# Infinite-color randomly reinforced urns with dominant colors

Hristo Sariev a, Sandra Fortini b, Sonia Petrone b
a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
b Department of Decision Sciences, Università Commerciale Luigi Bocconi
Corresponding author<EMAIL_ADDRESS>
###### Abstract

We define and prove limit results for a class of dominant Pólya sequences,
which are randomly reinforced urn processes with color-specific random weights
and unbounded number of possible colors. Under fairly mild assumptions on the
expected reinforcement, we show that the predictive and the empirical
distributions converge almost surely (a.s.) in total variation to the same
random probability measure $\tilde{P}$; moreover, $\tilde{P}(\mathcal{D})=1$
a.s., where $\mathcal{D}$ denotes the set of dominant colors for which the
expected reinforcement is maximum. In the general case, the predictive
probabilities and the empirical frequencies of any $\delta$-neighborhood of
$\mathcal{D}$ converge a.s. to one. That is, although non-dominant colors
continue to be regularly observed, their distance to $\mathcal{D}$ converges
in probability to zero. We refine the above results with rates of convergence.
We further hint potential applications of dominant Pólya sequences in
randomized clinical trials and species sampling, and use our central limit
results for Bayesian inference.

Keywords: Reinforced processes, Pólya sequences, Random probability measures,
Species sampling, Bayesian nonparametrics

MSC2020 Classification: 60F15, 60B10, 60F05, 60G57

##  1 Introduction

Let $(\mathbb{X},\mathcal{X})$ be a measurable space. An $\mathbb{X}$-valued
sequence of random variables $(X_{n})_{n\geq 1}$ whose predictive structure is
given by $\mathbb{P}(X_{1}\in\cdot\,)=\nu(\cdot)$ and, for any $n\geq 1$,

$\mathbb{P}(X_{n+1}\in\cdot\,|X_{1},W_{1},\ldots,X_{n},W_{n})=\sum_{i=1}^{n}\frac{W_{i}}{\theta+\sum_{j=1}^{n}W_{j}}\delta_{X_{i}}(\cdot)+\frac{\theta}{\theta+\sum_{j=1}^{n}W_{j}}\nu(\cdot),$
(1.1)

for some non-negative random weights $(W_{n})_{n\geq 1}$, a probability
measure $\nu$ on $\mathbb{X}$, and a constant $\theta>0$, describes the
observation process of any urn scheme where only the color of the observed
ball is reinforced. The classical example is the $k$-color Pólya urn with
constant reinforcement $W_{n}=1$ and finite $|\mathbb{X}|=k$, in which case
(1.1) characterizes an exchangeable sequence $(X_{n})_{n\geq 1}$ with
$k$-dimensional Dirichlet prior distribution. The important extension of Pólya
urns to infinite colors is given by [8] who consider the model (1.1) with
$W_{n}=1$ and $\nu$ possibly diffuse. In that case, the urn scheme is better
known as a Pólya sequence and it generates an exchangeable process
$(X_{n})_{n\geq 1}$ whose directing random measure has a Dirichlet process
distribution with parameters $(\theta,\nu)$. Because of their simple
predictive structure and nonparametric nature, Pólya sequences play a
fundamental role in the construction of more complex procedures for
nonparametric Bayesian inference (see [13] for a review). In general, an
exchangeable process with prediction rule $\eqref{eq:intro:rrps}$ is a member
of the class of species sampling sequences [24], which entails, by [17,
Theorem 1], that $W_{n}/(\theta+\sum_{j=1}^{n}W_{j})$ and
$(X_{1},\ldots,X_{n})$ are conditionally independent given the partition of
$\\{1,\ldots,n\\}$ generated by $(X_{1},\ldots,X_{n})$ and the
$\sigma$-algebra generated by the sets $\\{X_{i}\mbox{ is an atom of }\nu\\}$.

Bassetti and coauthors [3] define a class of generalized species sampling
sequences that have predictive distributions of the kind
$\eqref{eq:intro:rrps}$, but the process they generate is conditionally
identically distributed (c.i.d.). By [6], a stochastic process is c.i.d. if
and only if its sequence of predictive distributions is a measure-valued
martingale. Thus, any exchangeable process is c.i.d., while the converse is
generally not true; indeed, a sequence of random variables is exchangeable if
and only if it is stationary and c.i.d. [19]. Nevertheless, c.i.d. processes
are asymptotically exchangeable, and so preserve main limit features of
exchangeable sequences, see [6]. In (1.1), the presence of the random weights
$W_{n}$ can lead to an imbalanced design, which may easily break the condition
of stationarity. Then it may be reasonable to assume that the process
$(X_{n})_{n\geq 1}$ is c.i.d., and a known sufficient condition is that
$W_{n}$ and $X_{n}$ are conditionally independent given
$(X_{1},W_{1},\ldots,X_{n-1},W_{n-1})$, see [3, 15].

In this paper we consider the predictive scheme (1.1) for which the random
reinforcement $W_{n}$ depends explicitly on the color $X_{n}$, and the process
$(X_{n})_{n\geq 1}$ is in general neither exchangeable nor c.i.d. This case is
of interest in many contexts; for example, in problems of species sampling,
color-specific random weights capture different levels of species
adaptability. Then we expect that the colors which have a higher expected
reinforcement will tend to dominate the observation process, just as better-
adapted species have a greater chance at survival. To the best of our
knowledge, existing studies consider only the model with $k$ colors, which is
known as a randomly reinforced urn (RRU), and show that the probability of
drawing a color whose expected weight is maximum goes indeed to one, see,
e.g., [4, 21, 22, 26]. We thus explore the extension of RRUs to an unbounded
set of possible colors, which parallels the extension provided by exchangeable
Pólya sequences for $k$-color Pólya urns.

We study the limit behavior of the proposed class of randomly reinforced Pólya
sequences under minimal assumptions on the reinforcement. Let $\bar{w}$ be the
sup of the expected reinforcement as it varies across colors, and
$\mathcal{D}\in\mathcal{X}$ the set of dominant colors whose expected weight
is $\bar{w}$. Let us denote the predictive and the empirical distributions of
$(X_{n})_{n\geq 1}$ by

$P_{n}(\cdot)\equiv\mathbb{P}(X_{n+1}\in\cdot\,|X_{1},W_{1},\ldots,X_{n},W_{n})\qquad\mbox{and}\qquad\hat{P}_{n}(\cdot)\equiv\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}(\cdot).$

We show in Theorem 3.3 that, under certain conditions of compactness, the
probability of observing non-dominant colors may not vanish, though their
distance to $\mathcal{D}$ converges in probability to zero. However, if
$\bar{w}$ is significantly larger than the expected reinforcement of any
$x\in\mathcal{D}^{c}$, then there exists a random probability measure
$\tilde{P}$ on $\mathbb{X}$ with $\tilde{P}(\mathcal{D})=1$ a.s. such that

$\|P_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0\qquad\mbox{and}\qquad\|\hat{P}_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0,$
(1.2)

where $\|\alpha-\beta\|\equiv\sup_{A\in\mathcal{X}}|\alpha(A)-\beta(A)|$ is
the total variation distance between two probability measures $\alpha$,
$\beta$ on $\mathbb{X}$. The latter implies, in particular, that $\tilde{P}$
is a.s. discrete on the set $\mathcal{D}$. On the other hand, one can show
from (1.2) that $(X_{n})_{n\geq 1}$ is asymptotically exchangeable with limit
directing random measure $\tilde{P}$; that is, as $n\rightarrow\infty$,
$(X_{n+1},X_{n+2},\ldots)$ converges in distribution to an exchangeable
sequence with directing random measure $\tilde{P}$, see [1, Lemma 8.2].

In addition, we provide set-wise rates of convergence for $P_{n}$ and
$\hat{P}_{n}$ in the form of a central limit theorem; namely, Theorem 4.2
gives conditions under which, for every $A\in\mathcal{X}$ such that
$A\cap\mathcal{D}\neq\emptyset$,

$\sqrt{n}\bigl{(}\hat{P}_{n}(A)-P_{n}(A)\bigr{)}\qquad\mbox{and}\qquad\sqrt{n}\bigl{(}P_{n}(A)-\tilde{P}(A)\bigr{)},$

converge to non-degenerate Gaussian limits in the sense of stable convergence
and a.s. conditional convergence, respectively. We illustrate the statistical
use of these results by obtaining marginal asymptotic credible intervals for
$\tilde{P}$ under a Bayesian approach.

The paper is organized as follows. The formal definition of the proposed Pólya
sequences with dominant colors is given in Section 2. Their first-order
convergence properties are derived in Section 3, which also includes examples
of their potential use in species sampling and clinical trials. Section 4
contains results on rates of convergence and central limit theorems. The
proofs are postponed to Section 5.

##  2 Pólya sequence with dominant colors

Let $(\mathbb{X},d)$ be a complete separable metric space endowed with its
Borel $\sigma$-algebra $\mathcal{X}$, and let $(X_{n})_{n\geq 1}$ be a
sequence of $\mathbb{X}$-valued random variables that are defined on some
probability space $(\Omega,\mathcal{H},\mathbb{P})$ and have predictive
distributions (1.1). We assume that the random reinforcement in (1.1) is
color-specific in the sense that

$W_{n}=h(X_{n},U_{n}),$

where $(U_{n})_{n\geq 1}$ is an independent and identically distributed
sequence of real-valued random variables such that $U_{n}$ is independent of
$(X_{1},\ldots,X_{n})$, and $h$ is a measurable function from
$\mathbb{X}\times\mathbb{R}$ into $\mathbb{R}_{+}$.

Let us define the expected weight of color $x\in\mathbb{X}$ by

$w(x)=\mathbb{E}[W_{n}|X_{n}=x],$

and denote

$\bar{w}=\sup_{x\in\mathbb{X}}w(x).$

We further let $\mathcal{F}_{0}=\\{\Omega,\emptyset\\}$,
$\mathcal{F}_{n}=\sigma(X_{1},U_{1},\ldots,X_{n},U_{n})$, $n\geq 1$, and

$\displaystyle\mathcal{D}=\\{x\in\mathbb{X}:w(x)=\bar{w}\\},\qquad\mathcal{D}_{\delta}=\\{x\in\mathbb{X}:d(x,\mathcal{D})<\delta\\},$

for every $\delta>0$. The set $\mathcal{D}$ is called the set of _dominant_
colors. We make the following assumptions:

$\displaystyle 0\leq W_{n}\leq\beta\mbox{ for some constant }\beta;$ (2.1)
$\displaystyle\bar{w}\in\mbox{supp}(\nu_{w});$
$\displaystyle\bar{w}>\sup_{x\in\mathcal{D}_{\delta}^{c}}w(x)\textnormal{ for
every }\delta>0,$

where $\nu_{w}$ is the image measure of $\nu$ under $w$, and
$\mbox{supp}(\nu_{w})$ is the support of $\nu_{w}$ on $\mathbb{R}_{+}$,
defined by

$\mbox{supp}(\nu_{w})=\\{u\geq
0:\nu_{w}((u-\epsilon,u+\epsilon))>0,\forall\epsilon>0\\}.$

The assumptions (2.1) essentially require that the colors $x$ for which the
expected weight $w(x)$ is arbitrarily close to $\bar{w}$ lie in a compact set
that we are able to sample from. Notice that when $\mathcal{D}$ is non-empty
and $\nu(\mathcal{D})>0$, then $\bar{w}\in\mbox{supp}(\nu_{w})$ is
automatically satisfied. In those cases, we will be interested in the
relationship between $\bar{w}$ and the next highest value in the support of
$\nu_{w}$, namely,

$\bar{w}^{c}=\sup\\{u\geq 0:u\in\mbox{supp}((\nu_{|\mathcal{D}^{c}})_{w})\\},$

where $\nu_{|\mathcal{D}^{c}}(\cdot)=\nu(\cdot\,|\mathcal{D}^{c})$. We call
any process characterized by (1.1) a dominant Pólya sequence (DPS) if it
satisfies the assumptions (2.1), and proceed to study its asymptotic
properties.

##  3 First-order convergence

We start by looking at the DPS with a constant expected weight function
$w(x)\equiv\bar{w}$. Then all colors are dominant and one can show that
$W_{n}$ and $X_{n}$ are conditionally uncorrelated given $\mathcal{F}_{n-1}$.
Recall from Section 1 that $(X_{n})_{n\geq 1}$ is c.i.d. whenever $W_{n}$ and
$X_{n}$ are conditionally independent. In that case, the predictive and the
empirical distributions evaluated at any set $A\in\mathcal{X}$ converge a.s.
to the same random limit, see [6]. Here, $(X_{n})_{n\geq 1}$ is generally not
c.i.d. because the weights depend on the corresponding colors and
$(P_{n})_{n\geq 1}$ may not be a measure-valued martingale. Nevertheless,
$P_{n}$ and $\hat{P}_{n}$ converge to a common random probability measure, and
convergence is in total variation as in (1.2).

######  Proposition 3.1.

Let $\mathbb{X}=\mathcal{D}$. Then there exists a random probability measure
$\tilde{P}$ on $\mathbb{X}$ such that

$\|P_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0\qquad\mbox{and}\qquad\|\hat{P}_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0.$

We now consider the opposite extreme case of a strictly increasing $w$
function on $\mathbb{X}=[0,1]$. Then “$1$” is the only dominant color and we
show that the predictive and the empirical distributions converge weakly on a
set of probability one to a probability measure degenerate at “$1$”. Still,
the probability of drawing a non-dominant color may not vanish.

######  Proposition 3.2.

Let $\mathbb{X}=[0,1]$ and $(X_{n})_{n\geq 1}$ be a DPS with a continuous and
strictly increasing expected weight function $w$. Then, as
$n\rightarrow\infty$,
$\frac{1}{n}\sum_{i=1}^{n}W_{i}\overset{a.s.}{\longrightarrow}w(1)$ and

$P_{n}\overset{w}{\longrightarrow}\delta_{1}\quad\mbox{a.s.}[\mathbb{P}]\qquad\mbox{and}\qquad\hat{P}_{n}\overset{w}{\longrightarrow}\delta_{1}\quad\mbox{a.s.}[\mathbb{P}].$

Moreover, there exists a $[0,1]$-valued random variable $\eta$ such that
$P_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}\eta$.

The sampling scheme implied by a strictly increasing $w$ can be described as
follows: colors are picked and reinforced until a new color that is closer to
“$1$” is drawn. This color is then increasingly preferred by the model until a
better alternative appears, which happens almost surely. In fact, as we will
show in Proposition 4.1, non-dominant colors are forgotten quicker the more
distant they are from the dominant color “$1$”.

