# Greedy Blockwise Residual Learning
for Scalable Monaural Source Separation

# BLOOM-Net: Blockwise Optimization for Masking Networks
Toward Scalable and Efficient Speech Enhancement

###### Abstract

In this paper, we present a blockwise optimization method for masking-based
networks (BLOOM-Net) for training scalable speech enhancement networks. Here,
we design our network with a residual learning scheme and train the internal
separator blocks sequentially to obtain a scalable masking-based deep neural
network for speech enhancement. Its scalability lets it dynamically adjust the
run-time complexity depending on the test time environment. To this end, we
modularize our models in that they can flexibly accommodate varying needs for
enhancement performance and constraints on the resources, incurring minimal
memory or training overhead due to the added scalability. Our experiments on
speech enhancement demonstrate that the proposed blockwise optimization method
achieves the desired scalability with only a slight performance degradation
compared to corresponding models trained end-to-end.

Index Terms—  Speech Enhancement, ResNet, Model Compression, Scalability

##  1 Introduction

Deep learning-based supervised methods have dramatically boosted single-
channel source separation performances in recent years. Typically, effective
and dominating deep learning solutions operate by estimating masks, either on
the time-frequency (TF) representations, such as the short-time Fourier
transform (STFT) [1, 2, 3] or recent models that learn a separator module that
applies masks in the latent feature space. The latter models have improved the
state of the art as the learned feature space allows them to bypass limits
imposed by TF-domain solutions (e.g., time-frequency resolution trade-off,
using noisy phase or dealing with phase estimation, etc.) in addition to the
advanced separator module’s architecture. Various architectures have been
proposed with each progressive model showing relative improvements: fully-
convolutional Conv-TasNet [4] that initially popularized the time-domain
approach, dual-path recurrent neural networks (DPRNN) that enabled long-term
sequence modeling [5], and Transformer and Conformer-based models that
overcome limitations from convolutional neural networks (CNN) and RNN based
approaches (e.g., limitations of receptive fields and extensive recurrent
connections) [6, 7, 8, 9].

However, a major drawback of aforementioned deep learning solutions is the
complexity of models. Heavy memory occupancy and especially their exorbitant
computational cost makes them impractical for deployment onto resource-
constrained devices.

Model compression methods offer effective solutions to this problem by
reducing complexity of neural network architectures while minimizing their
drop in generalization performance. There are various model compression
methods such as quantizing model parameters using low-bit resolution fixed-
point representations and/or pruning less important network components [10,
11] , simplifying convolutional operations [12], grouping RNN’s intermediate
tensor representations into smaller blocks [13], multi-resolution features via
successive downsampling and resampling [14], distilling knowledge from a
larger network to improve the performance of its corresponding compressed
model [15, 16], etc. These compressed models are typically designed to
minimize the inference complexity targeting the low-resource environment, thus
not being able to scale up to challenging separation problems. Eventually, in
order for a legacy system to be scalable, a range of versions need to be
retained in a device, increasing the total spatial complexity.

We argue that a scalable and efficient system must cover a wide resource-
related diversity in edge computing via an adaptive model architecture rather
than simply enumerating various model architectures. The scalable, thus
adaptive systems can be commonly found in coding applications. In [17], the
cross-module residual learning scheme enabled greedy module-wise neural codec
learning, where a deep autoencoder is trained to model the residual signal
that its preceding autoencoder fails to model. The system can preserve the
order of relative importance of the participating autoencoder modules, that
gives scalability to the system, i.e., the first part of the bitstream is more
important than the rest. SoundStream audio codec features bitrate scalability,
allowing the codec to adapt to the network conditions that can vary while
transmitting signals, too [18]. However, these models are specifically for
signal compression, seeking scalability within their resulting bitstreams,
thus not suitable for other applications. Meanwhile, the once-for-all (OFA)
scheme provides a general-purpose adaptive training mechanism that learns
multiple compressed variants of a model via a single training task [19].
However, it does not provide a single architecture that scales to different
test environments freely as we propose.

Likewise, we envision a scalable speech enhancement model that changes its
operation modes, ranging from an energy-efficient version to a performance-
boosted one. In doing so, instead of preparing each different model in the
device, a scalable model provides a flexible structure that adjusts its
performance and complexity per resource constraint. To this end, we propose a
scalable time-domain architecture for speech enhancement: BLOckwise
Optimization for Masking networks (BLOOM-Net). It is a greedy residual
learning strategy to train individual blocks sequentially. Since each block is
trained to improve the previous block’s speech enhancement result further, the
deployment can optionally choose to use only the first few blocks depending on
the available computational resources. Although it has been known that a
sequence of two heterogeneous speech enhancement processes is effective [20,
21], our approach differs from that literature in that (a) ours focuses on the
architectural innovation rather than a concatenation of two heterogeneous
models (b) the proposed model eventually achieves the scalability in the
feature space rather than in the raw signal domain. BLOOM-Net shows
competitive performance compared to their end-to-end counterparts, while
providing the additional advantage that its modularized blocks are easy to
attach or detach for scalability.

