ and
$S\in\\{0,\delta,2\delta,...\\}$ denote

$N_{u}=\lceil\frac{u\delta_{u}}{S\Delta(u)}\rceil,\ \ \ \
t_{j}=\frac{\Delta(u)jS}{u},\ \ \ \
\Delta_{j,S,u}=[t_{u}+t_{j},t_{u}+t_{j+1}]_{\delta/u},\ \ \ \
j\in[-N_{u}-1,N_{u}],$

where $\lceil\cdot\rceil$ is the ceiling function. We have by Bonferroni
inequality that

(27) $\displaystyle\sum\limits_{-N_{u}\leq j\leq
N_{u}-1}p_{j,S,u}-\sum\limits_{-N_{u}-1\leq i\neq j\leq
N_{u}}p_{i,j;S,u}\leq\mathbb{P}\left(\sup\limits_{t\in
I(t_{u})}X_{u}(t)>m(u)\right)\leq\sum\limits_{-N_{u}-1\leq j\leq
N_{u}}p_{j,S,u},$

where

$\displaystyle
p_{j,S,u}=\mathbb{P}\left(\sup\limits_{t\in\Delta_{j,S,u}}X_{u}(t)>m(u)\right)\
\ \ \ \text{ and }\ \ \ \
p_{i,j;S,u}=\mathbb{P}\left(\sup\limits_{t\in\Delta_{j,S,u}}X_{u}(t)>m(u),\sup\limits_{t\in\Delta_{i,S,u}}X_{u}(t)>m(u)\right).$

By [8] (_proof of lower bound of_ $\pi_{T_{u}}(u)$, p. 288) we have as
$u\to\infty$ and then $S\to\infty$

$\sum\limits_{-N_{u}-1\leq i\neq j\leq
N_{u}}p_{i,j;S,u}=o\left(\frac{u}{m(u)\Delta(u)}\Psi(m(u))\right).$

Hence from the asymtotics of $\sum\limits_{-N_{u}\leq j\leq N_{u}-1}p_{j,S,u}$
given in (28) and (29) we obtain that as $u\to\infty$ and then $S\to\infty$

$\mathbb{P}\left(\sup\limits_{t\in
I(t_{u})}X_{u}(t)>m(u)\right)\sim\sum\limits_{-N_{u}\leq j\leq
N_{u}}p_{j,S,u}$

and we need to calculate the asymptotics of the sum above. That can be done
via uniform approximation of $p_{j,S,u}$ for all $-N_{u}-1\leq j\leq N_{u}$
separately for cases $\varphi\in(0,\infty)$ and $\varphi=\infty$.

_Case $\varphi\in(0,\infty)$._ Let $\Delta(u)=1$,
$N_{u}=\lceil\frac{\sqrt{u}\ln u}{S}\rceil$, $t_{j}=\frac{jS}{u}$ and
$\Delta_{j,S,u}=[t_{u}+t_{j},t_{u}+t_{j+1}]_{\delta/u}$. We have for any
$\varepsilon>0,\ 0\leq j\leq N_{u}$ for all $u$ large enough with
$m^{-}_{j}(u)=\frac{m(u)}{1-(1-\varepsilon)\frac{B}{2A}(\frac{jS}{u})^{2}}$

$\displaystyle p_{j,S,u}=\mathbb{P}\left(\exists
t\in\Delta_{j,S,u}:\overline{X}_{u}(t)>\frac{m(u)}{\sigma_{u}(t)}\right)$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\left(\exists
t\in\Delta_{j,S,u}:\overline{X}_{u}(t)>\frac{m(u)}{1-(1-\varepsilon)\frac{B}{2A}(t-t_{u})^{2}}\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in\Delta_{j,S,u}}\overline{X}_{u}(t)>m^{-}_{j}(u)\right)$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,S]_{\delta}}\overline{X}_{u}(\frac{t+ut_{u}+ut_{j}}{u})>m_{j}(u)\right)$
$\displaystyle=:$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,S]_{\delta}}\overline{X}_{u}^{\prime}(t)>m_{j}(u)\right).$

By Lemma 5.1 we have that
$1-r_{\overline{X}^{\prime}_{u}}(t,s)\sim\frac{\sigma^{2}(|t-s|)}{2\sigma^{2}(ut_{*})},\
u\to\infty,\ t,s\in[0,S]$. Thus, by Lemma 1 in [4] the last probability above
is asymptotically equal to
$\mathcal{H}_{\eta^{\prime}}([0,S]_{\delta})\Psi(m^{-}_{j}(u)),$ as
$u\to\infty$, where $\eta^{\prime}$ is a centered Gaussian process with
stationary increments, a.s. continuous sample paths and variance (asymptotics
of $m(u)$ is given in (16))

$\sigma^{2}_{\eta^{\prime}}(t)=\lim_{u\to\infty}\frac{m^{2}(u)}{2\sigma^{2}(ut_{*})}\sigma^{2}(t)=\lim_{u\to\infty}\frac{u^{2}(1+ct_{*})^{2}}{2t_{*}\sigma^{2}(ut_{*})\sigma^{2}(u)}\sigma^{2}(t)=\frac{2c^{2}}{\varphi^{2}}\sigma^{2}(t).$

Note that $\eta^{\prime}$ and $\eta$ defined in (8) have the same
distributions. Thus,

$\sum\limits_{0\leq j\leq
N_{u}}p_{j,S,u}\leq\mathcal{H}_{\eta}([0,S]_{\delta})\sum\limits_{0\leq j\leq
N_{u}}\Psi(m^{-}_{j}(u)),\ \ \ u\to\infty.$

Next, as $u\to\infty$, (set $C_{-}=\frac{(1-\varepsilon)B}{2A}$) using the
same reasoning as in case $\varphi=0$, we have

$\displaystyle\frac{\sum\limits_{0\leq j\leq
N_{u}}\Psi(m^{-}_{j}(u))}{\Psi(m(u))}$ $\displaystyle\sim$
$\displaystyle\sum\limits_{0\leq j\leq
N_{u}}e^{-\frac{m^{2}(u)}{2}(\frac{1}{(1-C_{-}(\frac{jS}{u})^{2})^{2}}-1)}$
$\displaystyle\sim$ $\displaystyle\sum\limits_{0\leq j\leq
N_{u}}e^{-\frac{m^{2}(u)}{2}2C_{-}(\frac{jS}{u})^{2}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\frac{Sm(u)}{u}\in[0,\frac{N_{u}Sm(u)}{u}]_{\frac{Sm(u)}{u}}}e^{-C_{-}(\frac{jSm(u)}{u})^{2}}$
$\displaystyle=$
$\displaystyle\frac{u}{Sm(u)}\Big{(}\frac{Sm(u)}{u}\sum\limits_{t\in[0,\frac{m(u)\ln
u}{\sqrt{u}}]_{\frac{Sm(u)}{u}}}e^{-C_{-}t^{2}}\Big{)}.$

Due to (16) we have $\frac{u}{Sm(u)}\to 0$ and $\frac{m(u)\ln
u}{\sqrt{u}}\to\infty$, as $u\to\infty$. Thus, the sum above converges to
$\int\limits_{0}^{\infty}e^{-C_{-}t^{2}}dt=\frac{\sqrt{\pi}}{2\sqrt{C}_{-}}$
as $u\to\infty$. Similar calculation can be done for $j<0$. Hence, we have as
$u\to\infty$ and then $S\to\infty$

$\sum\limits_{-N_{u}\leq j\leq
N_{u}}p_{j,S,u}\leq\mathcal{H}_{\eta}([0,S]_{\delta})\frac{u\Psi(m(u))}{Sm(u)}\frac{\sqrt{\pi}}{\sqrt{C}_{-}}.$

By Lemma 5.3 we know that
$\frac{\mathcal{H}_{\eta}([0,S]_{\delta})}{S}\to\mathcal{H}^{\delta}_{\eta}\in(0,\infty)$
as $S\to\infty$. Hence, letting $S\to\infty$, we have

$\sum\limits_{-N_{u}\leq j\leq
N_{u}}p_{j,S,u}\leq\mathcal{H}^{\delta}_{\eta}\frac{u\Psi(m(u))}{m(u)}\frac{\sqrt{\pi}}{\sqrt{C}_{-}}(1+o(1)),\
\ \ \ u\to\infty.$

By the same arguments we have the asymptotic lower bound

$\sum\limits_{-N_{u}\leq j\leq
N_{u}}p_{j,S,u}\geq\mathcal{H}^{\delta}_{\eta}\frac{u\Psi(m(u))}{m(u)}\frac{\sqrt{\pi}}{\sqrt{C}_{+}}(1+o(1)),\
\ \ \ u\to\infty,$

with $C_{+}=\frac{(1+\varepsilon)B}{2A}$. Hence letting $\varepsilon\to 0$ we
have that, as $S\to\infty$ and then $u\to\infty$,

(28) $\displaystyle\sum\limits_{-N_{u}\leq j\leq
N_{u}}p_{j,S,u}\sim\mathcal{H}^{\delta}_{\eta}\frac{u}{m(u)}\sqrt{\frac{2\pi
A}{B}}\Psi(m(u)),$

which completes the proof of the case $\varphi\in(0,\infty)$.