Given Propositions 3.1 and 3.2, we are ready to state our main result of this
section, which considers a DPS with an arbitrary expected weight function
$w(\cdot)$ and a general space of colors $\mathbb{X}$. We show that, when
restricted to $\mathcal{D}^{c}$, the process behaves similarly to the case in
LABEL:{results:dps:mon} of a strictly increasing $w$ function. If further
$\bar{w}>\bar{w}^{c}$, then the probability of observing non-dominant colors
converges a.s. to zero. Thus, neglecting the occasional non-dominant
observation, we have asymptotically a constant expected weight function.

######  Theorem 3.3.

For any DPS $(X_{n})_{n\geq 1}$, as $n\rightarrow\infty$,
$\frac{1}{n}\sum_{i=1}^{n}W_{i}\overset{a.s.}{\longrightarrow}\bar{w}$ and

$P_{n}(\mathcal{D}_{\delta}^{c})\overset{a.s.}{\longrightarrow}0\qquad\mbox{for
every }\delta>0.$

Moreover, there exists a $[0,1]$-valued random variable $\eta$ such that

$P_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\eta.$

If, in addition, it holds that $\bar{w}>\bar{w}^{c}$, then $\eta=0$ and there
exists a random probability measure $\tilde{P}$ on $\mathbb{X}$ with
$\tilde{P}(\mathcal{D})=1$ a.s.$[\mathbb{P}]$ such that

$\|P_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0\qquad\mbox{and}\qquad\|\hat{P}_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0.$

If $\bar{w}=\bar{w}^{c}$, then
$P_{n}(\mathcal{D}_{\delta})\overset{a.s.}{\longrightarrow}1$ and, by Theorem
3.3, $\liminf_{n}P_{n}(\mathcal{D}_{\delta}\backslash\mathcal{D})$ can be
strictly positive. In that case, _non_ -dominant colors continue to be
regularly observed yet get closer and closer to $\mathcal{D}$. In fact, using
dominated convergence,

$d(X_{n},\mathcal{D})\overset{p}{\longrightarrow}0.$ (3.1)

If $\bar{w}>\bar{w}^{c}$, then, in particular,
$P_{n}\overset{w}{\longrightarrow}\tilde{P}$ a.s.$[\mathbb{P}]$ and
$\hat{P}_{n}\overset{w}{\longrightarrow}\tilde{P}$ a.s.$[\mathbb{P}]$. It
follows from a generalized martingale convergence theorem (see [7, Theorem 2])
that the conditional distribution of $X_{n+1}$ given $(X_{1},\ldots,X_{n})$
converges weakly on a set of probability one to $\tilde{P}$. Thus, by [1,
Lemma 8.2], $(X_{n})_{n\geq 1}$ is asymptotically exchangeable with limit
directing random measure $\tilde{P}$.

The reinforced urn scheme (1.1) implies that there will be ties in a sample
$(X_{1},\ldots,X_{n})$ of size $n$ with positive probability. Let us denote
the number of distinct colors by

$L_{n}=\max\bigl{\\{}k\in\\{1,\ldots,n\\}:X_{k}\notin\\{X_{1},\ldots,X_{k-1}\\}\bigr{\\}}.$

The next result shows that $L_{n}$ is approximately $\bar{w}^{-1}\theta\log n$
for large $n$.

######  Proposition 3.4.

For any DPS $(X_{n})_{n\geq 1}$, if $\nu$ is diffuse, then, as
$n\rightarrow\infty$,

$\frac{L_{n}}{\log n}\overset{a.s.}{\longrightarrow}\frac{\theta}{\bar{w}}.$

###  3.1 Examples

We close this section with a few examples and hints of potential areas of
application.

######  Example 3.5 ($k$-color RRU).

Let $F_{1},\ldots F_{k}$ be probability distribution functions on $[0,\beta]$.
Denote $m_{j}=\int tdF_{j}(t)$, $j=1,\ldots,k$, and suppose that, for some
$k_{0}\in\\{1,\ldots,k\\}$,

$m_{1}=\cdots=m_{k_{0}}=\bar{m}>\bar{m}^{c}\equiv\max_{k_{0}<j\leq k}m_{j}.$
(3.2)

The $k$-color RRU is a DPS that associates $W_{n}|\\{X_{n}=j\\}\sim F_{j}$ to
color $j=1,\ldots,k$. Then $w(j)=m_{j}$, $j=1,\ldots,k$ and
$\mathcal{D}=\\{1,\ldots,k_{0}\\}$. In [21, 22] the authors look at the case
$k_{0}=1$ and $k_{0}=k$, whereas [4, 26] consider $1<k_{0}<k$, although [4]
assumes (3.2) only asymptotically.

Define by
$P_{n}(\cdot\,|\mathcal{D})=P_{n}(\cdot\cap\mathcal{D})\bigl{/}P_{n}(\mathcal{D})$
the predictive distribution of $X_{n+1}$ conditional on being in
$\mathcal{D}$. It follows from Proposition 3.1 that
$P_{n}(\\{j\\}|\mathcal{D})\overset{a.s.}{\longrightarrow}\tilde{p}_{j}$,
$j=1,\ldots,k_{0}$, for some non-negative random variables
$\tilde{p}_{1},\ldots,\tilde{p}_{k_{0}}$ such that
$\sum_{j=1}^{k_{0}}\tilde{p}_{j}=1$ a.s.$[\mathbb{P}]$. As
$P_{n}(\mathcal{D})\overset{a.s.}{\longrightarrow}1$ by Theorem 3.3, then

$P_{n}(\\{j\\})=P_{n}(\mathcal{D})P_{n}(\\{j\\}|\mathcal{D})+P_{n}(\mathcal{D}^{c})P_{n}(\\{j\\}|\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\tilde{p}_{j}.$

Therefore, $\|P_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0$ and,
similarly, $\|\hat{P}_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0$, where
$\tilde{P}=\sum_{j=1}^{k_{0}}\tilde{p}_{j}\delta_{j}$.

######  Example 3.6 (Unimodal DPS).

Suppose that there is a unique dominant color, i.e., $\mathcal{D}=\\{x_{0}\\}$
for some $x_{0}\in\mathbb{X}$. It follows from (3.1) that

$X_{n}\overset{p}{\longrightarrow}x_{0}.$

Moreover,

$P_{n}\overset{w}{\longrightarrow}\delta_{x_{0}}\qquad\mbox{a.s.}[\mathbb{P}].$

Indeed, for any open set $G\in\mathcal{X}$, we have that, if $x_{0}\in G$,
then there exists $\delta>0$ such that $x_{0}\in\mathcal{D}_{\delta}\subseteq
G$; therefore, by Theorem 3.3,

$\liminf_{n\rightarrow\infty}P_{n}(G)\geq\liminf_{n\rightarrow\infty}P_{n}(\mathcal{D}_{\delta})=1=\delta_{x_{0}}(G)\qquad\mbox{a.s.}[\mathbb{P}].$

As $\delta_{x_{0}}(G)=0$ otherwise, a.s. weak convergence of $P_{n}$ to
$\delta_{x_{0}}$ follows.

Unimodal DPSs can be used in the context of response-adaptive randomization
procedures for clinical trials (see [25] for a general reference) to provide
an ethical design that can deal with an unbounded number or even a continuum
of potential treatments. Consider a stylized trial for determining the optimal
dose of a drug. Let $X_{n}$ represent the dose given to the $n$th patient who
enters the trial, and denote by $W_{n}$ the patient’s response to it. Then the
reinforced predictive scheme (1.1) with a unimodal $w(\cdot)$ function
provides a response-adaptive rule for selecting the dose for the next patient.
It follows from Theorem 3.3 that such a design is ethical in the sense that,
for any $\delta>0$,
$P_{n}(\mathcal{D}_{\delta})\overset{a.s.}{\longrightarrow}1$; that is, even
in situations where it is practically impossible to select $x_{0}$ exactly
(e.g., $\nu(\\{x_{0}\\})=0$ with $\nu$ is diffuse), the probability of
assigning the next patient a dose that is arbitrarily close to $x_{0}$
converges a.s. to one.

######  Example 3.7 (Species sampling model with dominant species).

Exchangeable species sampling models are of interest in many fields and, in
particular, in the broad area of Bayesian nonparametrics [24], yet forms of
competition between species can easily break the symmetry imposed by
exchangeability. Dominant Pólya sequences can thus provide an extension to the
framework of species sampling models by addressing situations where
exchangeability holds only asymptotically and a restricted set of dominant
species tends to prevail. The latter is closely related to the notion of
sparsity in random graph theory, where conciliating between exchangeability
and graph density is a relevant issue, see [9].

Consider again the sampling scheme (1.1). In this case, interest lies on the
categorical species and $X_{n}$ plays the role of a label chosen, say, in
$\mathbb{X}=[0,1]$ from a uniform $\nu$. Take $0<w_{1}<w_{2}<\infty$ and
$p\in(0,1)$. Let $W_{n}=w(X_{n})$ where

$w(x)=w_{1}\cdot 1\\!\\!1_{[0,p)}(x)+w_{2}\cdot
1\\!\\!1_{[p,1]}(x)\quad\mbox{for }x\in\mathbb{X}.$

Then $(X_{n})_{n\geq 1}$ is a DPS with $\mathcal{D}=[p,1]$. According to this
scheme, when a new species appears, it has probability $p$ to be non-dominant
and $(1-p)$ to be dominant, independently over $n$. As $p>0$, then
$\sum_{i=1}^{n}\delta_{X_{i}}(\mathcal{D}^{c})$ diverges a.s. and we discover
non-dominant species infinitely often. However, by Theorem 3.3,
$\hat{P}_{n}([p,1])$ converges a.s. to one, so the discovery rate of non-
dominant species is of order less than $n$ (see also Proposition 4.1).

##  4 Rates and central limit results

Let $(X_{n})_{n\geq 1}$ be a DPS such that $\bar{w}>\bar{w}^{c}$, and let
$\tilde{P}$ be a random probability measure as in Theorem 3.3. It follows
that, for every $A\in\mathcal{X}$,

$P_{n}(A)-\tilde{P}(A)\overset{a.s.}{\longrightarrow}0\qquad\mbox{and}\qquad\hat{P}_{n}(A)-P_{n}(A)\overset{a.s.}{\longrightarrow}0.$

We proceed to study the rate of convergence of the above quantities by first
looking at the convergence rate on $A=\mathcal{D}^{c}$.

###  4.1 Rate of convergence for dominated colors

Assume $\bar{w}>\bar{w}^{c}$. Then Theorem 3.3 implies that both
$\hat{P}_{n}(\mathcal{D}^{c})$ and $P_{n}(\mathcal{D}^{c})$ converge a.s. to
zero. The next result shows that, in fact, $\hat{P}_{n}(\mathcal{D}^{c})$ and
$P_{n}(\mathcal{D}^{c})$ are $O(n^{-(\bar{w}-\bar{w}^{c})/\bar{w}})$ as
$n\rightarrow\infty$.

######  Proposition 4.1.

Suppose $\bar{w}>\bar{w}^{c}$. Let $\gamma=1-\bar{w}^{c}/\bar{w}$. Then there
exists a random variable $\xi$ with $\xi\in[0,\infty)$ such that

$n^{\gamma}\cdot\hat{P}_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\xi\qquad\mbox{and}\qquad
n^{\gamma}\cdot
P_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\frac{\bar{w}^{c}}{\bar{w}}\xi.$

Moreover,
$n^{\psi}\cdot\hat{P}_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\infty$
and $n^{\psi}\cdot
P_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\infty$ for every
$\psi>\gamma$.

###  4.2 Central limit theorem

Let $A\in\mathcal{X}$ be such that $A\cap\mathcal{D}\neq\emptyset$. We here
give conditions under which

$C_{n}(A)=\sqrt{n}\bigl{(}\hat{P}_{n}(A)-P_{n}(A)\bigr{)}\qquad\textnormal{and}\qquad
D_{n}(A)=\sqrt{n}\bigl{(}P_{n}(A)-\tilde{P}(A)\bigr{)},$

converge to non-degenerate Gaussian limits. We will denote by
$\mathcal{N}(\mu,\sigma^{2})$ a Gaussian law with parameters
$(\mu,\sigma^{2})$, where $\mathcal{N}(\mu,0)=\delta_{\mu}$. The results are
given in terms of stable and almost sure (a.s.) conditional convergence, which
we briefly recall.

Almost sure conditional convergence. Let $\mathcal{G}=(\mathcal{G}_{n})_{n\geq
0}$ be a filtration, and $\tilde{Q}$ a random probability measure on
$\mathbb{X}$. A sequence $(Y_{n})_{n\geq 1}$ that is non-necessarily adapted
to $\mathcal{G}$ is said to converge to $\tilde{Q}$ in the sense of a.s.
conditional convergence w.r.t. $\mathcal{G}$ if

$\mathbb{P}(Y_{n}\in\cdot\,|\mathcal{G}_{n})\overset{w}{\longrightarrow}\tilde{Q}(\cdot)\quad\mbox{a.s.}[\mathbb{P}].$

We refer to [11] for more details.

Stable convergence. Stable convergence is a strong form of convergence in
distribution, albeit weaker than a.s. conditional convergence. A sequence
$(Y_{n})_{n\geq 1}$ converges stably to $\tilde{Q}$ if

$\mathbb{E}\bigl{[}V\,f(Y_{n})\bigr{]}\longrightarrow\mathbb{E}\Bigl{[}V\int_{\mathbb{X}}f(x)\tilde{Q}(dx)\Bigr{]},$

for any continuous bounded function $f$ and any integrable random variable
$V$. The main application of stable convergence is in central limit theorems
that allow for mixing variables in the limit. See [18] for a complete
reference on stable convergence.

We will assume $\bar{w}>2\bar{w}^{c}$; thus, $\hat{P}_{n}(\mathcal{D}^{c})$
and $P_{n}(\mathcal{D}^{c})$ are $o(n^{-1/2})$ as $n\rightarrow\infty$, by
Proposition 4.1. Let us denote

$P_{n}(A|\mathcal{D})=\frac{P_{n}(A\cap\mathcal{D})}{P_{n}(\mathcal{D})}\qquad\mbox{and}\qquad\hat{P}_{n}(A|\mathcal{D})=\frac{\hat{P}_{n}(A\cap\mathcal{D})}{\hat{P}_{n}(\mathcal{D})}.$

Then
$C_{n}(A)-\sqrt{n}(\hat{P}_{n}(A|\mathcal{D})-P_{n}(A|\mathcal{D}))\overset{a.s.}{\longrightarrow}0$
and
$D_{n}(A)-\sqrt{n}(P_{n}(A|\mathcal{D})-\tilde{P}(A))\overset{a.s.}{\longrightarrow}0$,
so we can work with the sequence $(X_{n})_{n\geq 1}$ conditional on being in
$\mathcal{D}$, in which case the expected weight function is constant.