((a))

((b))

Fig. 1: (a) The baseline separation model (b) a weak separation block

##  2 Methodologies

###  2.1 Baseline 1: The Time-Domain Separation Model

Our baseline source separation model adopts the common structure found in
time-domain source separation networks that consists of an encoder, separator,
masker, and decoder as shown in Fig. 1(a). First, the input
$\bm{x}\in\mathbb{R}^{T}$ with $T$ samples in the time domain is fed to the
encoder, which is a function that transforms the input into a latent
representation: $\text{Enc}:\bm{x}\rightarrow\bm{h}\in\mathbb{R}^{D}$. We
posit that it is a small neural network module, e.g., a 1D convolutional layer
followed by a nonlinear activation function. Then, the latent representation
is fed into the separator module:
$\text{Sep}:\bm{h}\rightarrow\bm{z}\in\mathbb{R}^{K}$, whose output $\bm{z}$
is used as input to the mask estimation module
$\text{Mas}:\bm{z}\rightarrow\bm{m}\in\mathbb{R}^{D}$. Finally, the mask is
applied to the encoded mixture representation $\bm{h}$ to retrieve the source-
specific estimate of the latent representation $\tilde{\bm{h}}$ (e.g., for the
speech source), which is then decoded back to the time-domain estimate of the
target source,
$\text{Dec}:\tilde{\bm{h}}\rightarrow\hat{\bm{s}}\in\mathbb{R}^{T}$.

In the time-domain separation models, it is common to employ a large separator
module with repeating structures indexed by $l$. It is also popular to merge
the input and output of each block as input to the next one, performing
residual learning as proposed in ResNet [22]. These blocks are learned
altogether in the state-of-the-art models as well as in Baseline 1, limiting
the models’ scalability.

((a)) The time-domain blockwise optimization model (Baseline 2).

((b)) The proposed BLOOM-Net architecture.

Fig. 2: The scalable speech enhancement models.

###  2.2 Baseline 2: Time-Domain Blockwise Optimization

To build up our proposed BLOOM-Net model, we begin with a naïve concatenation
approach as our second baseline. Fig. 1(b) shows a weak source separation
block as a compromised version of Baseline 1 in Fig. 1(a). Although it is with
only one ResNet block for the separator module, it is still a legitimate
stand-alone separation model. We assume that there are $L$ such weak
separation blocks, each of which enhances its previous block’s results as
shown in Fig. 2(a). Let ${\mathcal{F}}^{(l)}(\cdot)$ be the $l$-th weak
separation block. It performs speech enhancement on $\hat{\bm{s}}^{(l-1)}$, an
output of the $(l-1)$-th block:
${\mathcal{F}}^{(l)}:\hat{\bm{s}}^{(l-1)}\rightarrow\hat{\bm{s}}^{(l)}$. Note
that the notion of a “weak” separation block comes from the boosting methods
that incrementally add weak learners [23].

The second baseline provides a mechanism to serialize the speech enhancement
models with an order of significance. If the $(l-1)$-th weak separation block
leaves room for improvement, i.e., when the loss
${\mathcal{L}}^{(l-1)}(\bm{s},\hat{\bm{s}}^{(l-1)})$ is not sufficiently
small, the next block ${\mathcal{F}}^{(l)}(\hat{\bm{s}}^{(l-1)})$ focuses on
that sample and tries to improve. During the test time, suppose the device can
afford only up to $\ell$ blocks. Since each of the $L$ blocks is sequentially
trained with its own reconstruction loss
${\mathcal{L}}^{(l)}(\bm{s},\hat{\bm{s}}^{(l)})$, scalability is achieved by
performing the inference on only the first $l\leq\ell$ weak separation
blocks—the result is still a legitimate source estimate. In contrast, Baseline
1 would need to redundantly prepare all the scaled variants of the model in
order to achieve the desired scalability.

###  2.3 BLOOM-Net: Blockwise Optimization in the Latent Space

Baseline 2 exhibits a redundancy issue. During the test time, the chosen block
should execute all four submodules to deliver a time-domain signal to its
successive block. Instead, we propose a blockwise optimization scheme that
works in the latent space, so that the serialization is done among the
separator submodules. In that way, the system can avoid unnecessary masking,
decoding, and encoding operations that repeat at every weak separation block
in Baseline 2.