Case $\varphi=\infty$. In this case we have that $\Delta(u),N_{u}\to\infty$
and $\Delta(u)/u\to 0$ as $u\to\infty$. Moreover, for
$m_{k}(u)=\frac{m(u)}{1-C_{+}|t_{k}|^{2}},\ \ -N_{u}\leq k\leq N_{u}$ for
large $S,u$

$\displaystyle p_{k,S,u}$ $\displaystyle\geq$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[t_{k},t_{k+1}]_{\delta/u}}\overline{X}_{u}(t)>m_{k}(u)\right)$
$\displaystyle\geq$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[t_{k},t_{k+1}]_{\frac{\delta[\varepsilon_{1}\Delta(u)]}{u}}}\overline{X}_{u}(t)>m_{k}(u)\right)$
$\displaystyle\geq$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,S-\frac{\delta[\varepsilon_{1}\Delta(u)]}{\Delta(u)}]_{\delta\varepsilon_{1}\frac{[\varepsilon_{1}\Delta(u)]}{\varepsilon_{1}\Delta(u)}}}\overline{X}_{u}(\frac{t\Delta(u)}{u}+t_{k})>m_{k}(u)\right)$
$\displaystyle\geq$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,S(1-\varepsilon_{2})]_{\delta\varepsilon_{1}}}\overline{X}_{u}(\frac{t[\Delta(u)\varepsilon_{1}]}{u\varepsilon_{1}}+t_{k})>m_{k}(u)\right)$
$\displaystyle=:$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,S(1-\varepsilon_{2})]_{\delta\varepsilon_{1}}}Y_{u}(t)>m_{k}(u)\right),\
\ \ u\to\infty,$

where $\varepsilon_{1},\varepsilon_{2}$ are any small positive numbers. As
follows from Lemma 5.1, correlation of $Y_{u}$, as $u\to\infty$, has expansion

$\displaystyle
r_{Y_{u}}(t,s)=1-\frac{\sigma^{2}(\Delta(u))|t-s|^{2\alpha}}{2\sigma^{2}(u)t_{*}^{2\alpha}}+o\left(\frac{\sigma^{2}(\Delta(u))|t-s|^{2\alpha}}{\sigma^{2}(u)}\right).$

Next by Lemma 5.1 in [8] using the notation introduced therein with index set
$K$ consisting of 1 element and

$g(u)=m_{k}(u),\quad\theta(u,s,t)=|t-s|^{2\alpha},\quad
V=B_{\alpha},\quad\sigma_{V}(t)=|t|^{2\alpha}$

we have uniformly for $-N_{u}\leq k\leq N_{u}$

$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,(1-\varepsilon_{2})S]_{\delta\varepsilon_{1}}}Y_{u}(t)>m_{k}(u)\right)\sim\mathcal{H}_{B_{\alpha}}([0,S(1-\varepsilon_{2})]_{\delta\varepsilon_{1}})\Psi(m_{k}(u)),\
\ \ u\to\infty.$

Thus, for large $S$, as $u\to\infty$,

$\displaystyle\sum\limits_{-N_{u}\leq k\leq
N_{u}}p_{k,S,u}\geq\mathcal{H}_{B_{\alpha}}([0,(1-\varepsilon_{2})S]_{\delta\varepsilon_{1}})\sum\limits_{-N_{u}\leq
k\leq N_{u}}\Psi(m_{k}(u))(1+o(1)).$

Next we calculate the sum above. Similarly to the previous cases we have as
$u\to\infty$ with
$\widehat{C}_{+}=\frac{C_{+}(1+ct_{*})^{2}}{(t_{*})^{2\alpha}}$ and
$l_{u}=\frac{S\Delta(u)}{\sigma(u)}\to 0$, that

$\displaystyle\frac{\sum\limits_{k=-N_{u}}^{N_{u}}\Psi(m_{k}(u))}{\Psi(m(u))}$
$\displaystyle\sim$
$\displaystyle\sum\limits_{k=-N_{u}}^{N_{u}}e^{-\frac{m^{2}(u)}{2}(\frac{1}{(1-C_{+}t_{k}^{2})^{2}}-1)}$
$\displaystyle\sim$
$\displaystyle\sum\limits_{k=-N_{u}}^{N_{u}}e^{-C_{+}m^{2}(u)t_{k}^{2}}\sim\sum\limits_{k=-\frac{\sigma(u)\ln
u}{S\Delta(u)}}^{\frac{\sigma(u)\ln
u}{S\Delta(u)}}e^{-\frac{C_{+}(1+ct_{*})^{2}}{t_{*}^{2\alpha}}(\frac{kS\Delta(u)}{\sigma(u)})^{2}}$
$\displaystyle=$ $\displaystyle\sum\limits_{kl_{u}\in(-\ln u,\ln
u)_{l_{u}}}e^{-\widehat{C}_{+}(kl_{u})^{2}}=\frac{1}{l_{u}}\Big{(}l_{u}\sum\limits_{t\in(-\ln
u,\ln u)_{l_{u}}}e^{-\widehat{C}_{+}t^{2}}\Big{)}.$

Since $l_{u}\to 0$ as $u\to\infty$, the expression in the parentheses above
converges as $u\to\infty$ to

$\displaystyle\int\limits_{\mathbb{R}}e^{-\widehat{C}_{+}t^{2}}dt=\frac{\sqrt{\pi}}{\sqrt{\widehat{C}_{+}}}=\sqrt{\frac{2A\pi}{B}}\frac{t_{*}^{\alpha}}{1+ct_{*}}\frac{1}{\sqrt{1+\varepsilon}}.$

Thus, summarizing the calculations above we have as $u\to\infty$ for large $S$

$\displaystyle\sum\limits_{-N_{u}\leq k\leq N_{u}}p_{k,S,u}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\sqrt{1+\varepsilon}}\frac{1}{S}\mathcal{H}_{B_{\alpha}}([0,(1-\varepsilon_{2})S]_{\delta\varepsilon_{1}})\Psi(m(u))\frac{\sigma(u)}{\Delta(u)}\sqrt{\frac{2A\pi}{B}}\frac{t_{*}^{\alpha}}{1+ct_{*}}(1+o(1)).$

Letting $S\to\infty$ then $\varepsilon_{2}\to 0$ and then $\varepsilon_{1}\to
0$, in view of Lemma 12.2.7 ii) and Remark 12.2.10 in [13] we obtain
$\frac{1}{S}\mathcal{H}_{B_{\alpha}}([0,(1-\varepsilon_{2})S]_{\delta\varepsilon_{1}})\to\mathcal{H}_{B_{\alpha}}$.
Letting then $\varepsilon\to 0$ we get the asymptotic lower bound

(29) $\displaystyle\sum\limits_{-N_{u}\leq k\leq N_{u}}p_{k,S,u}$
$\displaystyle\geq$
$\displaystyle\mathcal{H}_{B_{\alpha}}\Psi(m(u))\frac{\sigma(u)}{\Delta(u)}\sqrt{\frac{2A\pi}{B}}\frac{t_{*}^{\alpha}}{1+ct_{*}}(1+o(1)),\
\ \ u\to\infty.$

Similarly, we have for $\varepsilon>0$ with
$m_{k}^{-}(u)=\frac{m(u)}{1-C_{-}|t_{k}|^{2}}$ (recall,
$C_{-}=\frac{B(1-\varepsilon)}{2A}$) as $S\to\infty$

$\displaystyle\sum\limits_{-N_{u}\leq k\leq N_{u}}p_{k,S,u}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{1-\varepsilon}}\mathcal{H}_{B_{\alpha}}\Psi(m(u))\frac{\sigma(u)}{\Delta(u)}\sqrt{\frac{2A\pi}{B}}\frac{t_{*}^{\alpha}}{1+ct_{*}}(1+o(1)),\
\ \ \ u\to\infty.$

Letting $\varepsilon\to 0$ we obtain the right side in (29), thus, the claim
is established. $\Box$

###  5.2. Proof of Theorem 3.1.

By Lemma 5.2 we have

(30)
$\displaystyle\psi_{T,\delta}^{\sup}(u)\sim\mathbb{P}\left(\sup\limits_{t\in[0,T/u]_{\delta/u},s-t\in
I(t_{u})}Z_{u}(t,s)>m(u)\right),\ \ \ u\to\infty.$

As in the proof of Theorem 2.1 next we consider three cases: $\varphi=0$,
$\varphi\in(0,\infty)$ and $\varphi=\infty$.

_Case $\varphi=0$._ We have by Bonferroni inequality

$\displaystyle\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sum\limits_{s\in
I(t_{u})}\\!\\!\\!\mathbb{P}\left(Z_{u}(t,s)\\!>\\!m(u)\right)\\!\\!\\!$
$\displaystyle\geq$
$\displaystyle\\!\\!\\!\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}},s\in
I(t_{u})}\\!\\!\\!Z_{u}(t,s)>m(u)\right)$ $\displaystyle\geq$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sum\limits_{s\in
I(t_{u})}\\!\\!\\!\mathbb{P}\left(Z_{u}(t,s)\\!>\\!m(u)\right)-\\!\\!\\!\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sum\limits_{\overset{s_{1},s_{2}\in
I(t_{u})}{s_{1}\neq
s_{2}}}\\!\\!\\!\\!\\!\mathbb{P}\left(Z_{u}(t,s_{1}),Z_{u}(t,s_{2})\\!>\\!m(u)\right)\\!.$

For the last term above we have

(31)
$\displaystyle\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sum\limits_{\overset{s_{1},s_{2}\in
I(t_{u})}{s_{1}\neq
s_{2}}}\\!\\!\\!\\!\\!\mathbb{P}\left(Z_{u}(t,s_{1}),Z_{u}(t,s_{2})\\!>\\!m(u)\right)\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!(1+[\frac{T}{\delta}])\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sum\limits_{\overset{s_{1},s_{2}\in
I(t_{u})}{s_{1}\neq
s_{2}}}\\!\\!\\!\mathbb{P}\left(X_{u}(s_{1}-t),X_{u}(s_{2}-t)>m(u)\right)$
$\displaystyle\leq$
$\displaystyle\\!\\!\mathbb{C}(1+[\frac{T}{\delta}])\ln^{2}(u)u^{1/2-\varepsilon^{\prime}}\Psi(m(u)),$

where the last inequality above follows from (23). Next we have

$\displaystyle\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sum\limits_{s\in
I(t_{u})}\mathbb{P}\left(Z_{u}(t,s)>m(u)\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sum\limits_{s\in
I(t_{u})}\mathbb{P}\left(X_{u}(s-t)>m(u)\right)$ $\displaystyle\sim$
$\displaystyle(1+[\frac{T}{\delta}])\sum\limits_{\tau\in
I(t_{u})}\mathbb{P}\left(X_{u}(\tau)>m(u)\right),\ \ \ u\to\infty.$

The asymptotics of the last sum above was calculated in (24), hence

$\displaystyle\sum\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sum\limits_{s_{1}\neq
s_{2}\in
I(t_{u})}\mathbb{P}\left(Z_{u}(t,s)>m(u)\right)\sim(1+[\frac{T}{\delta}])\frac{\sqrt{2\pi\alpha}u\Psi(m(u))}{\delta
c(1-\alpha)^{3/2}m(u)},\ \ \ \ u\to\infty.$

By the line above combined with (31) we obtain

$\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}},s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\sim(1+[\frac{T}{\delta}])\frac{\sqrt{2\pi\alpha}u\Psi(m(u))}{\delta
c(1-\alpha)^{3/2}m(u)},\ \ \ \ u\to\infty$

and the claim follows by (30).