Lastly, we can show using standard arguments that
$\mathbb{E}[f(X_{n+1})|\mathcal{F}_{n}]\overset{a.s.}{\longrightarrow}\int_{\mathbb{X}}f(x)\tilde{P}(dx)$
for every $f\in M_{b}(\mathcal{X})$. Then the following limit exists
$\mathbb{P}$-a.s.

$q_{A}=\lim_{n\rightarrow\infty}\mathbb{E}[W_{n+1}^{2}\delta_{X_{n+1}}(A)|\mathcal{F}_{n}].$

######  Theorem 4.2.

Suppose $\bar{w}>2\bar{w}^{c}$. Let $A\in\mathcal{X}$ be such that
$A\cap\mathcal{D}\neq\emptyset$. Define

$V(A)=\frac{1}{\bar{w}^{2}}\bigl{\\{}(\tilde{P}(A^{c}))^{2}q_{A}+(\tilde{P}(A))^{2}q_{A^{c}}\bigr{\\}}\quad\mbox{and}\quad
U(A)=V(A)-\tilde{P}(A)\tilde{P}(A^{c}).$

Then

$C_{n}(A)\overset{stably}{\longrightarrow}\mathcal{N}\bigl{(}0,U(A)\bigr{)}\qquad\mbox{and}\qquad
D_{n}(A)\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\bigl{(}0,V(A)\bigr{)}\quad\mbox{w.r.t.
}\mathcal{F}.$ (4.1)

It follows from (4.1) and [5, Lemma 1] that

$\sqrt{n}\bigl{(}\hat{P}_{n}(A)-\tilde{P}(A)\bigr{)}=C_{n}(A)+D_{n}(A)\overset{stably}{\longrightarrow}\mathcal{N}\bigl{(}0,U(A)+V(A)\bigr{)}.$

Moreover, the a.s. conditional convergence of $D_{n}(A)$ implies that the
distribution of $\tilde{P}(A)$ has no point masses when $0<\nu(A)<1$ (see the
proof of [2, Theorem 3.2]). Then $0<\tilde{P}(A)<1$ a.s.$[\mathbb{P}]$, and so
$V(A)$ and $U(A)$ are $\mathbb{P}$-a.s. strictly positive.

In the case $\mathbb{X}=\mathcal{D}$, the assumptions of Theorem 4.2 are
automatically satisfied and (4.1) follows. Then, for any DPS with an arbitrary
space of colors $\mathbb{X}$, we get that, by conditioning on $\mathcal{D}$,

$\displaystyle\sqrt{n}\bigl{(}\hat{P}_{n}(A|\mathcal{D})-P_{n}(A|\mathcal{D})\bigr{)}\overset{stably}{\longrightarrow}\mathcal{N}\bigl{(}0,U(A)\bigr{)},$
(4.2)
$\displaystyle\sqrt{n}\bigl{(}P_{n}(A|\mathcal{D})-\tilde{P}(A)\bigr{)}\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\bigl{(}0,V(A)\bigr{)}\quad\mbox{w.r.t.
}\mathcal{F}.$

It follows that the assumption $\bar{w}>2\bar{w}^{c}$ in Theorem 4.2 cannot be
weakened. Indeed, under $\bar{w}\leq 2\bar{w}^{c}$, Proposition 4.1 implies
that

$D_{n}(A)-\sqrt{n}\bigl{(}P_{n}(A|\mathcal{D})-\tilde{P}(A)\bigr{)}=\sqrt{n}\;P_{n}(\mathcal{D}^{c})P_{n}(A|\mathcal{D})\overset{a.s.}{\longrightarrow}\infty;$

thus, $D_{n}(A)$ and, similarly, $C_{n}(A)$ fail to converge as it is (4.2)
that are convergent.

######  Example 4.3 ($k$-color RRU (Ctd.)).

Consider again the $k$-color RRU from Example 3.5. It follows that
$q_{j}=\lim_{n}\mathbb{E}[W_{n+1}^{2}\delta_{X_{n+1}}(\\{j\\})|\mathcal{F}_{n}]$,
$j=1,\ldots,k_{0}$ exists a.s.$[\mathbb{P}]$. Then, arguing as in (4.2), we
get

$\displaystyle\sqrt{n}\bigl{(}\hat{P}_{n}(\\{j\\}|\mathcal{D})-P_{n}(\\{j\\}|\mathcal{D})\bigr{)}\overset{stably}{\longrightarrow}\mathcal{N}\Bigl{(}0,\frac{\tilde{p}_{j}}{\bar{w}^{2}}\Bigl{\\{}(1-\tilde{p}_{j})^{2}q_{j}+\tilde{p}_{j}\sum_{i\leq
k_{0},i\neq
j}q_{i}\tilde{p}_{i}\Bigr{\\}}-\tilde{p}_{j}\bigl{(}1-\tilde{p}_{j}\bigr{)}\Bigr{)},$
$\displaystyle\sqrt{n}\bigl{(}P_{n}(\\{j\\}|\mathcal{D})-\tilde{p}_{j}\bigr{)}\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\Bigl{(}0,\frac{\tilde{p}_{j}}{\bar{w}^{2}}\Bigl{\\{}(1-\tilde{p}_{j})^{2}q_{j}+\tilde{p}_{j}\sum_{i\leq
k_{0},i\neq j}q_{i}\tilde{p}_{i}\Bigr{\\}}\Bigr{)}\quad\mbox{w.r.t.
}\mathcal{F}.$

In addition, $\mathbb{P}(\tilde{p}_{j}=p)=0$, $p\in[0,1]$, so that $p_{j}>0$
a.s.$[\mathbb{P}]$ and $\tilde{p}_{i}\neq\tilde{p}_{j}$ a.s.$[\mathbb{P}]$,
$i\neq j$.

###  4.3 An application: asymptotic credible intervals for $\tilde{P}$

We close with an application of Theorem 4.2 to a problem of statistical
inference on the limit structure of the process. We assume that the random
weights $W_{n}$ are observable, as it might be the case, for example, in the
context of clinical trials (see Example 3.6). Suppose $\mathbb{X}=\mathbb{R}$,
$\mathcal{D}=[a,b]$, $\bar{w}>2\bar{w}^{c}$. Define
$F_{n}(x)=P_{n}((-\infty,x])$, $x\in\mathbb{R}$, $n\geq 1$. By Theorem 3.3,
there exists a random distribution function $\tilde{F}$ on $[a,b]$ such that

$\|F_{n}-\tilde{F}\|\overset{a.s.}{\longrightarrow}0.$

The process $\tilde{F}$ is $\mathcal{F}_{\infty}$-measurable and thus cannot
be perfectly predicted. Still, one can do inference on it based on a finite
sample $(X_{i},W_{i})_{i=1}^{n}$; in particular, under a Bayesian approach,
one does so by computing the conditional distribution of $\tilde{F}$, given
$(X_{i},W_{i})_{i=1}^{n}$. The latter is usually summarized by a credible band
or, in practice, by plotting marginal credible intervals around $\tilde{F}(x)$
for $x$ varying on a fine grid. Let $x\in[a,b]$ be such that
$0<\nu((-\infty,x])<1$. The following limits exist $\mathbb{P}$-a.s.

$q_{1}=\lim_{n\rightarrow\infty}\mathbb{E}\bigl{[}W_{n+1}^{2}\delta_{X_{n+1}}([a,x])\bigr{|}\mathcal{F}_{n}\bigr{]},\qquad
q_{0}=\lim_{n\rightarrow\infty}\mathbb{E}\bigl{[}W_{n+1}^{2}\delta_{X_{n+1}}((x,b])\bigr{|}\mathcal{F}_{n}\bigr{]}.$
(4.3)

It follows from Theorem 4.2 that

$\sqrt{n}\bigl{(}F_{n}(x)-\tilde{F}(x)\bigr{)}\overset{a.s.cond.}{\longrightarrow}\mathcal{N}(0,V_{x})\quad\mbox{w.r.t.
}\mathcal{F},$

where $V_{x}=\bar{w}^{-2}((1-\tilde{F}(x))^{2}q_{1}+\tilde{F}(x)^{2}q_{0})$.
In order to obtain asymptotic credible intervals for $\tilde{F}(x)$, we need a
convergent estimator of $V_{x}$. Let us define, for $n\geq 1$,

$\displaystyle
V_{n,x}=\frac{1}{m_{n}^{2}}\bigl{(}(1-F_{n}(x))^{2}s_{n,1}+(F_{n}(x))^{2}s_{n,0}\bigr{)},$
$\displaystyle m_{n}=\frac{1}{n}\sum_{i=1}^{n}W_{i},\quad
s_{n,1}=\frac{1}{n}\sum_{i=1}^{n}W_{i}^{2}\delta_{X_{i}}([a,x]),\quad
s_{n,0}=\frac{1}{n}\sum_{i=1}^{n}W_{i}^{2}\delta_{X_{i}}((x,b]).$

By Theorem 3.3, $m_{n}\overset{a.s.}{\longrightarrow}\bar{w}$. Moreover,
$\sum_{n=1}^{\infty}\mathbb{E}[W_{n}^{2}]/n^{2}<\infty$, so [5, Lemma 2]
implies from (4.3) that $s_{n,1}\overset{a.s.}{\longrightarrow}q_{1}$ and
$s_{n,0}\overset{a.s.}{\longrightarrow}q_{0}$. Thus,

$V_{n,x}\overset{a.s.}{\longrightarrow}V_{x}.$

A generalized Slutsky’s theorem (see also [14, Theorem 6]) implies, for every
$t\in\mathbb{R}$,

$\mathbb{P}\Bigl{(}\frac{\tilde{F}(x)-F_{n}(x)}{\sqrt{V_{n,x}/n}}\leq
t\Bigl{|}\mathcal{F}_{n}\Bigr{)}\overset{a.s.}{\longrightarrow}{\mathcal{N}}(0,1)((-\infty,t]).$

As a consequence, denoting by $z_{\alpha}$ the $(1-\alpha/2)$-quantile of the
standard Normal distribution, we get

$\liminf_{n\rightarrow\infty}\mathbb{P}\Bigl{(}F_{n}(x)-z_{\alpha}\sqrt{\frac{V_{n,x}}{n}}\leq\tilde{F}(x)\leq
F_{n}(x)+z_{\alpha}\sqrt{\frac{V_{n,x}}{n}}\Bigr{|}\mathcal{F}_{n}\Bigr{)}\geq
1-\alpha.$

Thus, an asymptotic marginal credible interval for $\tilde{F}(x)$ at level
$1-\alpha$ is

$\Bigl{[}F_{n}(x)-z_{\alpha}\sqrt{\frac{V_{n,x}}{n}},\;F_{n}(x)+z_{\alpha}\sqrt{\frac{V_{n,x}}{n}}\Bigr{]}.$

##  5 Proofs

We shall use the following notation throughout this section,

$\displaystyle
N_{n}(A)=\theta\nu(A)+\sum_{i=1}^{n}W_{i}\delta_{X_{i}}(A),\quad
M_{n}(A)=\sum_{i=1}^{n}\delta_{X_{i}}(A)+1,\quad
N_{n}=\theta+\sum_{i=1}^{n}W_{i},$

so that $P_{n}(A)=N_{n}(A)/N_{n}$ for every $n\geq 0$ and $A\in\mathcal{X}$,
with the convention $\sum_{i=1}^{0}a_{i}=0$, and we will omit the parenthesis
when $A$ is an interval for the sake of clarity. In addition, we will denote
by
$M_{b}(\mathcal{X}),M_{0}(\mathcal{X}),M_{+}(\mathcal{X}),C_{b}(\mathbb{X})$
the collections of functions $f:\mathbb{X}\rightarrow\bar{\mathbb{R}}$ that
are bounded and $\mathcal{X}$-measurable, simple and $\mathcal{X}$-measurable,
non-negative and $\mathcal{X}$-measurable, and continuous and bounded,
respectively.

The following identity, given for any $A\in\mathcal{X}$ and $n\geq 0$, is a
consequence of the independence of $(U_{n})_{n\geq 1}$ and will be applied
repeatedly,

$\mathbb{E}[W_{n+1}\delta_{X_{n+1}}(A)|\mathcal{F}_{n}]=\mathbb{E}[w(X_{n+1})\delta_{X_{n+1}}(A)|\mathcal{F}_{n}].$
(5.1)

The proof of Proposition 3.1 uses the following general fact for DPSs with
$\mathbb{X}=\mathcal{D}$.