Fig. 2(b) describes the BLOOM-Net architecture, where $\text{Sep}^{(l)}$
performs residual learning in the sense that it relays the sum of its output
$\bm{z}^{(l)}$ and the input $\bar{\bm{z}}^{(l-1)}$ to the next separator
block:

$\bar{\bm{z}}^{(l)}=\bm{z}^{(l)}+\bar{\bm{z}}^{(l-1)},\text{ where
}\bm{z}^{(l)}=\text{Sep}^{(l)}\big{(}\bar{\bm{z}}^{(l-1)}\big{)}.$ (1)

Note that the encoder’s output $\bm{h}=\bm{z}^{(0)}$ is the input to
$\text{Sep}^{(1)}$. We also induce the input to $\text{Sep}^{(L)}$
recursively: $\bar{\bm{z}}^{(L-1)}=\sum_{l=0}^{L-1}\bm{z}^{(l)}$.

BLOOM-Net performs block-specific masking and decoding just to compute the
block-specific error, while the residual connections are defined among the
latent variables $\bm{z}^{(l)}$. The $l$-th masker and decoder works on the
separator output $\bm{z}^{(l)}$ as follows:

$\hat{\bm{s}}^{(l)}\\!=\\!\text{Dec}^{(l)}\big{(}\tilde{\bm{h}}^{(l)}\big{)},~{}\tilde{\bm{h}}^{(l)}\\!=\\!\bm{m}^{(l)}\odot\bm{h}^{(l)},~{}~{}\bm{m}^{(l)}\\!=\\!\text{Mas}^{(l)}\big{(}\bm{z}^{(l)}\big{)}.$
(2)

The blockwise output $\hat{\bm{s}}^{(l)}$ is then compared with the ground-
truth source $\bm{s}$ to compute the blockwise loss
${\mathcal{L}}^{(l)}(\bm{s},\hat{\bm{s}}^{(l)})$, ensuring the intermediate
output is usable. Note that Enc is shared and reused for all the sequence of
blocks such that all blocks learn to estimate denoising masks on the same
latent representation $\bm{h}$. Meanwhile, $\text{Mas}^{(l)}$ and
$\text{Dec}^{(l)}$ modules are block-specific and no longer updated once the
$l$-th block is trained.

During the test time, the actual inference involves Enc, $\text{Sep}^{(l)}$
where $1\\!\leq\\!l\\!\leq\\!\ell$, $\text{Mas}^{(l)}$, and
$\text{Dec}^{(l)}$, which are represented as shaded blocks in Fig. 2(b).
Compared to Baseline 2, BLOOM-Net saves the cost of $\ell-1$ encoding,
decoding, and masking operations. Although these blocks are lightweight,
removing them improves not only the computational efficiency but also feature
learning. The direct residual learning path in the latent feature space allows
BLOOM-Net to learn a hierarchy of latent representations, while Baseline 2 is
limited to concatenating shallow representations.

Fine-tuning can further refine a fully trained BLOOM-Net. While BLOOM-Net
provides the desired scalability via blockwise optimization, its greedy nature
prevents the model from learning from the full picture, thus underperforming
the theoretical upper bound. To this end, we propose to fine-tune BLOOM-Net,
where all modules in all $L$ blocks are updated using the combination of all
loss functions, i.e.,
$\sum_{l=1}^{L}{\mathcal{L}}^{(l)}(\bm{s},\hat{\bm{s}}^{(l)})$. This
comprehensive fine-tuning significantly improves BLOOM-Net, making its
performance near the theoretical upper bound.

##  3 Experimental Setup

###  3.1 Datasets

During training, we used clean speech samples from the Librispeech corpus [24]
and noise recordings from the MUSAN dataset [25]. We used train-clean-100 and
dev-clean subsets from Librispeech for training and validation respectively.
We split MUSAN’s free-sound subset at 80:20 ratio into training and validation
partitions. For testing, we used unseen speech samples from Librispeech’s
test-clean and noise from MUSAN’s sound-bible. Audio files are loaded at 16
kHz sampling rate and standardized to have a unit-variance. Noise samples are
scaled to random input SNR levels uniformly chosen between -5 and 10 dB and
added to speech signals to obtain noisy mixtures.