_Case $\varphi\in(0,\infty)$._ With the notation of Theorem 2.1 we have by
Bonferroni inequality for $u>0$

$\displaystyle\sum\limits_{j=-N_{u}}^{N_{u}}q_{j,S,u}\geq\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}},s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\geq\sum\limits_{j=-N_{u}}^{N_{u}}q_{j,S,u}-\sum\limits_{-N_{u}\leq
i<j\leq N_{u}}q_{i,j;S,u},$

where

$\displaystyle
q_{j,S,u}=\mathbb{P}\left(\\!\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}},s\in\Delta_{j,S,u}}\\!\\!\\!\\!\\!\\!\\!\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\right)\\!,\
q_{i,j;S,u}=\mathbb{P}\left(\\!\\!\exists
t\\!\in\\![0,\\!\frac{T}{u}]_{\frac{\delta}{u}}\\!\\!:\\!\\!\\!\sup\limits_{s\in\Delta_{j,S,u}}\\!\\!\\!Z_{u}(t,s)\\!>\\!m(u),\\!\\!\sup\limits_{s\in\Delta_{i,S,u}}\\!\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right)\\!.$

By [8] (_proof of lower bound of_ $\pi_{T_{u}}(u)$, p. 288) we have that as
$u\to\infty$ and then $S\to\infty$

$\displaystyle\sum\limits_{-N_{u}\leq i<j\leq
N_{u}}q_{i,j;S,u}=o\left(\frac{u}{m(u)\Delta(u)}\Psi(m(u))\right).$

Since

$\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}},s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\geq\mathbb{P}\left(\sup\limits_{t\in
I(t_{u})}X_{u}(t)>m(u)\right)\geq C\frac{u}{m(u)\Delta(u)}\Psi(m(u)),\quad
u\to\infty,$

we have that as $u\to\infty$ and then $S\to\infty$

$\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}},s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\sim\sum\limits_{j=-N_{u}}^{N_{u}}q_{j,S,u}$

and we need to calculate the asymptotics of the sum above. Next we uniformly
approximate each summand in the sum above. For $\varepsilon>0,j\geq 1,S>T$ and
$u$ large enough we have (recall,
$m_{j-1}^{-}(u)=\frac{m(u)}{1-(1-\varepsilon)\frac{B}{2A}(\frac{(j-1)S}{u})^{2}}$)

$\displaystyle q_{j,S,u}$ $\displaystyle=$
$\displaystyle\mathbb{P}\left(\\!\exists(t,s)\\!\in\\![0,\frac{T}{u}]_{\delta/u}\\!\times\\!\Delta_{j,S,u}\\!:\\!\overline{Z}_{u}(t,s)>\frac{m(u)}{\sigma_{u}(s-t)}\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{P}\left(\\!\exists(t,s)\\!\in\\![0,\frac{T}{u}]_{\delta/u}\\!\times\\!\Delta_{j,S,u}\\!:\\!\overline{Z}_{u}(t,s)\\!>\\!\frac{m(u)}{1\\!-\\!(1-\varepsilon)\frac{B}{2A}(s\\!-\\!t\\!-\\!t_{u})^{2}}\right)$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta},s\in[0,S]_{\delta}}\overline{Z}_{u}(t/u,t_{u}+s/u)>\frac{m(u)}{1-C_{-}(\frac{(j-1)S}{u})^{2}}\right)$
$\displaystyle=:$
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta},s\in[0,S]_{\delta}}\overline{Z}^{\prime}_{u}(t,s)>m^{-}_{j-1}(u)\right).$

By (18) correlation of $\overline{Z}_{u}^{\prime}$ satisfies

(32) $\displaystyle
1-r_{\overline{Z}^{\prime}_{u}}(t,s,t_{1},s_{1})\sim\frac{\sigma(|s-s_{1}|)^{2}+\sigma^{2}(|t-t_{1}|)}{2\sigma^{2}(ut_{*})},\
\ \ (t,s)\in[0,T]\times[0,S],\ \ \ u\to\infty.$

Applying Lemma 5.1 in [8] with

$\Phi:=\sup,\quad\theta(u,s,t):=\frac{2c^{2}}{\varphi^{2}}(\sigma^{2}(|s-s_{1}|)+\sigma^{2}(|t-t_{1}|)),\
\ \ \ V(t,s):=\frac{\sqrt{2}c}{\varphi}(X^{(1)}(t)+X^{(2)}(s)),$

where $X^{(1)}$ and $X^{(2)}$ are independent copies of $X$ we have

$\mathbb{P}\left(\sup\limits_{t\in[0,T]_{\delta},s\in[0,S]_{\delta}}\overline{Z}^{\prime}_{u}(t,s)>m_{j-1}^{-}(u)\right)\sim\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}([0,T]_{\delta})\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}([0,S]_{\delta})\Psi(m_{j-1}^{-}(u)).$

Finally, for $\varepsilon>0,j\geq 1,S>T$ and $u$ large we have

$\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\delta/u},s\in\Delta_{j,S,u}}Z_{u}(t,s)>m(u)\right)\leq\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}([0,T]_{\delta})\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}([0,S]_{\delta})\Psi(m_{j-1}^{-}(u))(1+o(1)).$

The rest of the proof is the same as in Theorem 2.1 case
$\varphi\in(0,\infty)$, thus, the claim is established.

_Case $\varphi=\infty$._ By Theorem 2.1 we have

$\displaystyle\psi_{T,\delta}^{\sup}(u)\geq\mathbb{P}\left(M_{\delta}>u\right)\sim\mathcal{H}_{B_{\alpha}}f(u)\Psi(m(u)),\quad
u\to\infty.$

By (30) we have

$\displaystyle\psi_{T,\delta}^{\sup}(u)\leq\mathbb{P}\left(\sup\limits_{t\in[0,T/u],s-t\in(-\delta_{u}+t_{u},t_{u}+\delta_{u})}Z_{u}(t,s)>m(u)\right)(1+o(1)),\quad
u\to\infty.$

From the proof of Theorem 3.1 in [8] it follows that the last probability
above does not exceed $(1+o(1))\mathcal{H}_{B_{\alpha}}f(u)\Psi(m(u)),\
u\to\infty$. Combining both bounds above we obtain the claim. $\Box$

Proof of Theorem 3.3. _Case $\varphi\in(0,\infty)$._ By Lemma 5.2 we have

$\psi_{T,\delta}^{\inf}(u)\sim\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\delta/u}}\sup\limits_{s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right),\ \ \ \ u\to\infty.$

With notation of Theorem 2.1 repeating the proof of Theorem 3.1 we have as
$u\to\infty$ and then $S\to\infty$

$\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\delta/u}}\sup\limits_{s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\sim\sum\limits_{j=-N_{u}}^{N_{u}}\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\delta/u}}\sup\limits_{s\in\Delta_{j,S,u}}Z_{u}(t,s)>m(u)\right).$

Next we uniformly approximate each summand in the sum above. For
$\varepsilon>0,j\geq 1,S>T$ and $u$ large enough similarly to the proof of
Theorem 3.1 we obtain (recall,
$m_{j-1}^{-}(u)=\frac{m(u)}{1-(1-\varepsilon)\frac{B}{2A}(\frac{(j-1)S}{u})^{2}}$)

$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\delta/u}}\sup\limits_{s\in\Delta_{j,S,u}}Z_{u}(t,s)>m(u)\right)\leq\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}\sup\limits_{s\in[0,S]_{\delta}}\overline{Z}^{\prime}_{u}(t,s)>m_{j-1}^{-}(u)\right),$

where $\overline{Z}_{u}^{\prime}(t,s)$ is a centered Gaussian process with
unit variance and correlation satisfying (32). Applying Lemma 5.1 in [8] with
(with $X^{(1)},X^{(2)}$ being independent copies of $X$)

$\Phi:=\inf\sup,\quad\theta(u,s,t):=\frac{2c^{2}}{\varphi^{2}}(\sigma^{2}(|s-s_{1}|)+\sigma^{2}(|t-t_{1}|)),\
\ \ \ V(t,s):=\frac{\sqrt{2}c}{\varphi}(X^{(1)}(t)+X^{(2)}(s))$

we have as $u\to\infty$

$\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}\sup\limits_{s\in[0,S]_{\delta}}\overline{Z}^{\prime}_{u}(t,s)>m_{j-1}^{-}(u)\right)\sim\mathbb{E}\left(\inf\limits_{t\in[0,T]_{\delta}}\sup\limits_{s\in[0,S]_{\delta}}e^{\sqrt{2}V(t,s)-\operatorname{Var}(V(t,s))}\right)\Psi(m_{j-1}^{-}(u)).$

Since $X^{(1)}(t)$ and $X^{(2)}(s)$ are independent we have

$\mathbb{E}\left(\inf\limits_{t\in[0,T]_{\delta}}\sup\limits_{s\in[0,S]_{\delta}}e^{\sqrt{2}V(t,s)-\operatorname{Var}(V(t,s))}\right)=\mathcal{G}_{\frac{\sqrt{2}c}{\varphi}X}([0,T]_{\delta})\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}([0,S]_{\delta}).$

Thus, as $u\to\infty$

$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\delta/u}}\sup\limits_{s\in\Delta_{j,S,u}}Z_{u}(t,s)>m(u)\right)\leq\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}^{\inf}([0,T]_{\delta})\mathcal{H}_{\frac{\sqrt{2}c}{\varphi}X}([0,S]_{\delta})\Psi(m_{j-1}^{-}(u)).$

The rest of the proof is the same as in the proof of Theorem 2.1. Thus, the
claim is established.