######  Lemma 5.1.

Under the conditions in Proposition 3.1, as $n\rightarrow\infty$,

$\frac{n}{N_{n}}\longrightarrow\frac{1}{\bar{w}}\qquad\mbox{a.s.}[\mathbb{P}]\mbox{
and in }L^{p}\mbox{ for all }p\geq 1.$

###### Proof.

By hypothesis, $\mathbb{E}[W_{n+1}|\mathcal{F}_{n}]=\bar{w}$ and
$\sum_{n=1}^{\infty}\mathbb{E}[W_{n}^{2}]/n^{2}\leq\sum_{n=1}^{\infty}\beta^{2}/n^{2}<\infty$,
so [5, Lemma 2] implies that $N_{n}/n\overset{a.s.}{\longrightarrow}\bar{w}$.
Define $N_{n}^{*}=\sum_{i=1}^{n}\\{W_{i}-\mathbb{E}[W_{i}]\\}$, $n\geq 1$. By
a classical martingale inequality (see [20, Lemma 1.5] and [5, Lemma 3]), we
have

$\mathbb{P}(|N_{n}^{*}|>x)\leq 2\,\exp\bigl{\\{}-x^{2}\bigr{/}2\beta
n^{2}\bigr{\\}}\qquad\mbox{for all }x>0.$

Fix $p>0$. It follows that

$\mathbb{E}[N_{n}^{-p}]=p\int_{\theta}^{\infty}\frac{1}{t^{p+1}}\mathbb{P}(N_{n}<t)dt\leq\frac{p}{\theta^{p+1}}\int_{\theta}^{\theta+\frac{n\bar{w}}{2}}\mathbb{P}(N_{n}<t)dt+p\int_{\theta+\frac{n\bar{w}}{2}}^{\infty}\frac{1}{t^{p+1}}dt.$

Clearly,
$p\int_{\theta+\frac{n\bar{w}}{2}}^{\infty}t^{-p-1}dt=(\theta+\frac{n\bar{w}}{2})^{-p}=O(n^{-p})$.
On the other hand, for $t<\theta+\frac{n\bar{w}}{2}$,

$\mathbb{P}(N_{n}<t)=\mathbb{P}(N_{n}^{*}<t-\theta-n\bar{w})\leq\mathbb{P}(|N_{n}^{*}|>\theta+n\bar{w}-t)\leq
2\exp\Bigl{\\{}-\frac{(\theta+n\bar{w}-t)^{2}}{2\beta^{2}n}\Bigr{\\}};$

thus, $\int_{\theta}^{\theta+\frac{n\bar{w}}{2}}\mathbb{P}(N_{n}<t)dt\leq
n\bar{w}\exp\\{-n\bar{w}^{2}/4\beta^{2}\\}$ and, ultimately,
$\mathbb{E}[N_{n}^{-p}]=O(n^{-p})$. ∎

######  Proof of Proposition 3.1.

Let $A\in\mathcal{X}$ and $n\geq 0$. Then

$\displaystyle P_{n+1}(A)-P_{n}(A)$
$\displaystyle=\frac{W_{n+1}/N_{n}}{1+W_{n+1}/N_{n}}\bigl{(}\delta_{X_{n+1}}(A)-P_{n}(A)\bigr{)}$
$\displaystyle\leq\frac{W_{n+1}}{N_{n}}P_{n}(A^{c})\delta_{X_{n+1}}(A)-\Bigl{(}\frac{W_{n+1}}{N_{n}}-\frac{W_{n+1}^{2}}{N_{n}^{2}}\Bigr{)}P_{n}(A)\delta_{X_{n+1}}(A^{c}),$

where we have used that $x-x^{2}\leq\frac{x}{1+x}\leq x$ for $0\leq x\leq 1$;
thus,

$\mathbb{E}[P_{n+1}(A)-P_{n}(A)|\mathcal{F}_{n}]\leq\beta^{2}P_{n}(A)P_{n}(A^{c})N_{n}^{-2}.$
(5.2)

Therefore, by Lemma 5.1,

$\sum_{n=0}^{\infty}\mathbb{E}[P_{n+1}(A)-P_{n}(A)|\mathcal{F}_{n}]\leq\sum_{n=1}^{\infty}\frac{\beta^{2}}{N_{n-1}^{2}}<\infty\qquad\mbox{a.s.}[\mathbb{P}].$
(5.3)

On the other hand,

$\sum_{n=0}^{\infty}\mathbb{E}\bigl{[}(P_{n+1}(A)-P_{n}(A))^{2}|\mathcal{F}_{n}\bigr{]}\leq\sum_{n=0}^{\infty}\mathbb{E}\biggl{[}\frac{W_{n+1}^{2}}{N_{n+1}^{2}}\Bigr{|}\mathcal{F}_{n}\biggr{]}\leq\sum_{n=1}^{\infty}\frac{\beta^{2}}{N_{n-1}^{2}}<\infty\quad\mbox{a.s.}[\mathbb{P}].$
(5.4)

Given (5.3) and (5.4), there exists by [23, Lemma 3.2] a random variable
$\tilde{p}_{A}$ such that

$P_{n}(A)\overset{a.s.}{\longrightarrow}\tilde{p}_{A}.$

Define $\tilde{Q}(A)=\tilde{p}_{A}$, $A\in\mathcal{X}$. Then
$\tilde{Q}(\mathbb{X})=1$, $\tilde{Q}(\emptyset)=0$ and $\tilde{Q}(A)\geq 0$,
$A\in\mathcal{X}$, a.s.$[\mathbb{P}]$. Moreover,
$\omega\mapsto\tilde{Q}(A)(\omega)$ is $\mathcal{H}$-measurable for every
$A\in\mathcal{X}$, and $\tilde{Q}(A_{1}\cup
A_{2})=\tilde{Q}(A_{1})+\tilde{Q}(A_{2})$ a.s.$[\mathbb{P}]$ for disjoint
$A_{1},A_{2}\in\mathcal{X}$. In addition, the map
$A\mapsto\mathbb{E}[\tilde{Q}(A)]$ is a probability measure on $\mathbb{X}$.
Indeed, let $\\{A_{m}\\}_{m\geq 1}\subseteq\mathcal{X}$ be such that
$A_{m}\cap A_{n}=\emptyset$, $m\neq n$. Put $A=\bigcup_{m=1}^{\infty}A_{m}$.
Using (5.2),

$\mathbb{E}[P_{n}(A_{m})]=\sum_{i=1}^{n}\mathbb{E}[P_{i}(A_{m})-P_{i-1}(A_{m})]+\nu(A_{m})\leq\beta^{2}\sum_{n=1}^{\infty}\mathbb{E}\bigl{[}P_{n}(A_{m})N_{n}^{-2}]+\nu(A_{m})\equiv
M(m).$

According to the Weierstrass M-test,
$\lim_{n\rightarrow\infty}\sum_{m=1}^{\infty}\mathbb{E}[P_{n}(A_{m})]=\sum_{m=1}^{\infty}\lim_{n\rightarrow\infty}\mathbb{E}[P_{n}(A_{m})]$,
whenever $\sum_{m=1}^{\infty}M(m)<\infty$. To that end, Lemma 5.1 implies

$\sum_{m=1}^{\infty}M(m)=\beta^{2}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\mathbb{E}\bigl{[}P_{n}(A_{m}){N_{n}^{-2}}\bigr{]}+\nu(A)\leq\beta^{2}\sum_{n=1}^{\infty}\mathbb{E}[N_{n}^{-2}]+\nu(A)<\infty,$

and, as a consequence,

$\mathbb{E}[\tilde{Q}(A)]=\lim_{n\rightarrow\infty}\sum_{m=1}^{\infty}\mathbb{E}[P_{n}(A_{m})]=\sum_{m=1}^{\infty}\lim_{n\rightarrow\infty}\mathbb{E}[P_{n}(A_{m})]=\sum_{m=1}^{\infty}\mathbb{E}[\tilde{Q}(A_{m})].$
(5.5)

By [16, Theorem 3.1], there exists a random probability measure $\tilde{P}$ on
$\mathbb{X}$ such that, for every $A\in\mathcal{X}$, we have
$\tilde{Q}(A)=\tilde{P}(A)$ a.s.$[\mathbb{P}]$, and so

$P_{n}(A)\overset{a.s.}{\longrightarrow}\tilde{P}(A).$

Moreover, using standard arguments,
$\mathbb{E}[f(X_{n+1})|\mathcal{F}_{n}]\overset{a.s.}{\longrightarrow}\int_{\mathbb{X}}f(x)\tilde{P}(dx)$
for every $f\in M_{b}(\mathcal{X})$, which implies that
$P_{n}\overset{w}{\longrightarrow}\tilde{P}$ a.s.$[\mathbb{P}]$ since
$\mathbb{X}$ is separable.

Define $S_{n}=\\{X_{1},\ldots,X_{n}\\}$, $n\geq 1$. Then
$S_{n}\uparrow\\{X_{1},X_{2},\ldots\\}\equiv S$ and, by Portmanteau theorem,

$\tilde{P}(S)=\lim_{n\rightarrow\infty}\tilde{P}(S_{n})\geq\lim_{n\rightarrow\infty}\limsup_{m\rightarrow\infty}P_{m}(S_{n})\geq\lim_{n\rightarrow\infty}\limsup_{m\rightarrow\infty}\Bigl{(}1-\frac{\theta}{N_{m}}\Bigr{)}=1\qquad\textnormal{a.s.}[\mathbb{P}],$

that is, $\tilde{P}$ is $\mathbb{P}$-a.s. discrete with random support $S$. On
the other hand, proceeding as in (5.3) and (5.4), we can show that
$P_{n}(\\{X_{m}\\})\overset{a.s.}{\longrightarrow}\tilde{p}_{m}$ for some
random variables $(\tilde{p}_{m})_{m\geq 1}$. By Portmanteau theorem,

$\tilde{P}(\\{X_{m}\\})\geq\limsup_{n\rightarrow\infty}P_{n}(\\{X_{m}\\})=\tilde{p}_{m}\qquad\textnormal{a.s.}[\mathbb{P}],\mbox{
for }m\geq 1.$

But
$1=\tilde{P}(S)=\sum_{m=1}^{\infty}\tilde{P}(\\{X_{m}\\})\geq\sum_{k=1}^{\infty}\tilde{p}_{m}$
a.s.$[\mathbb{P}]$ and, using the same arguments as in (5.5),

$\mathbb{E}\Bigl{[}\sum_{m=1}^{\infty}\tilde{p}_{m}\Bigr{]}=\sum_{m=1}^{\infty}\lim_{n\rightarrow\infty}\mathbb{E}[P_{n}(\\{X_{m}\\})]=\lim_{n\rightarrow\infty}\mathbb{E}[P_{n}(S)]=1\qquad\textnormal{a.s.}[\mathbb{P}],$

so $\sum_{m=1}^{\infty}\tilde{p}_{m}=1$ a.s.$[\mathbb{P}]$, and thus
$\tilde{P}(\\{X_{m}\\})=\tilde{p}_{m}$ a.s.$[\mathbb{P}]$, $m\geq 1$. As a
result, we have

$\mathbb{P}\bigl{(}P_{n}(\\{x\\})\rightarrow\tilde{P}(\\{x\\})\mbox{ for all
}x\in
S\bigr{)}=\mathbb{P}\bigl{(}P_{n}(\\{X_{m}\\})\rightarrow\tilde{P}(\\{X_{m}\\})\mbox{
for all }m\geq 1\bigr{)}=1,$

and, by Scheffe’s lemma,

$\|P_{n}-\tilde{P}\|\leq\sup_{x\in
S}\bigl{|}P_{n}(\\{x\\})-\tilde{P}(\\{x\\})\bigr{|}+P_{n}(S^{c})\leq\sum_{x\in
S}\bigl{|}P_{n}(\\{x\\})-\tilde{P}(\\{x\\})\bigr{|}+\theta_{n}\nu(S^{c})\overset{a.s.}{\longrightarrow}0.$

Finally, [5, Lemma 2] implies that
$\hat{P}_{n}(\\{X_{m}\\})\overset{a.s.}{\longrightarrow}\tilde{P}(\\{X_{m}\\})$,
$m\geq 1$, and so $\|\hat{P}_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0$.
∎

The proof of Proposition 3.2 makes use of the following three preliminary
results.

######  Lemma 5.2.

Under the conditions in Proposition 3.2, as $n\rightarrow\infty$,
$N_{n}(t,1]\overset{a.s.}{\longrightarrow}\infty$ for every $t\in(0,1)$.

###### Proof.

Fix $t\in(0,1)$. Denote $A=(t,1]$. It follows from (2.1) that
$P_{n}(A)\geq\theta\nu(A)\bigr{/}(\theta+n\beta)>0$, so
$\sum_{n=1}^{\infty}P_{n}(A)=\infty$, and thus
$\sum_{n=1}^{\infty}\delta_{X_{n}}(A)=\infty$ a.s.$[\mathbb{P}]$ by [12,
Theorem 1]. Since $N_{n}(A)=\sum_{i=1}^{n}(N_{i}(A)-N_{i-1}(A))+\theta\nu(A)$,
then, by [10, Theorem 1], $N_{n}(A)\overset{a.s.}{\longrightarrow}\infty$ if
$\sum_{n=1}^{\infty}\mathbb{E}[N_{n}(A)-N_{n-1}(A)|\mathcal{F}_{n-1}\vee\sigma(X_{n})]=\infty$
a.s.$[\mathbb{P}]$. Indeed, from the monotonicity of $w$,

$\sum_{i=1}^{n}\mathbb{E}[N_{i}(A)-N_{i-1}(A)|\mathcal{F}_{i-1}\vee\sigma(X_{i})]=\sum_{i=1}^{n}w(X_{i})\delta_{X_{i}}(A)\geq
w(t)\sum_{i=1}^{n}\delta_{X_{i}}(A)\overset{a.s.}{\longrightarrow}\infty.$

∎

######  Lemma 5.3.

Under the conditions in Proposition 3.2, for every $t\in(0,1)$, there exist
$s\in(t,1)$ and $\lambda_{t}\in(0,1)$ such that, for all
$\lambda\in(\lambda_{t},1]$, the ratio
$N_{n}[0,t]\bigl{/}N_{n}(s,1]^{\lambda}$ converges $\mathbb{P}$-a.s. to a
finite limit.

###### Proof.

Fix $t\in(0,1)$. Put $\lambda_{t}=(w(1)+w(t))/2w(1)$. Take
$\lambda\in(\lambda_{t},1]$ and $s\in(t,1)$ such that $w(s)>w(t)/\lambda_{t}$;
such an $s$ exists as $w([0,1])$ is connected and
$w(t)<w(t)/\lambda_{t}<w(1)$. Then, by (5.1),

$\displaystyle\mathbb{E}\biggl{[}\frac{N_{n+1}[0,t]}{N_{n+1}(s,1]^{\lambda}}-\frac{N_{n}[0,t]}{N_{n}(s,1]^{\lambda}}\Bigr{|}\mathcal{F}_{n}\biggr{]}$
$\displaystyle=\frac{N_{n}[0,t]}{N_{n}(s,1]^{\lambda}}\mathbb{E}\biggl{[}\frac{N_{n+1}[0,t]}{N_{n}[0,t]}\Bigl{(}1-\frac{W_{n+1}1\\!\\!1_{\\{X_{n+1}>s\\}}}{N_{n+1}(s,1]}\Bigr{)}^{\lambda}-1\Bigr{|}\mathcal{F}_{n}\biggr{]}$
$\displaystyle\leq\frac{N_{n}[0,t]}{N_{n}(s,1]^{\lambda}N_{n}}\Bigl{\\{}w(t)-\lambda
w(s)\frac{N_{n}(s,1]}{N_{n}(s,1]+\beta}\Bigr{\\}},$

where we have used that $(1+x)^{\lambda}\leq 1+\lambda x$ for $-1\leq x\leq
0$. Let us define, for $n,m\geq 0$,

$R_{n}=w(t)-\lambda
w(s)\frac{N_{n}(s,1]}{N_{n}(s,1]+\beta}\qquad\mbox{and}\qquad
T_{m}=\inf\\{n\geq m:R_{n}\geq 0\\}.$