###  3.2 Architecture and Training Details

To investigate its application to time-domain audio separation networks [4, 5,
6, 7, 26], our ResNet-based models are implemented as a simplified form of
Conv-TasNet [4]. We use the same encoder and decoder design as in [4] with the
same hyperparameters. Each separator module is defined as a residual block
using 1-D convolutional layers similar to those in [4], but without dilation
and intermediate skip-connections. Each residual block consists of a $1\times
1$ convolution operation followed by a depthwise convolution operation, with
parametric ReLU (PReLU) [27] and global layer normalization (gLN) added after
each convolution operation. We used the ConvTasNet implementation available in
Asteroid [28] and tuned the hyperparameters to construct our time-domain
ResNet models.

We used the negative scale-invariant signal to distortion ratio (SI-SDR) as
the loss function [29], defined as

${\mathcal{L}}(\bm{s},\hat{\bm{s}})=-\text{SI-
SDR}(\bm{s},\hat{\bm{s}})=-10\log_{10}\bigg{(}\frac{||\alpha\bm{s}||^{2}}{||\alpha\bm{s}-\hat{\bm{s}}||^{2}}\bigg{)}$
(3)

where $\alpha=\frac{\hat{\bm{s}}^{\top}\bm{s}}{||\bm{s}||^{2}}$ is a scaling
factor. We train all models on 1-second long segments using a mini-batch size
of 64. Adam is used as the optimizer with learning rate initialized to
$1\times 10^{-4}$. Early stopping is applied and models with the lowest
validation losses are used for final evaluation on the test mixtures.

###  3.3 Training Configurations

The experiments are on two types of baselines and our proposed BLOOM-Net
method. We additionally examine the impact of fine-tuning on BLOOM-Net.

  * •

Baseline 1 - Full: The time-domain separation model (Sec. 2.1) trained in a
conventional end-to-end manner. It employs $L=6$ separator blocks from the
beginning and learns all of them together. Although there are $L=6$
separators, removing one of them to reduce the complexity will cause a
complete break down as the model is not scalable.

  * •

Baseline 1 - Int.: We also learn $L=6$ different versions of Baseline 1, each
of which is an intermediate version, containing up to $L=\\{1,\ldots,6\\}$
separators, respectively, e.g., when $L=3$ there are three separators that are
trained altogether. Each of these models is the upper bound of its
corresponding BLOOM-Net with a matching separator number.

  * •

Baseline 2: The time-domain weak separation blocks (Sec. 2.2). Individual
blocks are trained as a stand-alone separation model one after another to
promote scalability.

  * •

BLOOM: BLOOM-Net in its basic setup (Sec. 2.3).

  * •

BLOOM-FT: BLOOM-Net, fine-tuned to minimize the combination of losses from all
blocks. It overcomes the suboptimal performance caused by BLOOM’s greedy
training.

Table 1: SI-SDR improvements of the competing models. Evaluations are with
respect to the number of blocks $\ell$ chosen for inference.

Method | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$ | $\ell=6$
---|---|---|---|---|---|---
Baseline 1 - Full | -0.60 | -0.86 | -0.12 | 0.40 | 0.97 | 8.89
Baseline 1 - Int. | 4.55 | 6.83 | 7.65 | 8.30 | 8.58 | 8.89
Baseline 2 | 4.55 | 5.01 | 5.17 | 5.25 | 5.25 | 5.25
BLOOM | 4.55 | 6.13 | 6.92 | 7.40 | 7.61 | 7.75
BLOOM-FT | 4.74 | 6.55 | 7.44 | 8.14 | 8.51 | 8.72

((a)) $\bm{x}_{1}^{~{}}$ (SI-SDR: 7.63 dB)

((b)) $\hat{\bm{s}}_{1}^{(2)}$ (SI-SDR: 12.12 dB)

((c)) $\hat{\bm{s}}_{1}^{(4)}$ (SI-SDR: 13.13 dB)

((d)) $\hat{\bm{s}}_{1}^{(6)}$ (SI-SDR: 13.29 dB)

((e)) $\bm{s}_{1}^{~{}}$

((f)) $\bm{x}_{2}^{~{}}$ (SI-SDR: 3.05 dB)

((g)) $\hat{\bm{s}}_{2}^{(2)}$ (SI-SDR: 14.22 dB)

((h)) $\hat{\bm{s}}_{2}^{(4)}$ (SI-SDR: 19.93 dB)

((i)) $\hat{\bm{s}}_{2}^{(6)}$ (SI-SDR: 19.98 dB)

((j)) $\bm{s}_{2}^{~{}}$

Fig. 3: Denoising output samples from intermediate blocks. Each row represents
a different example. The columns represent noisy mixture, estimated
reconstructions from $l$-th blocks, and the corresponding ground-truth target
clean speech.