_Case $\varphi=\infty$._ Let $R(s,t)=X(s)-X(t)-c(s-t),\ t,s\geq 0.$ Using the
idea from [4], (the next equation after (2)) we write

$\displaystyle\frac{\psi^{\inf}_{T,\delta}(u)}{\psi^{\sup}_{T,\delta}(u)}$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,T]_{\delta}}\sup\limits_{t\leq
s\in
G_{\delta}}R(s,t)>u\Big{|}\sup\limits_{t\in[0,T]_{\delta}}\sup\limits_{t\leq
s\in G_{\delta}}R(s,t)>u\right)$ $\displaystyle\geq$ $\displaystyle
1-\sum\limits_{t\in[0,T]_{\delta}}\left(1-\frac{\mathbb{P}\left(\sup\limits_{t\leq
s\in
G_{\delta}}R(s,t)>u\right)}{\mathbb{P}\left(\sup\limits_{a\in[0,T]_{\delta}}\sup\limits_{a\leq
b\in G_{\delta}}R(a,b)>u\right)}\right).$

By Theorems 2.1 and 3.1 we have that the right part of the expression above
tends to 1 as $u\to\infty$. Thus, we have
$\psi^{\inf}_{T,\delta}(u)\sim\psi_{T,\delta}^{\sup}(u)\sim\mathcal{H}_{B_{\alpha}}f(u)\Psi(m(u)),\
u\to\infty$ and the claim follows. $\Box$

Proof of Proposition 3.4. Since $T\geq\delta$ with any
$K\in[\delta,T]_{\delta}$, $a\in(0,t_{*}),b>0$ and
$J(t_{u})=[-a+t_{u},t_{u}+b]$ we have

$\displaystyle\psi^{\inf}_{\delta,T}(u)$ $\displaystyle\leq$
$\displaystyle\psi^{\inf}_{\delta,K}(u)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\left(\exists s_{1},s_{2}\in J(t_{u})\cap
G_{\frac{\delta}{u}}:Z_{u}(0,s_{1})>m(u),Z_{u}(\frac{K}{u},s_{2})>m(u)\right)$
$\displaystyle+\mathbb{P}\left(\exists s\notin
J(t_{u}):Z_{u}(0,s)>m(u)\right)$ $\displaystyle=:$ $\displaystyle
p_{1}(u)+p_{2}(u).$

_Estimation of $p_{1}(u)$._ Fix some $s_{1},s_{2}\in J(t_{u})\cap
G_{\frac{\delta}{u}}$ and let
$(W_{1},W_{2})=(Z_{u}(0,s_{1}),Z_{u}(\frac{K}{u},s_{2})).$ We have that
$(W_{1},W_{2})$ is a centered Gaussian vector with $Var(W_{1}),Var(W_{2})\leq
1$ and correlation $r_{W_{u}}(s_{1},s_{2})$ satisfying (see (18))

$1-r_{W_{u}}(s_{1},s_{2})\geq\frac{\sigma^{2}(u|s_{1}-s_{2}|)+\sigma^{2}(K)}{2\sigma^{2}(ut_{*})}(1+o(1))\geq(1+o(1))\frac{\sigma^{2}(K)}{2t_{*}^{2\alpha}\sigma^{2}(u)},\
\ \ \ \ s_{1},s_{2}\in J(t_{u}).$

Thus, by Lemma 2.3 in [17]

$\displaystyle\Psi(m(u))^{-1}\mathbb{P}\left(W_{1}>m(u),W_{2}(u)>m(u)\right)$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle
3\Psi\big{(}m(u)\sqrt{\frac{1-r_{W_{u}}(s_{1},s_{2})}{2}}\big{)}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle
3\Psi\Big{(}(1+o(1))\frac{\sigma(K)m(u)}{2t_{*}^{\alpha}\sigma(u)}\Big{)}\\!=\\!3\Psi\Big{(}\\!(1+o(1))\frac{\sigma(K)(1+ct_{*})u}{2t_{*}^{2\alpha}\sigma^{2}(u)}\Big{)}.$

Note that $\varphi=0$ implies $\frac{u}{\sigma^{2}(u)}\to\infty$ as
$u\to\infty$. Thus, since there are less than $Cu^{2}\ln^{2}u$ points in
$(J(t_{u})\cap G_{\frac{\delta}{u}})\times(J(t_{u})\cap G_{\frac{\delta}{u}})$
we have with any $\mathbb{C}_{K}<\frac{\sigma(K)(1+ct_{*})}{2t_{*}^{2\alpha}}$
by (14) as $u\to\infty$

$\displaystyle p_{1}(u)$ $\displaystyle\leq$ $\displaystyle
Cu^{2}(\ln^{2}u)\cdot
3\Psi(m(u))\Psi\Big{(}(1+o(1))\frac{\sigma(K)(1+ct_{*})u}{2t_{*}^{2\alpha}\sigma^{2}(u)}\Big{)}\leq\Psi(m(u))\Psi\Big{(}\mathbb{C}_{K}\frac{u}{\sigma^{2}(u)}\Big{)}.$

_Estimation of $p_{2}(u)$._ Since $Z_{u}(0,s)\overset{d}{=}X_{u}(s),\ s\geq
0$, it follows from the estimation of $R_{1}(u)$ and $R_{2}(u)$ in the proof
of Lemma 5.2 below (see (35) and (36), respectively) that for appropriately
chosen $a\in(0,t_{*}),b>0$ and small $\varepsilon>0$

$p_{2}(u)\leq\mathbb{P}\left(\exists
s\in[0,a]:X_{u}(s)>m(u)\right)+\mathbb{P}\left(\exists
s\in[t_{*}+b,\infty):X_{u}(s)>m(u)\right)\leq\Psi(m(u))\mathbb{C}e^{-\frac{m^{2}(u)}{u^{2\varepsilon}}}.$

Combing this inequality with the upper bound of $p_{1}(u)$ we obtain that for
any $K\in[\delta,T]_{\delta}$ it holds that

$\psi^{\inf}_{\delta,T}(u)\leq\Psi(m(u))\Psi\Big{(}\mathbb{C}_{K}\frac{u}{\sigma^{2}(u)}\Big{)},\
\ \ u\to\infty$

and taking the supremum with respect to $K$ we obtain the claim. $\Box$

Proof of Corollary 4.3. We start with the proof of the first statement. Since
$Z(t)$ is a Gaussian process with stationary increments, a.s. sample paths and
variance satisfying A applying Theorem 2.1 with parameters

$\displaystyle\varphi=\lim_{u\to\infty}\frac{\sigma_{\zeta}^{2}(u)}{u}=\frac{2}{G}\in(0,\infty),\
\alpha=1/2,\ t_{*}=1/c,\ \Delta(u)=1,\ A=2\sqrt{C},\ D=c^{2}\sqrt{c}/2,$
$\displaystyle\
m(u)=\sqrt{2Gc}\sqrt{u}+\frac{c^{3/2}G^{3/2}D}{\sqrt{2}}u^{-1/2}+o(u^{-1/2}),\
f(u)=\frac{2\sqrt{\pi}}{c\sqrt{cG}}\sqrt{u}+O(u^{-1}),\
\eta(t)=\frac{cG}{\sqrt{2}}Z(t)$

we have $\Psi(m(u))\sim e^{-ucG-c^{2}G^{2}D}\frac{1}{2\sqrt{\pi
Gcu}},u\to\infty$ implying

$\displaystyle\mathbb{P}\left(\exists
t\in[0,\infty):Z(t)-ct>u\right)\sim\mathcal{H}^{\delta}_{cGZ(t)/\sqrt{2}}\frac{1}{c^{2}G}e^{-ucG-c^{2}G^{2}D},\
\ \ \ \ u\to\infty$

and the first claim follows. Applying Theorems 3.1 and 3.3 with the same
parameters we obtain the second and third claims, respectively. $\Box$

Proof of Lemma 5.2. We start with the first claim. We have

(33)
$\displaystyle\mathbb{P}\left(\\!\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sup\limits_{s\in
I(t_{u})}\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right)\\!\\!\\!$
$\displaystyle\leq$ $\displaystyle\\!\\!\\!\psi_{\delta,T}^{\sup}(u)$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!\mathbb{P}\left(\\!\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sup\limits_{s\in
I(t_{u})}\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right)\\!+\\!\mathbb{P}\left(\\!\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sup\limits_{s\in(G_{\frac{\delta}{u}}\backslash
I(t_{u}))}\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right).$

Our first aim is to show that

(34)
$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sup\limits_{s\in(G_{\frac{\delta}{u}}\backslash
I(t_{u}))}Z_{u}(t,s)>m(u)\right)=o(\Psi(m(u))),\ \ \ \ \ u\to\infty.$

Since for any fixed $0\leq t\leq s$ random variables $Z_{u}(t,s)$ and
$X_{u}(s-t)$ have the same distributions we have with
$I^{\prime}(t_{u})=(-\frac{\delta_{u}}{2}+t_{*},\frac{\delta_{u}}{2}+t_{*})\cap
G_{\delta/u}$