Each $T_{m}$ is a $\mathcal{F}$-stopping time and $(N_{n\wedge
T_{m}}[0,t]\bigr{/}N_{n\wedge T_{m}}(s,1]^{\lambda})_{n\geq m}$ a non-negative
supermatingale, so it converges to a random limit $L_{m}<\infty$, say, on
$\Omega_{m}:\mathbb{P}(\Omega_{m})=1$. Put
$\Omega^{*}=\\{\liminf_{n}R_{n}<0\\}\cap\bigcup_{m}\Omega_{m}$. By Lemma 5.2,
$\liminf_{n}R_{n}<0$ a.s.$[\mathbb{P}]$ and $\mathbb{P}(\Omega^{*})=1$. Take
$\omega\in\Omega^{*}$. There exists $m_{0}\in\mathbb{N}$ such that
$R_{n}(\omega)<0$, $n\geq m_{0}$. Then $T_{m_{0}}(\omega)=\infty$, so
$N_{n}[0,t](\omega)\bigr{/}N_{n}(s,1](\omega)^{\lambda}\rightarrow
L_{m_{0}}(\omega)$. Therefore,
$\lim_{n\rightarrow\infty}N_{n}[0,t]\bigr{/}N_{n}(s,1]^{\lambda}<\infty$
a.s.$[\mathbb{P}]$. ∎

######  Lemma 5.4.

Under the conditions in Proposition 3.2, if $\nu(t,1)>0$ for every
$t\in(0,1)$, then
$N_{n}(\\{1\\}^{c})\bigr{/}M_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}w(1).$

###### Proof.

Fix $t\in(0,1)$. It follows from $\nu(t,1)>0$ and a slightly modified version
of Lemma 5.3 that
$N_{n}[0,t)\bigl{/}N_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}0$. Then

$\liminf_{n\rightarrow\infty}\frac{1}{P_{n}(\\{1\\}^{c})}\mathbb{E}[W_{n+1}\delta_{X_{n+1}}(\\{1\\}^{c})|\mathcal{F}_{n}]\geq\liminf_{n\rightarrow\infty}w(t)\frac{P_{n}[t,1)}{P_{n}(\\{1\\}^{c})}=w(t)\qquad\mbox{a.s.}[\mathbb{P}].$

Since
$\limsup_{n\rightarrow\infty}\mathbb{E}[W_{n+1}\delta_{X_{n+1}}(\\{1\\}^{c})|\mathcal{F}_{n}]\bigr{/}P_{n}(\\{1\\}^{c})\leq
w(1)$ a.s.$[\mathbb{P}]$, letting $t\uparrow 1$, we get

$\frac{1}{P_{n}(\\{1\\}^{c})}\mathbb{E}[W_{n+1}\delta_{X_{n+1}}(\\{1\\}^{c})|\mathcal{F}_{n}]\overset{a.s.}{\longrightarrow}w(1).$
(5.6)

Define $T_{0}=0$ and $T_{n}=\inf\\{m>T_{n-1}:X_{m}\neq 1\\}$, $n\geq 1$. It
follows from $\mathbb{P}(X_{n}\neq 1\textnormal{ i.o.})=1$ that each $T_{n}$
is a $\mathcal{F}$-stopping time such that $T_{n}<\infty$ a.s.$[\mathbb{P}]$
and $T_{n}\overset{a.s.}{\longrightarrow}\infty$. We further let, for $n\geq
1$,

$X^{*}_{n}=X_{T_{n}},\quad W^{*}_{n}=W_{T_{n}},\quad
P^{*}_{n}(\cdot)=\frac{P_{T_{n+1}-1}(\cdot\cap\\{1\\}^{c})}{P_{T_{n+1}-1}(\\{1\\}^{c})},\quad\mathcal{F}^{*}_{n}=\sigma(X^{*}_{1},W^{*}_{1},\ldots,X^{*}_{n},W^{*}_{n}),$

which are well-defined on the probability space
$(\Omega^{*},\mathcal{H}^{*},\mathbb{P}^{*})\equiv(\Omega^{*},\mathcal{H}\cap\Omega^{*},\mathbb{P}(\cdot\,|\Omega^{*}))$,
where $\Omega^{*}\equiv\bigcap_{n=0}^{\infty}\\{T_{n}<\infty\\}$ with
$\mathbb{P}(\Omega^{*})=1$ from before. Fix $n\in\mathbb{N}$. Take
$B\in\mathcal{B}[0,1]$ and $A\in\mathcal{F}^{*}_{n}$. Then

$P^{*}_{n}(B)=\frac{\theta\nu(B\cap\\{1\\}^{c})+\sum_{i=1}^{T_{n+1}-1}W_{i}\delta_{X_{i}}(B\cap\\{1\\}^{c})}{\theta\nu(B\cap\\{1\\}^{c})+\sum_{i=1}^{T_{n+1}-1}W_{i}\delta_{X_{i}}(\\{1\\}^{c})}=\frac{\theta\nu(B\cap\\{1\\}^{c})+\sum_{k=1}^{n}W^{*}_{k}\delta_{X^{*}_{k}}(B)}{\theta\nu(\\{1\\}^{c})+\sum_{k=1}^{n}W^{*}_{k}};$

thus, $P^{*}_{n}(B)$ is $\mathcal{F}^{*}_{n}$-measurable. Since
$1\\!\\!1_{A}=f(X^{*}_{1},W^{*}_{1},\ldots,X^{*}_{n},W^{*}_{n})$ for some
measurable $f$, we can show that

$\displaystyle\mathbb{P}^{*}\bigl{(}A\cap\\{X^{*}_{n+1}\in$ $\displaystyle
B\\}\bigr{)}=\mathbb{E}\bigl{[}f(X_{T_{1}},W_{T_{1}},\ldots,X_{T_{n}},W_{T_{n}})1\\!\\!1_{\\{X_{T_{n+1}}\in
B\\}\cap\Omega^{*}}\bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}f(X_{T_{1}},W_{T_{1}},\ldots,X_{T_{n}},W_{T_{n}})\frac{P_{T_{n+1}-1}(B\cap\\{1\\}^{c})}{P_{T_{n+1}-1}(\\{1\\}^{c})}1\\!\\!1_{\Omega^{*}}\Bigr{]}=\mathbb{E}^{*}[1\\!\\!1_{A}\cdot
P^{*}_{n}(B)].$

Therefore, $P^{*}_{n}$ is a version of the conditional distribution of
$X^{*}_{n+1}$ given $\mathcal{F}^{*}_{n}$. Then, by (5.1) and (5.6),

$\mathbb{P}^{*}(\mathbb{E}^{*}[w(X^{*}_{n+1})|\mathcal{F}^{*}_{n}]\rightarrow
w(1))=1.$

Moreover, $\sum_{n=1}^{\infty}\mathbb{E}^{*}[(W^{*}_{n})^{2}]/n^{2}<\infty$,
so $\mathbb{P}^{*}(\frac{1}{n}\sum_{k=1}^{n}W^{*}_{k}\rightarrow w(1))=1$ by
[5, Lemma 2]. As a result,

$\frac{N_{n}(\\{1\\}^{c})}{M_{n}(\\{1\\}^{c})}=\frac{\theta\nu(\\{1\\}^{c})}{M_{n}(\\{1\\}^{c})}+\frac{\sum_{k=1}^{M_{n}(\\{1\\}^{c})-1}W_{T_{k}}}{M_{n}(\\{1\\}^{c})}\overset{a.s.}{\longrightarrow}w(1).$

∎

######  Proof of Proposition 3.2.

_Part I._ ($P_{n}\overset{w}{\longrightarrow}\delta_{1}$,
$\hat{P}_{n}\overset{w}{\longrightarrow}\delta_{1}$,
$\frac{N_{n}}{n}\rightarrow w(1)$, a.s.[$\mathbb{P}$]). Let $t\in(0,1)$. Take
$s\in(t,1)$ and $\lambda\in(0,1)$ as in the statement of Lemma 5.3. Then
Lemmas 5.2 and 5.3 imply that

$\limsup_{n\rightarrow\infty}P_{n}[0,t]\leq\limsup_{n\rightarrow\infty}\frac{N_{n}[0,t]}{N_{n}(t,1]}\leq\limsup_{n\rightarrow\infty}\frac{1}{N_{n}(s,1]^{1-\lambda}}\frac{N_{n}[0,t]}{N_{n}(s,1]^{\lambda}}=0\quad\textnormal{a.s.}[\mathbb{P}];$

thus, $P_{n}(t,1]\overset{a.s.}{\longrightarrow}1$ for every $t\in(0,1)$, and
so $P_{n}\overset{w}{\longrightarrow}\delta_{1}$ a.s.$[\mathbb{P}]$. By [5,
Lemma 2], $\hat{P}_{n}(t,1]\overset{a.s.}{\longrightarrow}1$ for every
$t\in(0,1)$ and, similarly,
$\hat{P}_{n}\overset{w}{\longrightarrow}\delta_{1}$ a.s.$[\mathbb{P}]$.

Finally, (5.1) and the a.s. weak convergence of $P_{n}$ to $\delta_{1}$ imply
that $\mathbb{E}[W_{n+1}|\mathcal{F}_{n}]\overset{a.s.}{\longrightarrow}w(1)$.
As
$\sum_{n=1}^{\infty}\mathbb{E}[W_{n}^{2}]/n^{2}\leq\sum_{n=1}^{\infty}\beta^{2}/n^{2}<\infty$,
then, by [5, Lemma 2],

$\frac{N_{n}}{n}\overset{a.s.}{\longrightarrow}w(1).$ (5.7)

_Part II._ ($P_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}\eta$). Suppose
$0<\nu(\\{1\\})<1$; else, if $\nu(\\{1\\})=0$ (or $\nu(\\{1\\})=1$), then
$P_{n}(\\{1\\}^{c})=1$ ($P_{n}(\\{1\\}^{c})=0$), and so
$P_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}\eta$ with $\eta=1$
($\eta=0$). In that case, $P_{n}(\\{1\\}^{c})>0$ for every $n\geq 0$, so
$M_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}\infty$ by [12, Theorem 1],
and hence $\mathbb{P}(X_{n}\neq 1\textnormal{ i.o.})=1$.

_Case 1: $\nu(t,1)=0$ for some $t\in(0,1)$_. It follows that
$P_{n}(\\{1\\}^{c})=P_{n}[0,t]$, so Portmanteau theorem implies
$\limsup_{n\rightarrow\infty}P_{n}[0,t]\leq\delta_{1}([0,t])=0$, and thus
$P_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}0$.

_Case 2: $\nu(t,1)>0$ for each $t\in(0,1)$_. By Lemma 5.4,
$N_{n}(\\{1\\}^{c})\bigl{/}M_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}w(1)$
and, similarly,
$N_{n}(\\{1\\})\bigl{/}M_{n}(\\{1\\})\overset{a.s.}{\longrightarrow}w(1)$. Let
$\psi\in(0,1)$. Then

$\displaystyle\mathbb{E}\Bigl{[}\frac{M_{n+1}(\\{1\\})^{\psi}}{M_{n+1}(\\{1\\}^{c})}-\frac{M_{n}(\\{1\\})^{\psi}}{M_{n}(\\{1\\}^{c})}\Bigr{|}\mathcal{F}_{n}\Bigr{]}$
$\displaystyle=\frac{M_{n}(\\{1\\})^{\psi}}{M_{n}(\\{1\\}^{c})}\mathbb{E}\biggl{[}\Bigl{(}1+\frac{\delta_{X_{n+1}}(\\{1\\})}{M_{n}(\\{1\\})}\Bigr{)}^{\psi}\frac{M_{n}(\\{1\\}^{c})}{M_{n+1}(\\{1\\}^{c})}-1\Bigr{|}\mathcal{F}_{n}\biggr{]}$
$\displaystyle\leq\frac{M_{n}(\\{1\\})^{\psi}}{M_{n}(\\{1\\}^{c})N_{n}}\Bigl{\\{}\psi\frac{N_{n}(\\{1\\})}{M_{n}(\\{1\\})}-\frac{N_{n}(\\{1\\}^{c})}{M_{n}(\\{1\\}^{c})+1}\Bigr{\\}},$

using that $(1+x)^{\psi}\leq 1+\psi x$ for $0\leq x\leq 1$. Since

$\limsup_{n\rightarrow\infty}\Bigl{(}\psi\frac{N_{n}(\\{1\\})}{M_{n}(\\{1\\})}-\frac{N_{n}(\\{1\\}^{c})}{M_{n}(\\{1\\}^{c})+1}\Bigr{)}<0\qquad\mbox{a.s.}[\mathbb{P}],$

we can show, arguing as in the proof of Lemma 5.3, that
$(M_{n}(\\{1\\})^{\psi}\bigr{/}M_{n}(\\{1\\}^{c}))_{n\geq 1}$ converges
a.s.$[\mathbb{P}]$. But $\psi$ is arbitrary, so it holds that

$\frac{M_{n}(\\{1\\})^{\psi}}{M_{n}(\\{1\\}^{c})}=M_{n}(\\{1\\})^{\frac{\psi-1}{2}}\frac{M_{n}(\\{1\\})^{\frac{1+\psi}{2}}}{M_{n}(\\{1\\}^{c})}\overset{a.s.}{\longrightarrow}0.$

Analogously,
$M_{n}(\\{1\\}^{c})^{\psi}\bigl{/}M_{n}(\\{1\\})\overset{a.s.}{\longrightarrow}0$.
Therefore, for every $\psi<1$,

$\frac{n^{\psi}}{N_{n}(\\{1\\})}=\Bigl{(}\frac{M_{n}(\\{1\\}^{c})+M_{n}(\\{1\\})-2}{M_{n}(\\{1\\})^{1/\psi}}\Bigr{)}^{\psi}\frac{M_{n}(\\{1\\})}{N_{n}(\\{1\\})}\overset{a.s.}{\longrightarrow}0\quad\mbox{and}\quad\frac{n^{\psi}}{N_{n}(\\{1\\}^{c})}\overset{a.s.}{\longrightarrow}0.$
(5.8)

Define $Z_{n}=\log\frac{N_{n}(\\{1\\}^{c})}{N_{n}(\\{1\\})}$, $n\geq 0$. By
[23, Lemma 3.2], $(Z_{n})_{n\geq 0}$ converges $\mathbb{P}$-a.s. in
$[-\infty,\infty)$ if

$\sum_{n=0}^{\infty}\mathbb{E}[Z_{n+1}-Z_{n}|\mathcal{F}_{n}]<\infty\;\;\textnormal{a.s.}[\mathbb{P}]\quad\textnormal{and}\quad\sum_{n=0}^{\infty}\mathbb{E}\bigl{[}(Z_{n+1}-Z_{n})^{2}\bigr{|}\mathcal{F}_{n}\bigr{]}<\infty\;\;\textnormal{a.s.}[\mathbb{P}].$
(5.9)

In that case, $(N_{n}(\\{1\\}^{c})\bigr{/}N_{n}(\\{1\\}))_{n\geq 0}$ converges
$\mathbb{P}$-a.s. in $[0,\infty)$ and there exists a random variable
$\eta\in[0,1)$ such that

$P_{n}(\\{1\\}^{c})=\frac{N_{n}(\\{1\\}^{c})}{N_{n}(\\{1\\}^{c})+N_{n}(\\{1\\})}\overset{a.s.}{\longrightarrow}\eta.$

To prove (5.9), observe that

$\displaystyle\mathbb{E}[Z_{n+1}-Z_{n}|\mathcal{F}_{n}]$
$\displaystyle=\mathbb{E}\biggl{[}\log\frac{N_{n+1}(\\{1\\}^{c})}{N_{n}(\\{1\\}^{c})}\delta_{X_{n+1}}(\\{1\\}^{c})-\log\frac{N_{n+1}(\\{1\\})}{N_{n}(\\{1\\})}\delta_{X_{n+1}}(\\{1\\})\Bigr{|}\mathcal{F}_{n}\biggr{]}$
$\displaystyle=\mathbb{E}\biggl{[}\int_{0}^{W_{n+1}}\frac{\delta_{X_{n+1}}(\\{1\\}^{c})}{N_{n}(\\{1\\}^{c})+t}dt-\int_{0}^{W_{n+1}}\frac{\delta_{X_{n+1}}(\\{1\\})}{N_{n}(\\{1\\})+t}dt\Bigr{|}\mathcal{F}_{n}\biggr{]}$
$\displaystyle\begin{aligned}
\leq\mathbb{E}\biggl{[}\delta_{X_{n+1}}(\\{1\\}^{c})\Bigl{(}\frac{W_{n+1}}{N_{n}(\\{1\\}^{c})}&-\frac{W_{n+1}^{2}}{2N_{n}(\\{1\\}^{c})^{2}}+k_{1}\frac{W_{n+1}^{3}}{3N_{n}(\\{1\\}^{c})^{3}}\Bigr{)}\Bigr{|}\mathcal{F}_{n}\biggr{]}-\\\
&-\mathbb{E}\biggl{[}\delta_{X_{n+1}}(\\{1\\})\Bigl{(}\frac{W_{n+1}}{N_{n}(\\{1\\})}-k_{2}\frac{W_{n+1}^{2}}{2N_{n}(\\{1\\})^{2}}\Bigr{)}\Bigr{|}\mathcal{F}_{n}\biggr{]}\end{aligned}$
$\displaystyle\leq\frac{1}{N_{n}}\Bigl{(}k_{1}\frac{\beta^{3}}{3N_{n}(\\{1\\}^{c})^{2}}+k_{2}\frac{\beta^{2}}{2N_{n}(\\{1\\})}\Bigr{)},$

where we have used Taylor series approximation of the function $f(x)=1/(x+t)$
with some constants $k_{1},k_{2}\geq 0$. By (5.7) and (5.8),
$\sum_{n=0}^{\infty}\mathbb{E}[Z_{n+1}-Z_{n}|\mathcal{F}_{n}]<\infty$
a.s.$[\mathbb{P}]$. On the other hand,

$\displaystyle\mathbb{E}\bigl{[}(Z_{n+1}-Z_{n})^{2}\bigr{|}\mathcal{F}_{n}\bigr{]}$
$\displaystyle\leq\mathbb{E}\biggl{[}\frac{\delta_{X_{n+1}}(\\{1\\}^{c})}{N_{n}(\\{1\\}^{c})^{2}}\Bigl{(}\int_{0}^{W_{n+1}}dt\Bigr{)}^{2}+\frac{\delta_{X_{n+1}}(\\{1\\})}{N_{n}(\\{1\\})^{2}}\Bigl{(}\int_{0}^{W_{n+1}}dt\Bigr{)}^{2}\Bigr{|}\mathcal{F}_{n}\biggr{]}$
$\displaystyle\leq
P_{n}(\\{1\\}^{c})\frac{\beta^{2}}{N_{n}(\\{1\\}^{c})^{2}}+P_{n}(\\{1\\})\frac{\beta^{2}}{N_{n}(\\{1\\})^{2}},$

and thus
$\sum_{n=0}^{\infty}\mathbb{E}[(Z_{n+1}-Z_{n})^{2}|\mathcal{F}_{n}]<\infty$
a.s.$[\mathbb{P}]$. ∎

######  Proof of Theorem 3.3.

Let us define $g(x)=w(x)/\bar{w}$, $x\in\mathbb{X}$ and, for $n\geq 1$ and
$A\in\mathcal{B}[0,1]$,

$\tilde{X}_{n}=g(X_{n}),\qquad\tilde{W}_{n}=W_{n},\qquad\tilde{P}_{n}(A)=P_{n}(g^{-1}(A)),\qquad\tilde{\nu}(A)=\nu(g^{-1}(A)).$

Then $\mathbb{E}[\tilde{W}_{n}|\tilde{X}_{n}]=\bar{w}\tilde{X}_{n}$. In
addition, we have from (2.1) that, for every $t\in(0,1)$,

$\tilde{\nu}(t,1]=\nu(\\{x\in\mathbb{X}:w(x)>t\bar{w}\\})=\nu_{w}(t\bar{w},\bar{w}]>0;$

thus, $1\in\mbox{supp}(\tilde{\nu})$. On the other hand,

$\tilde{P}_{n}(A)=\sum_{i=1}^{n}\frac{\tilde{W}_{i}}{\theta+\sum_{j=1}^{n}\tilde{W}_{j}}\delta_{\tilde{X}_{i}}(A)+\frac{\theta}{\theta+\sum_{j=1}^{n}\tilde{W}_{j}}\tilde{\nu}(A).$

Therefore, $(\tilde{X}_{n})_{n\geq 1}$ is111Strictly speaking,
$(\tilde{X}_{n})_{n\geq 1}$ differs from the definition of a DPS in that
$\tilde{W}_{n}$ is a function of $(X_{n},U_{n})$ but not of
$(\tilde{X_{n}},U_{n})$, and $\tilde{P}_{n}$ is the conditional distribution
of $\tilde{X}_{n+1}$ given $\mathcal{F}_{n}$ instead of
$\sigma(\tilde{X}_{1},U_{1},\ldots,\tilde{X_{n}},U_{n})$. Nevertheless, the
conclusions of Proposition 3.2 continue to hold for the process
$(\tilde{X}_{n})_{n\geq 1}$ since
$\mathbb{E}[\tilde{W}_{n}|\mathcal{F}_{n-1}\vee\sigma(X_{n})]=\tilde{w}(\tilde{X}_{n})$.
a $[0,1]$-valued DPS with the continuous and strictly increasing expected
weight function $\tilde{w}:[0,1]\rightarrow[0,\bar{w}]$, given by
$\tilde{w}(t)=\bar{w}t$, $t\in[0,1]$. Then Proposition 3.2 implies

$\frac{1}{n}\sum_{i=1}^{n}W_{i}=\frac{1}{n}\sum_{i=1}^{n}\tilde{W}_{i}\overset{a.s.}{\longrightarrow}\tilde{w}(1)=\bar{w}.$

_Part I._ ($P_{n}(\mathcal{D}_{\delta}^{c})\overset{a.s.}{\longrightarrow}0$).
Fix $\delta>0$. Let
$\epsilon=(\bar{w}-\sup_{x\in\mathcal{D}^{c}_{\delta}}w(x))/2$. Then
$\epsilon>0$ by (2.1), so Proposition 3.2 implies

$P_{n}(\mathcal{D}_{\delta}^{c})\leq\mathbb{P}\bigl{(}w(X_{n+1})<\bar{w}-\epsilon\,|\mathcal{F}_{n}\bigr{)}=\tilde{P}_{n}[0,1-\epsilon/\bar{w})\overset{a.s.}{\longrightarrow}0.$

_Part II._ ($P_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\eta$). By
Proposition 3.2, there exists a random variable $\eta$ with $\eta\in[0,1]$
such that

$P_{n}(\mathcal{D}^{c})=\tilde{P}_{n}(\\{1\\}^{c})\overset{a.s.}{\longrightarrow}\eta.$

Suppose $\bar{w}>\bar{w}^{c}$. Then there exists $t\in(0,1)$ such that
$\nu_{w}(t\bar{w},\bar{w})=0$. But
$\tilde{\nu}(t,1)=\nu_{w}(t\bar{w},\bar{w})=0$, so
$P_{n}(\mathcal{D}^{c})=\tilde{P}_{n}[0,t]\overset{a.s.}{\longrightarrow}0$.

_Part III._ ($\|P_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0$,
$\|\hat{P}_{n}-\tilde{P}\|\overset{a.s.}{\longrightarrow}0$). Suppose
$\bar{w}>\bar{w}^{c}$. It follows from Part II that
$P_{n}(\mathcal{D})\overset{a.s.}{\longrightarrow}1$ and, by [5, Lemma 2],
$\hat{P}_{n}(\mathcal{D})\overset{a.s.}{\longrightarrow}1$. Then [12, Theorem
1] implies $\mathbb{P}(X_{n}\in\mathcal{D}\textnormal{ i.o.})=1$.

Let us define $T=0$ and, for $n\geq 1$,

$T_{n}=\inf\\{m\in\mathbb{N}:m>T_{n-1},X_{m}\in\mathcal{D}\\}\qquad\textnormal{and}\qquad
P_{n}(\cdot\,|\mathcal{D})=P_{n}(\cdot\cap\mathcal{D})\bigr{/}P_{n}(\mathcal{D}).$

Then $T_{n}$ is a $\mathcal{F}$-stopping time such that $T_{n}<\infty$
a.s.$[\mathbb{P}]$ and $T_{n}\overset{a.s.}{\longrightarrow}\infty$ from
$\mathbb{P}(X_{n}\in\mathcal{D}\textnormal{ i.o.})=1$. Put
$\Omega^{*}=\bigcap_{n=0}^{\infty}\\{T_{n}<\infty\\}$. Arguing as in Lemma
5.4, one can show that $P_{n}^{*}(\cdot)\equiv
P_{T_{n}-1}(\cdot\,|\mathcal{D})$ is the conditional distribution of
$X_{T_{n}}$ given $\sigma(X_{T_{1}},W_{T_{1}},\ldots,X_{T_{n-1}},W_{T_{n-1}})$
on the probability space
$(\Omega^{*},\mathcal{H}^{*}\mathbb{P}^{*})\equiv(\Omega^{*},\mathcal{H}\cap\Omega^{*},\mathbb{P}(\cdot\,|\Omega^{*}))$.
Moreover, $(X_{T_{n}})_{n\geq 1}$ is a DPS such that
$\mathbb{E}^{*}[W_{T_{n}}|X_{T_{n}}]=\bar{w}$, $n\geq 1$, so there exists by
Proposition 3.1 a random probability measure $\tilde{P}^{*}$ on $\mathcal{D}$
such that

$\mathbb{P}^{*}(\|P_{n}^{*}-\tilde{P}^{*}\|\rightarrow 0)=1.$

Define $\tilde{P}(A)(\omega)=\tilde{P}^{*}(A\cap\mathcal{D})(\omega)$,
$\omega\in\Omega^{*}$ and $\tilde{P}(A)(\omega)=0$, $\omega\notin\Omega^{*}$,
$A\in\mathcal{X}$. As $\mathbb{P}(\Omega^{*})=1$, then $\tilde{P}$ is a random
probability measure on $\mathbb{X}$ such that $\tilde{P}(\mathcal{D})=1$
a.s.$[\mathbb{P}]$. On the other hand, $P_{n}(\cdot\,|\mathcal{D})\equiv
P_{m_{n}}^{*}(\cdot)$ on $\Omega^{*}$, with
$m_{n}=\sum_{j=1}^{n}1\\!\\!1_{\\{T_{j}\leq n\\}}$, $n\geq 1$, so

$\mathbb{P}(\|P_{n}(\cdot\,|\mathcal{D})-\tilde{P}(\cdot)\|\rightarrow
1)=\mathbb{P}^{*}(\|P_{m_{n}}^{*}-\tilde{P}^{*}\|\rightarrow 1)=1.$

Therefore,

$\|P_{n}-\tilde{P}\|\leq
2\,P_{n}(\mathcal{D}^{c})+\|P_{n}(\cdot\,|\mathcal{D})-\tilde{P}(\cdot)\|\overset{a.s.}{\longrightarrow}0.$

Finally, let
$\hat{P}_{n}(\cdot\,|\mathcal{D})=\hat{P}_{n}(\cdot\cap\mathcal{D})\bigr{/}\hat{P}_{n}(\mathcal{D})$,
$n\geq 0$, which is well-defined for large $n$. Then
$\hat{P}_{n}(\cdot\,|\mathcal{D})$ denotes the relative frequency of
$(X_{n})_{n\geq 1}$ conditional on being in $\mathcal{D}$, so, arguing as
before, we have

$\|\hat{P}_{n}-\tilde{P}\|\leq
2\,\hat{P}_{n}(\mathcal{D}^{c})+\|\hat{P}_{n}(\cdot\,|\mathcal{D})-\tilde{P}(\cdot)\|\overset{a.s.}{\longrightarrow}0.$

∎

######  Proof of Proposition 3.4.

By Theorem 3.3, $N_{n}/n\overset{a.s.}{\longrightarrow}\bar{w}$. Define
$\theta_{n}=\theta/N_{n}$ and $Z_{n}=L_{n}-L_{n-1}$, $n\geq 1$, with $L_{0}=0$
and $\theta_{0}=1$. As $\nu$ is diffuse and $Z_{n}\in\\{0,1\\}$, then
$L_{n}=\sum_{i=1}^{n}1\\!\\!1_{\\{Z_{i}=1\\}}$ and
$\theta_{n}=\mathbb{P}(Z_{n+1}=1|\mathcal{F}_{n})$. Moreover,
$\theta_{n}\geq\theta/(\theta+\beta n)$, so
$\sum_{n=1}^{\infty}\mathbb{P}(Z_{n}=1|\mathcal{F}_{n-1})=\infty$
a.s.$[\mathbb{P}]$ and, by [12, Theorem 1],

$\frac{L_{n}}{\sum_{k=1}^{n}\theta_{k-1}}=\frac{\sum_{i=1}^{n}1\\!\\!1_{\\{Z_{i}=1\\}}}{\sum_{k=1}^{n}\mathbb{P}(Z_{k}=1|\mathcal{F}_{k-1})}\overset{a.s.}{\longrightarrow}1.$

On the other hand, $\frac{1}{\log n}\sum_{k=1}^{n}k^{-1}\rightarrow 1$, so
$\frac{1}{\log n}\sum_{k=1}^{n}k^{-1}a_{k}\rightarrow a$ for any
$(a_{n})_{n\geq 1}\subseteq\mathbb{R}_{+}$ such that $a_{n}\rightarrow a$, and
thus