$\displaystyle\mathbb{P}\left(\sup\limits_{t\in[0,T/u]_{\delta/u}}\sup\limits_{s\in(G_{\delta/u}\backslash
I(t_{u}))}Z_{u}(t,s)>m(u)\right)$ $\displaystyle\leq$
$\displaystyle\sum\limits_{t\in[0,T/u]_{\delta/u}}\mathbb{P}\left(\sup\limits_{s\in(G_{\delta/u}\backslash
I(t_{u}))}Z_{u}(t,s)>m(u)\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{t\in[0,T/u]_{\delta/u}}\mathbb{P}\left(\sup\limits_{s\in(G_{\delta/u}\backslash
I(t_{u}))}X_{u}(s-t)>m(u)\right)$ $\displaystyle\leq$
$\displaystyle(1+[\frac{T}{\delta}])\mathbb{P}\left(\sup\limits_{s\in(G_{\delta/u}\backslash
I^{\prime}(t_{u}))}X_{u}(s)>m(u)\right).$

We have that for any chosen small $\varepsilon$ and large $M$ the last
probability above does not exceed

$\displaystyle\sum\limits_{t\in(G_{\frac{\delta}{u}}\backslash
I^{\prime}(t_{u}))}\\!\\!\\!\\!\\!\\!\\!\\!\mathbb{P}\left(X_{u}(t)\\!>\\!m(u)\right)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\sum\limits_{t\in(G_{\delta/u}\backslash
I^{\prime}(t_{u}))}\\!\\!\Psi(\frac{m(u)}{\sigma_{X_{u}}(t)})$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!2\Psi(m(u))\\!\Big{(}\\!\\!\sum\limits_{t\in[0,\varepsilon]_{\frac{\delta}{u}}}\\!\\!\\!\\!e^{\\!-\frac{m^{2}(u)}{2}(\frac{1}{\sigma_{X_{u}}^{2}\\!\\!(t)}-1)}\\!\\!+\\!\\!\\!\sum\limits_{t\in[M,\infty)_{\frac{\delta}{u}}}\\!\\!\\!\\!\\!\\!e^{\\!\\!-\frac{m^{2}(u)}{2}(\frac{1}{\sigma_{X_{u}}^{2}\\!\\!(t)}-1)}\\!+\\!\\!\\!\\!\sum\limits_{t\in([\varepsilon,M]_{\frac{\delta}{u}}\backslash
I^{\prime}(t_{u}))}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e^{\\!-\frac{m^{2}(u)}{2}(\frac{1}{\sigma_{X_{u}}^{2}\\!\\!(t)}-1)}\Big{)}$
$\displaystyle=:$ $\displaystyle 2\Psi(m(u))(R_{1}(u)+R_{2}(u)+R_{3}(u)).$

Thus, to establish (34) we need to prove that $R_{1}(u)+R_{2}(u)+R_{3}(u)\to
0$ as $u\to\infty$. By Lemma 5.1, $t_{u}$ is unique for large $u$ and we have

$\displaystyle\sigma_{X_{u}}(t)=\frac{\sigma(ut)}{u(1+ct)}m(u)=\frac{\sigma(ut)}{\sigma(ut_{u})}\frac{1+ct_{u}}{1+ct}.$

_Estimation of $R_{1}(u)$._ We have for all large $u$ and
$t\in[0,\varepsilon]_{\delta/u}$

$\sigma_{X_{u}}(t)\leq\mathbb{C}\frac{\sigma(ut)}{\sigma(ut_{u})}.$

i) Assume that $ut\geq\ln u$. Then with $h$ being a slowly varying function at
$\infty$ and $0<\epsilon<\alpha$ by Potter’s theorem (Theorem 1.5.6 in [16])
we have

$\frac{\sigma(ut)}{\sigma(ut_{u})}=(\frac{t}{t_{u}})^{\alpha}\frac{h(ut)}{h(ut_{u})}\leq\mathbb{C}t^{\alpha}(\frac{t_{u}}{t})^{\epsilon}\leq\mathbb{C}t^{\alpha-\epsilon}.$

ii) Assume that $ut<\ln u$. Since $t\in[0,\varepsilon]_{\delta/u}$ we have
$ut\geq\delta$ for $t\neq 0.$ Then for $\epsilon\in(0,\alpha)$ and large $u$

$\frac{\sigma(ut)}{\sigma(ut_{u})}\leq
u^{-(\alpha-\epsilon)}\sup\limits_{t\in[\delta,\ln u]}\sigma(t)\leq
u^{-(\alpha-\epsilon)}\ln u.$

Combining the above inequalities we have that for sufficiently small
$\varepsilon$ and for all $t\in[0,\varepsilon]_{\delta/u}$ uniformly for large
$u$ it holds that $\frac{1}{\sigma_{X_{u}}^{2}(t)}-1\geq 2$. Thus, for small
enough $\varepsilon>0$

(35) $\displaystyle R_{1}(u)\leq\mathbb{C}ue^{-m^{2}(u)}\to 0,\ \ \
u\to\infty.$

_Estimation of $R_{2}(u)$._ By Potter’s theorem we have for $M$ large enough
and $0<\epsilon^{\prime}<1-\alpha$

$\displaystyle\frac{\sigma(ut)}{\sigma(ut_{u})}\leq\mathbb{C}\left(\frac{t}{t_{u}}\right)^{\alpha+\epsilon^{\prime}}.$

Since $t_{u}\to t^{*}$ as $u\to\infty$ we have for some small $\epsilon>0$

$\sigma_{X_{u}}(t)\leq\mathbb{C}\frac{t^{\alpha+\epsilon^{\prime}}}{1+ct}\leq
t^{-\epsilon},$

hence for all $t>M$ uniformly for $u$ large it holds that
$\frac{1}{\sigma^{2}_{X_{u}}(t)}-1\geq 2t^{\epsilon}.$ Choosing $M$ large
enough and $\varepsilon$ sufficiently small we have as $u\to\infty$

(36) $\displaystyle
R_{2}(u)\leq\sum\limits_{t\in[M,\infty)_{\delta/u}}e^{-t^{\epsilon}m^{2}(u)}=\sum\limits_{t\in[M,\infty)_{\delta}}e^{-t^{\epsilon}\frac{m^{2}(u)}{u^{\varepsilon}}}\leq\mathbb{C}e^{-\frac{m^{2}(u)}{u^{\varepsilon}}}\to
0.$

_Estimation of $R_{3}(u)$._ We have by Lemma 5.1 that with some $\mathbb{C}>0$

$1-\sigma_{X_{u}}(t)\geq\mathbb{C}(t-t_{u})^{2},\ \ \ t\in[\varepsilon,M]$

and hence by (16) for $t\in[\varepsilon,M]_{\frac{\delta}{u}}\backslash
I(t_{u})$ it holds that

$m^{2}(u)(\frac{1}{\sigma_{X_{u}}^{2}(t)}-1)\geq
m^{2}(u)(1-\sigma_{X_{u}}(t))\geq\mathbb{C}(t-t_{u})^{2}m^{2}(u)\geq\mathbb{C}\ln^{2}u.$

Thus, for $t\in[\varepsilon,M]_{\frac{\delta}{u}}\backslash I(t_{u})$ it holds
that $e^{-\frac{m^{2}(u)}{2}(\frac{1}{\sigma_{u}^{2}(t)}-1)}\leq
u^{-\mathbb{C}\ln u},$ and we obtain

$\displaystyle R_{3}(u)\leq u\mathbb{C}_{1}u^{-\mathbb{C}_{2}\ln u}\to 0,\ \ \
u\to\infty.$

Combining the estimate above with (35) and (36) obtain (34). It follows from
the calculations in Theorem 3.1 that

$\mathbb{P}\left(\sup\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sup\limits_{s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\geq\Psi(m(u)),\ \ \ \ u\to\infty$

and the first claim follows by (33) and (34).

Next we show the second statement of the lemma. Again, by Bonferroni
inequality we have

$\displaystyle\mathbb{P}\left(\\!\inf\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sup\limits_{s\in
I(t_{u})}\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right)\\!\\!\\!$
$\displaystyle\leq$ $\displaystyle\\!\\!\\!\psi_{\delta,T}^{\inf}(u)$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!\mathbb{P}\left(\\!\inf\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sup\limits_{s\in
I(t_{u})}\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right)\\!+\\!\mathbb{P}\left(\\!\inf\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\\!\sup\limits_{s\in(G_{\frac{\delta}{u}}\backslash
I(t_{u}))}\\!\\!Z_{u}(t,s)\\!>\\!m(u)\\!\\!\right).$

It follows from the calculations in Theorem 3.1 that

$\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sup\limits_{s\in
I(t_{u})}Z_{u}(t,s)>m(u)\right)\geq\Psi(m(u)),\ \ \ \ u\to\infty$

and by (34) we obtain that

$\displaystyle\mathbb{P}\left(\inf\limits_{t\in[0,\frac{T}{u}]_{\frac{\delta}{u}}}\sup\limits_{s\in(G_{\frac{\delta}{u}}\backslash
I(t_{u}))}Z_{u}(t,s)>m(u)\right)=o(\Psi(m(u))),\ \ \ \ \ u\to\infty.$

Combining both statements above we obtain the second claim of the lemma.
$\Box$

Proof of Lemma 5.4. Fix some $t\neq s\in I(t_{u})$. Since
$\sigma_{X_{u}}(t),\sigma_{X_{u}}(s)\leq 1$ we have by Lemma 2.3 in [17], with
$r_{u}(t,s)=\operatorname{Corr}(\overline{X}_{u}(t),\overline{X}_{u}(s))$,