$\frac{L_{n}}{\log
n}=\frac{L_{n}}{\sum_{k=1}^{n}\theta_{k-1}}\biggl{(}\frac{1}{\log
n}+\frac{\theta}{\log
n}\sum_{k=1}^{n-1}\frac{1}{k}\Bigl{(}\frac{k}{N_{k}}\Bigr{)}\biggr{)}\overset{a.s.}{\longrightarrow}\frac{\theta}{\bar{w}}.$

∎

######  Proof of Proposition 4.1.

If $\nu(\mathcal{D}^{c})=0$, the result is immediate for $\xi=0$. Suppose
$\nu(\mathcal{D}^{c})>0$. Then
$\bar{w}^{c}\in\mbox{supp}((\nu_{|\mathcal{D}^{c}})_{w})$. Moreover,
$\nu(\\{x\in\mathcal{D}^{c}:w(x)>\bar{w}^{c}\\})=0$; otherwise,
$(\bar{w}^{c},\bar{w})\cap\mbox{supp}((\nu_{|\mathcal{D}^{c}})_{w})\neq\emptyset$
and there exists $u>\bar{w}^{c}$ such that
$u\in\mbox{supp}((\nu_{|\mathcal{D}^{c}})_{w})$, absurd. As a result,
$\bar{w}^{c}$ plays the same role for the subsequence of non-dominant
observations as $\bar{w}$ does for $(X_{n})_{n\geq 1}$; thus, by Theorem 3.3,
$N_{n}(\mathcal{D}^{c})\bigl{/}M_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\bar{w}^{c}$.
Then, proceeding as in Part II of the proof of Proposition 3.2, we can show
that, for every $\psi<\bar{w}^{c}/\bar{w}$,

$\frac{M_{n}(\mathcal{D})^{\psi}}{M_{n}(\mathcal{D}^{c})}\overset{a.s.}{\longrightarrow}0\qquad\mbox{and}\qquad\frac{M_{n}(\mathcal{D}^{c})^{\psi}}{M_{n}(\mathcal{D})}\overset{a.s.}{\longrightarrow}0,$

which implies
$M_{n}(\mathcal{D}^{c})/n^{\psi}\overset{a.s.}{\longrightarrow}\infty$ for
$\psi<\bar{w}^{c}/\bar{w}$ and
$M_{n}(\mathcal{D}^{c})/n^{\psi}\overset{a.s.}{\longrightarrow}0$ for
$\psi>\bar{w}^{c}/\bar{w}$. In turn,
$\log\frac{N_{n}(\mathcal{D}^{c})}{N_{n}(\mathcal{D})^{\bar{w}^{c}/\bar{w}}}$
converges $\mathbb{P}$-a.s. in $[-\infty,\infty)$, so there exists a random
variable $\xi\in[0,\infty)$ such that

$n^{\gamma}\cdot\hat{P}_{n}(\mathcal{D}^{c})=\frac{M_{n}(\mathcal{D}^{c})}{N_{n}(\mathcal{D}^{c})}\Bigl{(}\frac{N_{n}}{n}\frac{N_{n}(\mathcal{D}^{c})^{\bar{w}/\bar{w}^{c}}}{N_{n}(\mathcal{D})+N_{n}(\mathcal{D}^{c})}\Bigr{)}^{\bar{w}^{c}/\bar{w}}\overset{a.s.}{\longrightarrow}\xi\quad\mbox{and}\quad
n^{\gamma}\cdot
P_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}\frac{\bar{w}^{c}}{\bar{w}}\xi.$

∎

######  Proof of Theorem 4.2.

By Theorem 3.3, $P_{n}(\mathcal{D})\overset{a.s.}{\longrightarrow}1$, so we
can proceed as in Theorem 3.3 to derive results for the subsequence of
dominant observations. Let us define $T_{0}=0$ and, for $n\geq 1$

$T_{n}=\inf\\{n>T_{n-1}:X_{n}\in\mathcal{D}\\},\qquad
P_{n}(\cdot\,|\mathcal{D})=\frac{P_{n}(\cdot\cap\mathcal{D})}{P_{n}(\mathcal{D})},\qquad\hat{P}_{n}(\cdot\,|\mathcal{D})=\frac{\hat{P}_{n}(\cdot\cap\mathcal{D})}{\hat{P}_{n}(\mathcal{D})}.$

Put $m_{n}=M_{n}(\mathcal{D})-1$ and
$\mathcal{G}_{n}=\sigma(X_{T_{1}},U_{T_{1}},\ldots,X_{T_{m_{n}}},U_{T_{m_{n}}})$,
$n\geq 1$. With a slight abuse of notation, $(X_{T_{m_{n}}})_{n\geq 1}$ forms
a subsequence of dominant observations,

$P_{n}(\cdot\,|\mathcal{D})=\frac{\theta\nu(\cdot\cap\mathcal{D})+\sum_{k=1}^{m_{n}}W_{T_{k}}\delta_{X_{T_{k}}}(\cdot)}{\theta\nu(\mathcal{D})+\sum_{k=1}^{m_{n}}W_{T_{k}}}$
(5.10)

is the conditional distribution of $X_{T_{m_{n+1}}}$ given $\mathcal{G}_{n}$
(see the proof of Theorem 3.3), and $\hat{P}_{n}(\cdot\,|\mathcal{D})$ the
relative frequency, which is well-defined for large $n$. In addition,

$P_{n}(A|\mathcal{D})\overset{a.s.}{\longrightarrow}\tilde{P}(A)\qquad\mbox{and}\qquad\hat{P}_{n}(A|\mathcal{D})\overset{a.s.}{\longrightarrow}\tilde{P}(A).$

Observe that $P_{n}(\cdot\,|\mathcal{D})$ and
$\hat{P}_{n}(\cdot\,|\mathcal{D})$ remain unchanged unless $m_{n}>m_{n-1}$.
Then we can prove that

$\sqrt{m_{n}}\bigl{(}\hat{P}_{n}(A|\mathcal{D})-P_{n}(A|\mathcal{D})\bigr{)}\qquad\mbox{and}\qquad\sqrt{m_{n}}\bigl{(}P_{n}(A|\mathcal{D})-\tilde{P}(A)\bigr{)}$
(5.11)

converge to Gaussian limits by applying [5, Theorem 1] and [5, Proposition 1],
respectively. In fact, one can easily show that (5.11) satisfy the assumptions
of [5, Theorem 1, Proposition 1] using similar techniques to the ones in [4,
Theorem 4] and [5, Corollary 3]. As a result,

$\displaystyle\sqrt{m_{n}}\bigl{(}\hat{P}_{n}(A|\mathcal{D})-P_{n}(A|\mathcal{D})\bigr{)}\overset{stably}{\longrightarrow}\mathcal{N}\bigl{(}0,U(A)\bigr{)},$
$\displaystyle\sqrt{m_{n}}\bigl{(}P_{n}(A|\mathcal{D})-\tilde{P}(A)\bigr{)}\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\bigl{(}0,V(A)\bigr{)}\quad\textnormal{w.r.t.
}\mathcal{G}.$

But $M_{n}(\mathcal{D})/n\overset{a.s.}{\longrightarrow}1$, so, by a
generalized Slutsky’s theorem (see also [14, Theorem 6]),

$\displaystyle\sqrt{n}\bigl{(}\hat{P}_{n}(A|\mathcal{D})-P_{n}(A|\mathcal{D})\bigr{)}\overset{stably}{\longrightarrow}\mathcal{N}\bigl{(}0,U(A)\bigr{)},$
$\displaystyle\sqrt{n}\bigl{(}P_{n}(A|\mathcal{D})-\tilde{P}(A)\bigr{)}\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\bigl{(}0,V(A)\bigr{)}\quad\textnormal{w.r.t.
}\mathcal{G}.$

It follows from
$\hat{P}_{n}(A|\mathcal{D})\overset{a.s.}{\longrightarrow}\tilde{P}(A)$ that
the limit $\tilde{P}(A)$ is
$\sigma(X_{T_{m_{1}}},X_{T_{m_{2}}},\ldots)$-measurable. Moreover, using the
independence of $(U_{n})_{n\geq 1}$, we can show that
$(X_{T_{m_{n+1}}},X_{T_{m_{n+2}}},\ldots)$ and $\mathcal{F}_{n}$ are
conditionally independent given $\mathcal{G}_{n}$. Combining both results, we
get

$\sqrt{n}\bigl{(}P_{n}(A|\mathcal{D})-\tilde{P}(A)\bigr{)}\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\bigl{(}0,V(A)\bigr{)}\quad\textnormal{w.r.t.
}\mathcal{F}.$

On the other hand, Proposition 3.2 implies that

$\bigl{|}\sqrt{n}(P_{n}(A|\mathcal{D})-D_{n}(A)\bigr{|}=\sqrt{n}|P_{n}(A|\mathcal{D})-P_{n}(A)|\leq
2\sqrt{n}\cdot P_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}0.$

Then, using again the generalized Slutsky’s theorem,
$D_{n}(A)\overset{a.s.cond.}{\longrightarrow}\mathcal{N}\bigl{(}0,V(A)\bigr{)}$
w.r.t. $\mathcal{F}$.

Regarding the second result, we have

$\bigl{|}\sqrt{n}(\hat{P}_{n}(A|\mathcal{D})-P_{n}(A|\mathcal{D}))-C_{n}(A)\bigr{|}\leq
2\sqrt{n}\cdot
P_{n}(\mathcal{D}^{c})+2\sqrt{n}\cdot\hat{P}_{n}(\mathcal{D}^{c})\overset{a.s.}{\longrightarrow}0;$

therefore,
$C_{n}(A)\overset{stably}{\longrightarrow}\mathcal{N}\bigl{(}0,U(A)\bigr{)}$.
∎

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# Solid immersion metalens for directional single molecule emission with high
collection efficiency

Zhiheng Li  The Blackett Laboratory, Department of Physics, Imperial College
London, London SW7 2AZ, United Kingdom  Zequan Chen  The Blackett Laboratory,
Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom
Rupert F. Oulton  The Blackett Laboratory, Department of Physics, Imperial
College London, London SW7 2AZ, United Kingdom  Ming Fu  The Blackett
Laboratory, Department of Physics, Imperial College London, London SW7 2AZ,
United Kingdom  e-mail<EMAIL_ADDRESS>
###### Abstract

We present simulations of an efficient high numerical aperture solid immersion
metalens concept for fluorescence microscopy. The technique exploits the
preferential emission of interfacial dipoles into a high refractive index
substrate combined with a metalens and a conventional tube lens for imaging
them. We have thus simulated dipole emission and an all-dielectric metasurface
on opposite sides of a high refractive index substrate. Our calculations
predict dipole collection efficiencies of up to $87$ percent. The simulated
beam propagation through the imaging system shows excellent performance along
the optical axis, with aberrations accumulating with increasing field of view.
These aberrations can be controlled by using a metasurface with an optimized
non-hyperbolic phase profile. The high collection efficiency of dipole
emission suggests this compact solid immersion lens would be effective for
fluorescence imaging including single fluorescent centres for quantum optical
application.

## Introduction

Isolated fluorescent centres are widely used as sources for super-resolution
imaging [1, 2] as well as quantum optics [3], where both applications rely on
a high optical collection efficiency.[4] Since the radiation of dipole
emitters is omnidirectional, numerous elaborate techniques have been devised
to make the radiation more directional [5, 6, 7, 8, 9, 10]. However, even the
simple act of placing a dipole emitter at the interface between two dielectric
media can have a profound impact on its radiation pattern.[4, 11, 12] The
interface of air with a substrate of refractive index, $n_{S}$, can even be
viewed as a dielectric antenna, since it promotes dominant emission into the
substrate due to a higher substrate density of states, which scales with
$n_{S}^{3}$. When suitably optimised for dipole emitters orientated
perpendicular to the interface, Lee and colleagues achieved near-unity
collection efficiency.[4] However, the resulting high angle annular-shaped
emission still required a high numerical aperture (NA) solid immersion imaging
system. In this work, we show that high NA radiation patterns from interfacial
dipoles may be collected through the substrate using a metalens to collimate
the high angle annular emission patterns, and subsequently imaged using a tube
lens. The approach proves to be both efficient and effective for collecting
emission independently of dipole orientation.

## Results and Discussion

Figure 1: The schematic design of the high NA interfacial dipole solid
immersion metalens. The interfacial dipole radiation patterns are shown for a
substrate refractive index of $n_{S}=3$.

Metasurfaces are two-dimensional metamaterials composed of nanostructured
building blocks that enable a local control of transmission phase. For
example, a hyperbolic spatial phase profile metasurface defines a simple
metalens.[13, 14, 15, 16] The schematic design of the combined dielectric
antenna and metalens is presented in Figure 1. The necessary $2\pi$ phase
control over a metasurfaces is often achieved by a geometric phase associated
with circularly polarised light interacting with spatially varying anisotropic
building blocks[17, 18, 19]. This provides robust wavelength-independent phase
control, but its dependence on polarization limits its use for imaging of
incoherent light. Alternatively, metasurfaces composed of isotropic building
blocks are polarisation-independent, but suffer a wavelength-dependence, which
must be managed.[13, 20, 21] Since randomly oriented fluorescent dipoles
typically emit over a narrow spectral range, but with undefined polarization,
we employ the latter approach to fabricate our metalens.[13] In the following,
we have assumed dipole emission at a wavelength 0.6 $\mu{m}$ on a substrate
with thickness 100 $\mu{m}$ or 250 $\mu{m}$, and different refractive indices.

Figure 2: (a) The power ratios and (b) efficiencies as a function of the
refractive index for vertical and horizontal interfacial dipoles. Some
commonly used dielectric substrate materials including silicon dioxide,
silicon rich nitride, silicon and gallium phosphide are labeled.

We first discuss interfacial dipole radiation. It is well-known that a dipole
in a homogeneous medium has a doughnut-shaped radiation pattern. According to
Fermi’s golden rule, the radiative decay rate of a dipole emitter depends on
the density of states, which is proportional to the cube of the refractive
index for a homogeneous medium.[22] Therefore, placing a dipole emitter at a
dielectric interface preferentially directs emission into the high-index
substrate.[4] The problem of interfacial dipoles, solved by Engheta[23], shows
that this logic works well for dipoles oriented parallel to the interface; but
a rigorous description is required to treat vertical dipoles and to identify
the complicated radiation patterns, as shown in Figure 1. We define a dipole
oriented perpendicular to the interface as a vertical interfacial dipole,
while the parallel orientations are considered horizontal interfacial dipoles.
For the vertical interfacial dipole, the radiation into the dielectric half
space is rotationally symmetric and the radiation forms a narrow peak near the
critical angle $\theta_{C}$, where $\theta_{C}=\sin^{-1}{\left(1/n\right)}$.
Meanwhile, the horizontal dipole’s radiation pattern is not rotationally
symmetric, with peak emission at an azimuthal angle of $\phi=\pi/2$, and with
radiation dominant at angles above the critical angle. Although the
interfacial dipole radiation patterns are drastically distorted, the phase
fronts for both dipole orientations remain spherical and this enables the re-
focusing of the radiation patterns to form an image using a simple metalens.