$\displaystyle\mathbb{P}\left(X_{u}(t)>m(u),X_{u}(s)>m(u)\right)\leq\mathbb{P}\left(\overline{X}_{u}(t)>m(u),\overline{X}_{u}(s)>m(u)\right)\leq\Psi(m(u))\Psi\big{(}m(u)\sqrt{\frac{1-r_{u}(t,s)}{2}}\big{)}.$

If $\alpha<1/2$, then by Lemma 5.1 as $u\to\infty$ for some $\epsilon>0$ it
holds that $m(u)\sqrt{\frac{1-r_{u}(t,s)}{2}}\geq u^{\epsilon}$ hence
uniformly for $t\neq s\in I(t_{u})$

(37) $\displaystyle\Psi(m(u)\sqrt{\frac{1-r_{u}(t,s)}{2}})\leq
e^{-\frac{1}{3}u^{2\epsilon}},\ \ \ u\to\infty.$

If $\alpha=1/2$, then $t_{*}=1/c$ and by Lemma 5.1 and (13) we have that for
some $\varepsilon^{\prime}>0$ as $u\to\infty$

$\displaystyle\frac{m^{2}(u)}{2}\frac{1-r_{u}(t,s)}{2}\sim\frac{u^{2}(1+ct_{*})^{2}\sigma^{2}(u|s-t|)}{8\sigma^{4}(ut_{*})}\sim\frac{u^{2}c^{2}\sigma^{2}(u|s-t|)}{2\sigma^{4}(u)}\geq(1/2+\varepsilon^{\prime})\ln
u$

implying as $u\to\infty$

$\displaystyle\Psi(m(u)\sqrt{\frac{1-r_{u}(t,s)}{2}})\leq
u^{-1/2-\varepsilon^{\prime}}.$

Combining the line above with (37) we obtain the claim. $\Box$

Proof of Lemma 5.3. Let $a=K\delta$, where $K$ is a large natural number that
we shall choose later on. By the proofs of Theorem 15 and Lemma 16 in [18] we
have with $\sigma^{2}_{\eta}$ being the variance of $\eta$ that

$\liminf\limits_{S\to\infty}\frac{\mathcal{H}_{\eta}([0,S]_{a})}{S}\geq\frac{1}{a}\Big{(}1-\frac{2}{a}\int\limits_{0}^{\infty}e^{-\frac{\sigma_{\eta}^{2}(t)}{4}}dt\Big{)}.$

We have that for all $u$ large enough $\sigma_{\eta}^{2}(t)\geq\mathbb{C}t$
implying that
$\int\limits_{0}^{\infty}e^{-\frac{\sigma_{\eta}^{2}(t)}{4}}dt<\infty$.
Choosing sufficiently large $K$ we have

$\liminf\limits_{S\to\infty}\frac{\mathcal{H}_{\eta}([0,S]_{\delta})}{S}\geq\liminf\limits_{S\to\infty}\frac{\mathcal{H}_{\eta}([0,S]_{a})}{S}\geq\frac{1}{a}\cdot\frac{1}{2}>0.$

Next we prove that for $I(S):=\frac{\mathcal{H}_{\eta}([0,S]_{\delta})}{S}$ it
holds that for large $S\in G_{\delta}$

(38) $\displaystyle I(S)\geq I(S+\delta).$

We have

$\displaystyle(S+\delta)I(S+\delta)\leq\mathbb{E}\left(\sup\limits_{t\in\\{0,\delta,...,S\\}}e^{\sqrt{2}\eta(t)-\sigma^{2}_{\eta}(t)}\right)+\mathbb{E}\left(e^{\sqrt{2}\eta(S+\delta)-\sigma^{2}_{\eta}(S+\delta)}\right)F(S+\delta)=SI(S)+F(S+\delta),$

where
$F(M)=\mathbb{P}\left(\underset{t\in\\{0,\delta,...,M\\}}{\operatorname{argmax}}(\sqrt{2}\eta(t)-\sigma^{2}_{\eta}(t))=M\right)$
for $M\in G_{\delta}$. Thus, to claim (38) we need to show that for large $S$

(39) $\displaystyle\delta I(S)\geq F(S+\delta).$

Since $\liminf\limits_{S\to\infty}I(S)>0$, we have that $\delta
I(S)>\varepsilon$ for all $S$ and some positive $\varepsilon$, but on the
other hand as $S\to\infty$ it holds that

$\displaystyle
F(S+\delta)\leq\mathbb{P}\left(\sqrt{2}\eta(S+\delta)-\sigma^{2}_{\eta}(S+\delta)\geq\sqrt{2}\eta(0)-\sigma^{2}_{\eta}(0)\right)=\mathbb{P}\left(\sqrt{2}\eta(S+\delta)-\sigma^{2}_{\eta}(S+\delta)\geq
0\right)\to 0,$

consequently (39) holds and hence $I(S)$ is non-increasing for large $S$.
Thus, $\lim\limits_{S\to\infty}I(S)\in(0,\infty)$ and the claim holds. $\Box$

Acknowledgement: K. Dȩbicki was partially supported by NCN Grant No
2018/31/B/ST1/00370 (2019-2022). G. Jasnovidov was supported by the Ministry
of Science and Higher Education of the Russian Federation, agreement
075-15-2019-1620 date 08/11/2019 and 075-15-2022-289 date 06/04/2022.
The Authors would like to thank Professor Enkelejd Hashorva for fruitful
discussions that significantly improved the paper.

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11institutetext:  LOMC - UMR 6294, CNRS-Université du Havre, 25 rue Philippe
Lebon, BP 1123 - 76063 Le Havre cedex, France 22institutetext: Instituto de
Física Fundamental (CSIC). Calle Serrano 121-123, 28006, Madrid, Spain.
33institutetext: Department of Chemistry, Missouri University of Science and
Technology, Rolla, Missouri 65409, United States

#  CF+ excitation in the interstellar medium

Benjamin Desrousseaux 11 François Lique 11 Javier R. Goicoechea 22 Ernesto
Quintas-Sánchez 33 Richard Dawes 33

The detection of CF+ in interstellar clouds potentially allows astronomers to
infer the elemental fluorine abundance and the ionization fraction in
ultraviolet-illuminated molecular gas. Because local thermodynamic equilibrium
(LTE) conditions are hardly fulfilled in the interstellar medium (ISM), the
accurate determination of the CF+ abundance requires one to model its non-LTE
excitation via both radiative and collisional processes. Here, we report
quantum calculations of rate coefficients for the rotational excitation of CF+
in collisions with para- and ortho-H2 (for temperatures up to 150 K). As an
application, we present non-LTE excitation models that reveal population
inversion in physical conditions typical of ISM photodissociation regions
(PDRs). We successfully applied these models to fit the CF+ emission lines
previously observed toward the Orion Bar and Horsehead PDRs. The radiative
transfer models achieved with these new rate coefficients allow the use of CF+
as a powerful probe to study molecular clouds exposed to strong stellar
radiation fields.

###### Key Words.:

ISM, masers, scattering, molecular data, radiative transfer.

##  1 Introduction

Although fluorine is one of the most reactive species in the interstellar
medium (ISM; Neufeld et al. (2005); Neufeld & Wolfire (2009)), the chemistry
of the most abundant F-bearing molecules, HF and CF+, can be accurately
described by a few chemical reactions that depend on the amount of F atoms, H2
molecules, and C+ ions (Neufeld et al., 2005). CF+ is produced in ISM regions
bathed by stellar ultraviolet (UV) photons able to ionize carbon atoms in
diffuse interstellar clouds or at the illuminated surfaces of dense molecular
clouds (so-called photodissociation regions; PDRs). The simplicity of the
fluorine chemical network results in a great sensitivity of astrophysical
models to molecular data (reactive rate coefficients).

Interstellar CF+ was first detected in the Orion Bar PDR by Neufeld et al.
(2006). Since then, this molecular ion has been widely observed in other PDRs
and diffuse clouds of the Milky Way and beyond (Neufeld et al., 2006;
Kalenskii & Johansson, 2010a, b; Guzmán et al., 2012a; Liszt et al., 2014,
2015; Muller et al., 2016; Nagy et al., 2013, 2017). In these environments,
the CF+ abundance is a powerful proxy for C+ (closely related to the
ionization fraction) and for F atoms, which are both difficult to observe
(Guzmán et al., 2012a). Because local thermodynamic equilibrium (LTE)
conditions are hardly fulfilled in the ISM (Roueff & Lique, 2013), the
competition between radiative and collisional processes has to be taken into
account in the molecular line modeling in order to derive an accurate CF+
abundance and gas physical conditions. Collisional data for collisions between
CF+ and the most abundant species in the ISM (usually atomic and molecular
hydrogen) are then essential. Collisional rate coefficients with He, as a
template for H2, were first computed by Ajili & Hammami (2013) and recently
updated by Denis-Alpizar et al. (2018). However, it is well established
(Roueff & Lique, 2013) that He data are a bad surrogate for H2 in case of
collisions with an ion.

Denis-Alpizar & Rubayo-Soneira (2019) provided the first proper study of the
CF+–H2 system. They treated the rotational relaxation of CF+ by para-H2 using
a reduced-dimensional potential energy surface (PES) averaged over H2
rotation. The use of an averaged PES, however, only allows one to consider
collisions of CF+ with a spherical H2, neglecting anisotropy effects due to
the H2 rotation.

CF+ is mostly detected in warm molecular gas, where the H2 ortho-to-para ratio
is found to be large, therefore obtaining rate coefficients for collisional
excitation with ortho-H2 is crucial and cannot be achieved considering a
spherical structure-less H2.

In order to overcome this limitation, Desrousseaux et al. (2019) recently
presented a new highly accurate four-dimensional (4D) PES, by which the
calculation of rate coefficients for collisional excitation of CF+ by both
para- and ortho-H2 is made possible. In that work, preliminary cross-sections
at low collisional energy (up to 150 cm-1) were presented for both para- and
ortho-H2 collisions. In this paper, we describe further results using this
recently computed CF+–H2 4D PES (Desrousseaux et al., 2019) to carry out
scattering calculations of rate coefficients for the collisional excitation of
CF+ by H2. We also used the new collisional data to simulate the excitation of
CF+ in the ISM. We demonstrate that CF+ exhibits population inversion at
typical physical conditions of molecular clouds, and we show that CF+ is a
powerful tracer of the gas characteristics in diffuse interstellar clouds and
illuminated surfaces of dense molecular clouds.

The CF+–H2 PES was computed employing electronic structure data from coupled-
cluster theory extrapolated to the complete basis set (CBS) limit and
constructed using the recently released software-package autosurf (Quintas-
Sánchez & Dawes, 2019). On this PES, a global minimum of $-1230.2$ cm-1 is
found for a planar structure with the two monomers almost parallel (see
Desrousseaux et al. (2019) for more details). This large energy well-depth,
typical for ion-molecule collisional systems, makes employing state-of-the-art
methods to compute the cross-sections very challenging, and, to the best of
our knowledge, only four molecular ions (CN- (Kłos & Lique, 2011), HCO+ (Massó
& Wiesenfeld, 2014), C6H- (Walker et al., 2016), and SH+ (Dagdigian, 2019b,
a)) among the 38 detected in the ISM have been studied in collision with
nonspherical H2.

##  2 Scattering calculations

The molscat nonreactive scattering code (Hutson & Green 1994) was used to
carry out close-coupling calculations of the rotational (de)excitation cross-
sections for collisions between CF+ and both para- and ortho-H2. The cross-
sections were computed using the quantum time-independent close-coupling (CC)
approach and the hybrid log-derivative/Airy propagator implemented in the
molscat code. In the following, molecule-related parameters are labeled with
subscripts 1 and 2, referring to the CF+ and H2 molecules, respectively.

The expansion over angular functions of the potential was performed as
described by Green (1975):

$V(R,\theta_{1},\theta_{2},\varphi)=\sum_{l_{1},l_{2},l}v_{l_{1},l_{2},l}(R)A_{l_{1},l_{2},l}(\theta_{1},\theta_{2},\varphi),$
(1)

where $A_{l_{1},l_{2},l}(\theta_{1},\theta_{2},\varphi)$ is constructed from
coupled spherical functions $Y_{l_{i},m_{i}}(\theta_{i},\varphi)$ and the
rotational angular momenta of CF+ and H2. The potential was expanded including
$0\leq l_{1}\leq 24$ for the CF+ molecule, and $0\leq l_{2}\leq 6$ for the H2
molecule.

The two molecules were considered as rigid rotors. The molecular constants of
CF+ were set at the value given by Cazzoli et al. (2010): $B_{e}=1.720912$
cm-1, $\alpha_{e}=0.0189$ cm-1, and $D_{e}=63\times 10^{-6}$ cm-1. The
molecular constants of H2 were set as (Herzberg & Huber, 1979) $B_{e}=60.853$
cm-1, $\alpha_{e}=3.062$ cm-1, and $D_{e}=4.71\times 10^{-2}$ cm-1.

Cross-sections were obtained for the first 22 rotational levels of CF+ ($0\leq
j_{1}\leq 21$). As already discussed in Desrousseaux et al. (2019),
convergence of these cross-sections was ensured by including the 33 lowest
rotational levels of CF+ and only the lowest rotational levels of para-H2
($j_{2}=0$) and ortho-H2 ($j_{2}=1$) in the basis set.

At each collisional energy, the maximum value of the total angular momentum
$J_{\text{tot}}$ was automatically determined by the molscat code in order to
converge cross-sections to better than $1\times 10^{-4}$ Å2, going up to
$J_{\text{tot}}=120$ at the highest energies.

The determination of the thermal rate coefficients was achieved by averaging
the cross-sections $\sigma_{\alpha\rightarrow\beta}$ over the collisional
energy ($E_{c}$):

$k_{\alpha\rightarrow\beta}(T)=\left(\frac{8}{\pi\mu
k_{B}^{3}T^{3}}\right)^{\frac{1}{2}}\times\int_{0}^{\infty}\sigma_{\alpha\rightarrow\beta}\
E_{c}e^{-\frac{E_{c}}{k_{\text{B}}T}}dE_{c},$ (2)

where $\mu$ is the reduced mass of the colliding system and $k_{B}$ is the
Boltzmann’s constant. $\alpha$ and $\beta,$ respectively, designate the
initial and final rotational states of the CF+ molecule. Collisional energies
up to 1500 cm-1 were explored in order to allow the determination of the rate
coefficients for collisions between CF+ and both para- and ortho-H2 up to 150
K for the first 22 rotational levels of CF+.

Figure 1: Cross-sections (upper panel) and rate coefficients (lower panel) for
the collisional excitation of CF+ by para-H2($j_{2}=0$) (solid lines) and
ortho-H2($j_{2}=1$) (dashed lines) for selected rotational transitions
$j_{1}\rightarrow j_{1}^{\prime}$.

Cross-sections and corresponding rate coefficients for some selected
transitions are displayed in Figure 1. The cross-sections (upper panel)
present the same behavior for both para- and ortho-H2. At low energy (up to a
few hundreds cm-1), the cross-sections decrease with increasing energy,
following an almost linear dependence as a function of the energy logarithm.
This behavior is typical and expected, as predicted from Langevin theory for
ion–molecule collisions. The cross-sections also exhibit many resonances in
this energy region. This can be explained by the creation of quasi-bound
states within the deep van der Waals well of the CF+–H2 complex before its
dissociation, as already discussed by Denis-Alpizar et al. (2020). At higher
energies, it is generally observed that the cross sections are slowly
decreasing with increasing energy, beyond a slight initial increase for
transitions with low $\Delta j_{1}$. The magnitude of the cross-sections seems
to be approximately the same for collisions with either para- or ortho-H2,
except at very low collisional energies ($<10$ cm-1), where differences of up
to a factor of 2 can be seen for large $\Delta j_{1}$ transitions.

The corresponding rate coefficients (lower panel) show a relatively flat
temperature dependence, as predicted by the Langevin theory. The magnitude of
the rate coefficients decreases with increasing $\Delta j_{1}$, with
propensity rules in favor of transitions with $\Delta j_{1}=1$. Not
surprisingly, given the similar cross-sections, it is also observed that rate
coefficients for collisions with para- and ortho-H2 are similar in magnitude.

Figure 2: Comparison between rate coefficients (in units of cm3 s-1) for the
collisional excitation of CF+ by para- and ortho-H2. Rotational transitions
involving the first 22 rotational levels of CF+ are presented at four
different temperatures. The two dashed lines delimit the region where the rate
coefficients differ by less than a factor of 1.5.

This similarity between para- and ortho-H2 rate coefficients is explored
further in Figure 2, which compares para- and ortho-H2 collisional rate
coefficients for the rotational excitation of CF+ at four different
temperatures (10, 50, 100, and 150 K), and for all transitions involving the
first 22 rotational levels of CF+. As highlighted by the two dashed lines,
differences between the two sets of data are quite small and do not exceed
50%. The agreement improves with increasing temperature, with differences of
less than 10% being observed for temperatures above 50 K. It should also be
noted that for the most dominant transitions ($k>2\times 10^{10}$ cm3 s-1),
differences stay lower than 10% for all the temperatures explored. This global
agreement between para- and ortho-H2 rate coefficients confirms what has
already been observed for a wide variety of ion–molecule collisions (Massó &
Wiesenfeld, 2014; Walker et al., 2017; Denis-Alpizar et al., 2020; Balança et
al., 2020; Kłos & Lique, 2011).

Figure 3: Comparison between rate coefficients (in units of cm3 s-1) for the
collisional deexcitation of CF+ by para-H2 obtained in this work and those
obtained by Denis-Alpizar & Rubayo-Soneira (2019). Deexcitation transitions
involving the first seven rotational levels of CF+ are presented at four
different temperatures. The two dashed lines delimit the region where the rate
coefficients differ by less than a factor of 1.5.

In Figure 3, we present a comparison between rate coefficients for the
collisional deexcitation of CF+ by para-H2 obtained in this work and those
obtained by Denis-Alpizar & Rubayo-Soneira (2019). The agreement between the
two sets of data is good overall, with differences of less than 10% generally
observed at high temperatures ($\geq$40 K). At low temperatures, and, in
particular, for the most dominant transitions, differences up to 50% can be
seen. These differences can likely be attributed to the different levels of
electronic structure calculations used to generate the PESs, as well the use
of an averaged version of this PES, neglecting the H2 structure. Therefore, we
recommend the use of our new rate coefficients in astrophysical applications.

##  3 Excitation and radiative transfer study

Figure 4: Variation of excitation temperature (blue solid lines) and optical
depth (red dotted lines) as a function of H2-density for the $1\rightarrow 0$
(left panels) and $2\rightarrow 1$ (right panels) CF+ rotational transitions
at kinetic temperatures of 20 K (upper panels), 50 K (middle panels), and 100
K (lower panels), and a CF+ column density of $N($CF+$)=1\times 10^{12}$ cm-2.
In order to better display the large amplitude of its variation, the
excitation temperature is represented on this figure as
$Y=\frac{T_{\text{ex}}}{\left|T_{\text{ex}}\right|}\times\text{log}_{10}\left(1+\left|T_{\text{ex}}\right|\right)$.

|  | Critical densitya (105 cm-3) |  $W$ (mK.km.s-1)
---|---|---|---
Rotational line | Frenquency (GHz) | 20 K | 50 K | 100 K | Horsehead | Orion Bar
$1\rightarrow 0$ | 102.58748 | 0.28 | 0.39 | 0.48 | 150$\pm$20 | 86$\pm$10
$2\rightarrow 1$ | 205.17445 | 1.34 | 1.78 | 2.46 | 290$\pm$40 | 337$\pm$13
$3\rightarrow 2$ | 307.7443 | 5.16 | 5.80 | 7.04 | - | 428$\pm$34
$5\rightarrow 4$ | 512.8465 | 11.76 | 13.36 | 15.07 | - | 100$\pm$20

Table 1: Observed CF+ integrated line intensities in the Horsehead and Orion
Bar PDRs (Neufeld et al., 2006; Guzmán et al., 2012a; Nagy et al., 2017).