To estimate how efficiently the substrate collects dipole radiation, we have
simulated the power emitted into both vacuum ($P_{vac}$) and substrate
($P_{sub}$) to evaluate both the power ratio, $P_{sub}/P_{vac}$ and the
efficiency, $P_{sub}/(P_{sub}+P_{vac})$ as a function of the refractive index
of the substrate for both dipole orientations. Figure 2 shows power ratios for
both dipole orientations increase with substrate index. Notably, the
horizontal dipole follows the expected $n_{s}^{3}$ power law.[22] Both the
vertical and horizontal interfacial dipoles’ efficiencies saturate rapidly, so
extremely high refractive indices are not necessary. For example, we find
vertical and horizontal dipole substrate emission efficiencies of $0.8875$ and
$0.918$ respectively when the substrate index is just $2$, a value attainable
with many transparent materials.

Although close to 90 percent of a dipole’s energy is directed into the
substrate, a large proportion of the radiation will only be collected at high
NA. The advantage of metalenses are their capability to operate at high NA in
a planar configuration.[13] To demonstrate the importance of achieving a high
NA, we have compared the amount of dipole energy collected into two NAs of
$1.44$ and $2.00$ respectively, as presented in Table 1. In Table 1 the
efficiency is defined as the power collected by the metalens divided by the
total power emitted from the dipole. The efficiencies for the slightly reduced
NA drop to almost half. Let us now consider the construction of the metalens
and its transmission efficiency.

Table 1: The efficiencies collected by the metalens with two different NAs for the vertical/horizontal interfacial dipole. The thickness of the substrate is set to be 100 $\mu{m}$ Numerical aperture | Efficiency | Efficiency
---|---|---
| Vertical dipole | Horizontal dipole
$1.44$ | $0.55$ | $0.45$
$2.00$ | $0.85$ | $0.87$

The chosen metalens design is an array of cylindrical nanopillars arranged on
a square lattice. The isotropic geometry of the nanopillars results in a
polarization-insensitive transmission. The metalens should function as a solid
immersion lens and convert the spherical wave emission from interfacial
dipoles into a collimated beam. Consequently, the spatial phase profile of the
metalens must be hyperbolic. If we assume the beam propagation direction is
$z$ and the metalens is placed in the $z=0$ plane, then the phase profile of
the solid-immersion metalens is given by:

$\phi_{target}=-\frac{2\pi}{\lambda}n\left[\sqrt{x^{2}+y^{2}+f^{2}}-f\right]$
(1)

where $\lambda$ is the wavelength, $n$ is the refractive index of the
substrate, $x$ and $y$ are the coordinates in the metalens plane, and $f$ is
the focal length of the metalens. The focal length is set equal to the
thickness of the substrate, which is 100 $\mu{m}$. Here, we have set the nano-
pillars to be TiO2 with refractive index of 2.4.[24] The substrate is set to
be AlN with refractive index of 2.15.[25] The refractive index of the pillars
should be higher than the substrate to provide the $2\pi$ phase coverage with
fabrication limited nano-pillar sizes. The height of the nano-pillars is 0.6
$\mu{m}$ and the square unit cell size is 0.35 $\mu{m}$, as shown in the inset
of Figure 3a. The sub-wavelength periodic grating ensures that higher order
diffracted waves are evanescent and almost all the energy is directed into the
zeroth order.

We now vary the nano-pillar diameters range from 0.1 $\mu{m}$ to 0.312
$\mu{m}$. Therefore, the duty cycle (ratio of nano-pillar diameter to the unit
cell size) is from $0.29$ to $0.89$. In order to investigate the effect of the
phase and transmission relations on the field of view, we performed finite
difference time domain (FDTD) simulations to characterize the phase shift and
transmission as a function of the duty cycle at normal incidence and at the
field angle of 2 degrees. As shown in Figure 3a, the phase gradient for a
larger incident angle is nearly the same as that at normal incidence.
Similarly, as demonstrated in Figure 3b there are minor differences in
transmission for the two field angles. The total transmission of the metalens
for the field angles of 0 and 2 degrees are around $78.7$% and $79.8$%
respectively. From the simulation above, we can conclude that the variation of
the phase and transmission for different field angles does not limit the field
of view beyond conventional lens aberrations. The range of the duty cycle
should provide $2\pi$ phase coverage with as high a transmission as possible.
Noticeable phase fluctuations and transmission dips are apparent in Fig. 3a
and 3b, due to nano-pillar resonances.[20, 26] These could be narrowed or even
eliminated by increasing the nanopillar height or reducing the unit cell size,
but these operations lead to a larger aspect ratios which would be more
challenging to fabricate. In order to keep an acceptable aspect ratio and
avoid resonances, we chose six different duty cycles to avoid them. The chosen
diameters are indicated with dashed line in Figure 3. Given the phase relation
at normal incidence presented in Figure 3a, we can construct the metalens by
fitting the spatial phase profile to the duty cycle.

Figure 3: (a) The phase shift and (b) transmission as a function of the
nanopillar diameter at the field angles of $0$ and $2$ degrees. The inset
shows the scheme of one unit cell, with $n_{TiO_{2}}$ = 2.4 and $n_{AlN}$ =
2.15. $d$ is the nanopillar diameter. Figure 4: The beam propagation for (a)
the vertical interfacial dipole at $\phi=0$; (b) the horizontal interfacial
dipole at $\phi=0$ and (c) the horizontal interfacial dipole
$\phi=\frac{\pi}{2}$. The intensity distributions as a function of $y$ for (d)
the vertical interfacial dipoles and (e) the horizontal interfacial dipoles at
the field angles of $\pm{2}$ degrees, $\pm{1}$ degree and $0$ degree with the
metalens NA of $2.00$. (f) The intensity distribution as a function of $y$ for
the vertical interfacial dipoles at the field angles of $\pm{2}$ degrees,
$\pm{1}$ degree and $0$ degree with the metalens NA of $0.89$.

To evaluate the overall performance of our solid immersion metalens, we
simulated beam propagation from the vertical and horizontal interfacial
dipoles through to form an image by an ‘ideal’ tube lens. We assume the
metalens is placed at $z=0$ with the dielectric substrate ranging from z =
-100 $\mu{m}$ to z = 0 $\mu{m}$ and z $>$ 0 being vacuum. Consider first,
interfacial dipoles on the metalens’s optical axis at (0, 0, -100 $\mu{m}$).
Figures 4a, 4b and 4c present the beam propagation before and after the
metalens for the vertical dipole at $\phi$ = 0, the horizontal dipole at
$\phi$ = 0 and the horizontal dipole at $\phi$ = $\frac{\pi}{2}$,
respectively. The phase profile just after the lens is indeed constant showing
effective collimation by the metalens. Propagation beyond the metalens
considers the angular spectrum of plane waves at $z=0$ and also show a flat
phase profile at the field angle of $0^{o}$.

In order to show the effect of aberrations at larger field angles, we place a
tube lens at z = 20 $\mu{m}$. The focal length of the tube lens is set to be
5000 $\mu{m}$, so that the magnification of the whole optical system is
$108\times$. The tube lens is assumed to be perfect and thus introduces no
aberrations. We consider four more vertical or horizontal interfacial dipoles
at the positions (0, $\pm{1.75}$ $\mu{m}$, -100 $\mu{m}$) and (0, $\pm{3.49}$
$\mu{m}$, -100 $\mu{m}$). These dipoles correspond to the field angles
$\theta_{f}=$ $\pm{1}^{o}$ and $\pm{2}^{o}$. For the interfacial dipoles
displaced from the optical axis along the $y$-direction, the refracted angle
just after the metalens can be derived by the equation of the generalized laws
of reflection and refraction:[27]

$\sin\theta_{t}-n_{S}\sin\theta_{i}=\frac{\lambda_{0}}{2\pi}\frac{d\phi}{dy}$
(2)

where $\lambda_{0}$ is the wavelength of the incident light in vacuum,
$\theta_{i}$ is the incident angle in the substrate, $\theta_{t}$ is the
refracted angle in vacuum, and $d\phi$ is the phase difference between two
points separated by a distance of $dy$ at the interface. The calculated
refracted angle $\theta_{t}$ can then be used to predict beam propagation
beyond the lens. We have simulated the imaging of vertical and horizontal
interfacial dipole intensity distributions, as a function of position along
the $y$ axis, in the focal plane of the tube lens, as shown in Figure 4d and
4e, respectively. The intensity distributions for both dipole orientations at
$0^{o}$ field angle are diffraction limited Airy disks. However, for the field
angles of $\pm{1}^{o}$ and $\pm{2}^{o}$, significant aberration occurs.

The displacement of the central peaks from the optical axis in the focal plane
of the tube lens are calculated by the product of the displacement of the
dipole source and the magnification of the optical system. The displacements
of the central peaks are $\pm{375.9}$ $\mu{m}$ and $\pm{187.9}$ $\mu{m}$ for
the field angles of $\pm{2}^{o}$ and $\pm{1}^{o}$, respectively. From the
simulation, the largest peaks occur at $\pm{351.6}$ $\mu{m}$ and $\pm{169.7}$
$\mu{m}$ for the field angles of $\pm{2}^{0}$ and $\pm{1}^{o}$, which are in
good agreement with the theoretical expectations. The series of peaks
appearing at smaller displacement are attributed to lens aberration, in
particular coma.[28] Aberrations can be reduced by using a smaller NA. For
example, we have simulated the intensity distributions of five vertical
interfacial dipoles with a lower NA = 0.89 metalens ($90$ $\mu{m}$ in
diameter), as shown in Figure 4f. We can see the intensity distributions for
the field angles of $\pm{2}^{o}$ and $\pm{1}^{o}$ become more Airy disk-like.
While the lower NA successfully reduces the coma aberration, this is
unacceptable here as it drastically reduces the collection efficiency.

Figure 5: The design of singlet solid immersion metalens for abberation
correction. (a) Schematic of singlet metalens. The red circle represent the
interfacial dipole. The blue rectangle regime is the substrate. The green part
denote the metasurface. The red lines represent the light beam. The materials
of the substrate and nano pillors are AlN and TiO2 respectively. (b) The phase
profiles of the singlet non-hyperbolic metasurface and a normal hyperbolic
metasurface. (c) The ray tracing path diagram of the system with the incident
condition of 0, 1, and 1.7 degrees field angle. (d) Spot diagram of the
singlet metalens system. The blue dots indicate the image of interfacial
dipole. The circle outside the blue dots denotes the Airy disk, with the
radius of 14.6 $\mu{m}$.

In order to correct the abberation of the metalens for a range of field
angles, we have designed a singlet metalens with a non-hyperbolic phase
profile. The schematic of the system is shown in Figure 5a. The substrate is
chosen to be AlN with a refractive index of 2.15. In our simulation, the
thickness of the AlN substrate and focal length of the metalens are 0.25 mm.
The radius of the meatasurface is 0.3 mm. So the numerical aperture of the
metalens is 1.65. The focal length of the tubelens was chosen to be 12.25 mm,
so the objects can be magnified by $105\times$. The phase profile of the
singlet metalens is described by:

$\phi_{target}=-\frac{2\pi}{\lambda}n_{S}\left[\sqrt{r^{2}+f^{2}}-f\right]+\sum_{m=1}^{5}a_{m}(\frac{r}{R_{s}})^{2m}$
(3)

where $\lambda$ is the emission wavelength of the dipole (600 nm), $n_{S}$ is
the refractive index of substrate, $f$ is the focal distance (0.25 mm), $r$ is
the radial coordinate, $R_{s}=300$ $\mu$m is the normal radius, $a_{m}$ are
the optimization parameters. The optimised values, $a_{1}=-46.09$,
$a_{2}=16.46$, $a_{3}=-10.12$, $a_{4}=5.124$, and $a_{5}=-1.281$.

The phase profiles of our designed singlet non-hyperbolic metalens and the
normal hyperbolic metalens are shown in Figure 5b. Compared to the normal
hyperbolic lens, the additional terms increase the phase gradient at the edge
of the metalens. Figure 5c reveals the ray-tracing optical path diagram of
this system (Zemax). Here, the vertical double-arrow line represents the
metalens and the paraxial tube lens. The distance between them is zero. In
Zemax, we have used a Binary 2-surface to act as a metalens and choose three
field angles ($0^{o}$, $1^{o}$, $1.7^{o}$) to run the optimization. The
corresponding results are shown in Figure 5d. The information of the spot
sizes and the positions are shown in the Table 2. The black circle in Figure
5d denotes the Airy disk with a radius of 14.6 $\mu$m. As we can see in Figure
5d and the Table 2, all the root means square (RMS) spot radii of the incident
angle are inside the Airy disk circle. This means we can achieve
$\pm{1.7}^{o}$ field angle corresponding to $14.8$ $\mu$m field of view with
negligible image aberration.

Table 2: Spot size and the position of different field angles Field angle /degrees | 0 | 1 | 1.7
---|---|---|---
Source position /$\mu{m}$ | 0 | 4.36 | 7.42
Image position /$\mu{m}$ | 0 | -460.2 | -784
RMS Spot Radius /$\mu{m}$ | 5.76 | 1.86 | 10.6
Max Spot Radius /$\mu{m}$ | 6.82 | 2.49 | 14.6

## Conclusion

In this paper, we have reported an efficient and compact solid immersion
metalens to collect emission from dipole emission centres. Dipoles are placed
at the dielectric interface of vacuum and substrate so that most of the dipole
energy is directed into the substrate. A large proportion of this energy is
then collected by a high NA metalens on the opposite side of the substrate.
The thickness of the whole device is no more than 250 $\mu{m}$, and may be
varied to control the NA. According to the simulation result, the collection
efficiency of the collection device is $>85$%, for both dipole orientations.
While aberrations are pervasive for non-zero field angles, mitigation using a
non-hyperbolic gradient phase metalens is shown to be quite effective. We thus
report a high numerical aperture (NA=1.65), $105\times$ magnification, solid
immersion singlet metalens system with a low aberration over a ${14.8}$ $\mu$m
field of view.

## Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 844591. The authors thank M$\acute{o}$nica Mota for technical
support.

## Disclosures

The authors declare no conflicts of interest.

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