  * a

Critical density: $n_{\rm cr}=\frac{A_{ij}}{k_{ij}}$, where $i$ and $j,$
respectively, designate the initial and final rotational number, $A$ is the
Einstein coefficient, and $k$ is the collisional rate coefficient.

Figure 5: $\chi^{2}$-value as a function of the H2 column density and gas
kinetic temperature for the Horsehead (left panel) and Orion Bar (right panel)
PDRs. The solid black lines represent confidence contour levels of 63.3%,
90.0%, 99.0% and 99.9%. The sets of parameters ($T$, $n$(H2)) which give
minimum $\chi^{2}$ are indicated with a white + symbol.

In order to test the impact of the new collisional rate coefficients in
astrophysical applications, the radex (van der Tak et al., 2007) code was used
to perform non-LTE radiative transfer calculations111The collisional rate
coefficients are made available as a radex datafile in the LAMDA database
(Schöier et al., 2005). using the escape probability formalism, assuming an
isothermal and homogeneous medium.

The radiation field value was taken as the cosmic microwave background at 2.73
K, and the line width was set at 1 km/s. Energy levels, transition
frequencies, and Einstein $A$ coefficients were taken from the Cologne
Database for Molecular Spectroscopy (CDMS; Müller et al. (2005)).

Collisional deexcitation rate coefficients for the first 22 rotational levels
of CF+ (231 collisional transitions) obtained in this work were included for
temperatures up to 150 K. Note that because of the absence of nuclear spin-
dependence of the rate coefficients, we only considered collisional excitation
of CF+ through para-H2 collisions in the calculations.

It should also be noted that electron collisions were not included in these
radiative transfer calculations. To the best of our knowledge, there is no
available data for the collisional excitation of CF+ by electrons.
Nevertheless, while CF+ is detected in regions of high ionization fractions,
with the electron density given by $n_{e}=n_{H}\times X(\ch{C+})\sim
2n(\ch{H2})\times 10^{-4}$ cm-3, as for many molecular ions, CF+–H2
collisional rate coefficients are particularly large: $>10^{-10}$ cm3 s-1.
This implies that in order for electronic collisions to contribute to the
excitation process of CF+ molecules, electron rate coefficients should be
greater than a few $10^{-6}$ cm3 s-1, and even greater than $10^{-5}$ cm3 s-1
to dominate.

The H2-density variation of the excitation temperatures and population levels
were computed for a grid of kinetic temperatures ranging from 10 to 150 K and
column densities ranging from $10^{7}$ to $10^{15}$ cm-2. In Figure 4, we
represent the H2-density variation of the excitation
temperature222$n_{u}/n_{l}=\,g_{u}/g_{l}\,exp[-E_{ul}/k_{\rm B}\,T_{\rm ex}]$,
where $n_{u}$ ($n_{l}$) represents the population of the upper (lower) level,
$g_{u}$ ($g_{l}$) the degeneracy of the upper (lower) level, $E_{ul}$ the
difference of energy between the upper and lower levels, and $k_{\rm B}$ the
Boltzmann constant. $T_{\rm ex}=T_{\rm k}$ corresponds to LTE conditions. and
the optical depth $\tau$ for the $1\rightarrow 0$ and $2\rightarrow 1$ CF+
rotational lines. We explore kinetic temperatures of 20, 50, and 100 K, and a
CF+ column density of $10^{12}$ cm-2, corresponding to the typical value
inferred from previous observations (Neufeld et al., 2006; Guzmán et al.,
2012a; Nagy et al., 2017) in PDRs.

At low temperatures ($T_{\rm k}=20$ K), the $1\rightarrow 0$ rotational line
exhibits supra-thermal emission ($T_{\rm ex}>T_{\rm k}$) in the H2-density
range of $\sim 10^{4}-10^{6}$ cm-3. At higher kinetic temperatures (50 and 100
K), the excitation temperature becomes negative in approximately the same
H2-density range. The same behavior is observed for the $2\rightarrow 1$ line
in a narrower H2-density range and at higher temperatures: supra-thermal
excitation appears at $\sim$ 50 K in the H2-density range of $\sim
10^{5}-10^{6}$ cm-3, while negative excitation temperatures are observed at a
kinetic temperature of 100 K. This negative excitation temperature behavior
indicates a level population inversion ($n_{u}/g_{u}>\,n_{l}/g_{l}$). However,
as can be seen in Figure 4, at the relatively low CF+ column densities present
in prototypical PDRs, the opacities of the inverted lines remain very low
($-\tau<<1$) for all the explored physical conditions. This implies a
negligible maser amplification insufficient to exhibit an observable effect.

Finally, we used these radex models to infer the H2 gas density at the UV-
illuminated surfaces of the Horsehead and Orion Bar PDRs from the observed CF+
line emission. In particular, we tried to fit the CF+ emission spectra
reported by Neufeld et al. (2006), Guzmán et al. (2012a), Guzmán et al.
(2012b), and Nagy et al. (2017) summarized in Table 1.

In order to determine the set of parameters ($T$, $n$(H2) and $N$(CF+)) that
best reproduces the observations, we calculated the $\chi^{2}$-value for each
set of parameters as follows:

$\chi^{2}=\sum_{i=1}^{n}\left(\frac{W_{i}^{\rm obs}-W_{i}^{\rm
calc}}{\sigma_{i}}\right)^{2},$ (3)

where $n$ is the number of observed CF+ rotational lines, $W^{\rm calc}$ is
the integrated line intensity obtained from radex simulations, $W^{\rm obs}$
is the observed integrated line intensity reported in the papers, and $\sigma$
is the uncertainty on the observed value. The best fit is then obtained by
minimizing the $\chi^{2}$ values.

These fits implicitly assume that the observed CF+ emission is spatially
extended (i.e., it fills the beam of the telescope at each observed
frequency). This is a reasonable assumption because CF+ is expected to arise
from the extended C+ layers at the edge of PDRs (see models of Neufeld &
Wolfire 2009, Guzmán et al. 2012b).

For both sources, the best-fit value for the CF+ column density $N$(CF+)
allowing us to reproduce the observed line strengths were $1.63\times 10^{12}$
cm-2 and $1.90\times 10^{12}$ cm-2 for the Horsehead and Orion Bar regions,
respectively. In Figure 5, we present the $\chi^{2}$-value as a function of
the H2 gas density and gas kinetic temperature for both Horsehead and Orion
Bar PDRs at the corresponding CF+ column densities. In this Figure, we also
represent contour levels where the probability of enclosing the correct
parameters $T$, $n$(H2) and $N$(CF+) is 63.3%, 90.0%, 99.0%, and 99.9%. This
is achieved by considering the regions where $\chi^{2}\leq\chi^{2}_{\rm
min}+\Delta\chi^{2}$ for $\Delta\chi^{2}$ values of 2.3, 4.6, 9.2, and 13.8
(Lampton et al., 1976). The sets of parameters ($T$, $n$(H2)) that give
minimum $\chi^{2}$ (i.e., the best fit) are indicated with a white + symbol:
we obtain kinetic temperatures of 100 K and 50 K, and H2 gas densities of
$3.5\times 10^{4}$ cm-3 and $1.8\times 10^{5}$ cm-3, for the Horsehead and
Orion Bar PDRs, respectively.

In those two prototypical PDRs, the gas temperature of the C+-emitting layers
(likely similar to the CF+-emitting layers at the UV-illuminated cloud
surface) are observationally well constrained to $T_{\rm k}=60-100$ K in the
Horsehead (Pabst et al., 2017) and $T_{\rm k}=150-200$ K in the Orion Bar
(Goicoechea et al., 2016). We note that a more refined determination of the
gas density would require one to spatially resolve the true size of the CF+
-emitting regions (e.g., with ALMA) and carry out PDR-depth dependent
excitation models. If we assume that the CF+ emission in the Orion Bar arises
from a 10”-width filament (similar to other molecular ions such as SH+,
Goicoechea et al. 2017) and correct the observed line intensities by the
appropriate beam filling factors, we obtain reasonable fits consistent with
gas temperatures of about 100 K and H2 gas density n(H2)$\sim 7\times 10^{4}$
cm-3.

##  4 Summary

In summary, we computed rate coefficients for inelastic collisions of CF+ by
both para- and ortho-H2 for temperatures up to 150 K and rotational
transitions between the first 22 levels of the CF+ molecule ($0\leq j_{1}\leq
21$). We used these new rate coefficients in non-LTE excitation and radiative
transfer calculations that reveal inversion population and weak maser CF+
emission at gas physical conditions corresponding to those in regions where
this molecule can be observed. Finally, we were able to constrain gas density
at the surface of the Horsehead and Orion Bar PDRs by reproducing their
observed CF+ line intensities.

###### Acknowledgements.

F.L. acknowledges financial support from the Institut Universitaire de France.
We acknowledge the Programme National Physique et Chimie du Milieu
Interstellaire (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES.
This work was granted access to the Occigen HPC resources of CINES under the
allocation 2019 [A0070411036] made by GENCI. J.R.G. thanks the Spanish MICIU
for funding support under grants AYA2017-85111-P and PID2019-106110GB-I00.
R.D. is supported by the US Department of Energy Office of Science, Office of
Basic Energy Sciences (Award DE-SC0019740).

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