##  4 Experimental Results and Discussion

Table 1 presents the results of the competing systems. We first compare BLOOM-
Net with Baseline 1. At a glance, the model trained under Baseline 1 - Full
achieves the highest performance when it fully utilizes all 6 separator
blocks. However, the model’s scalability is limited, as it is evidently unable
to perform denoising when less than 6 separator blocks are used (columns from
$\ell=1$ to $5$). This behavior is expected since all 6 modules are trained
altogether in the conventional end-to-end fashion—its intermediate separator
outputs $\bm{z}^{(l)}$ are not suitable for the shared masker module to
compute the mask from, except for the final separator output $\bm{z}^{(6)}$.

Hence, it is unavoidable for Baseline 1 to train multiple versions of
different block configurations to scale to various application and hardware
requirements, which is the Baseline 1 - Int. setup. Since in this set up there
are totally six different end-to-end models, each of which specializes in each
choice of $\ell$, they form the performance upper bound. However, the system’s
total spatial complexity is the sum of all six versions, which is not the most
efficient option.

BLOOM-Net, on the other hand, exhibits desired scalability. Instead of
completely failing, BLOOM shows decent performance at $\ell=5$ case, which is
only a 0.1 dB drop from its $\ell=6$ setup. Its performance apparently drops
as the model complexity decreases more. In addition, BLOOM’s most powerful
setup $\ell=6$ is suboptimal compared to Baseline 1.

BLOOM-FT addresses this issue by fine-tuning the entire modules via the sum of
all blockwise loss functions. Compared against the oracle end-to-end Baseline
1 results (Baseline 1 - Int.), the fine-tuned results usually show less than
0.3 dB drop in all cases. Hence, we claim that the properly fine-tuned BLOOM-
Net is almost comparable to the traditional end-to-end models while it
provides unprecedented scalability and spatial efficiency.

Next, we draw attention towards the time-domain blockwise optimization method,
Baseline 2. Compared to BLOOM-Net that operates in the latent domain, Baseline
2 sequentially feeds time-domain inputs to the next stand-alone module. This
incurs an overhead of learning the feature transformation and its inverse
operation, constraining each block to only learn a shallow latent
representation; thus, the improvement by adding more weak separation blocks is
only minimal. It showcases the merit of the proposed BLOOM-Net algorithm that
performs blockwise optimization in the latent space.

Fig. 3 shows two denoising examples where the different model complexity
choices affect the quality of the output. The deeper the inputs are processed
within the BLOOM-Net, the higher the denoising quality. The depth of the
network can be decided based on the requirements of the deployed environment
or users’ preference. This demonstrates BLOOM-Net’s scalability feature that
offers flexible quality-complexity tradeoff to adjust to the test environment.

Finally, Table 2 shows the inference-time computational complexity of Baseline
1 - Int. and BLOOM-Net in terms of number of parameters and multiply-
accumulate (MAC) operations. Note that BLOOM and BLOOM-FT are equivalent in
this context. First, there is no difference in computational complexity
(MACs), as the active modules during inference are the same. However, BLOOM-
Net exhibits a significant advantage in terms of spatial complexity when we
assume a scalable model. For example, if the device can afford up to $\ell=2$,
the baseline has to prepare two different versions with $L=1$ and $L=2$ for
the best performance in both energy-efficient and performance-boosted use
cases. In doing so, although these two versions’ sizes are $0.28M$ and $0.42M$
parameters, respectively, their sum amounts to $0.70M$. Likewise, in order for
Baseline 1 to be scalable, it exponentially accumulates spatial complexity as
$\ell$ grows. Conversely, BLOOM-Net manages this scalability issue more
carefully: it increases the model size just by the amount of a single
separator block and its insignificantly small block-specific masker and
decoder modules (about 0.21M parameters). Hence, for example, our scalable
BLOOM-Net with five separators (1.13M) is smaller than the baseline that
covers three complexity profiles (1.26M).

Table 2: Computational requirements of time-domain ResNet models trained under
Baseline 1 and our proposed BLOOM-Net method. The number of parameters
reported encompasses the entire model parameters needed to implement the
scalable model. MACs are computed given 1-second inputs.

| Method | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$ | $\ell=6$
---|---|---|---|---|---|---|---
MACs (G) | Baseline 1 | 0.53 | 0.80 | 1.07 | 1.33 | 1.60 | 1.87
BLOOM
Params (M) | Baseline 1 | 0.28 | 0.70 | 1.26 | 1.95 | 2.78 | 3.64
BLOOM | 0.28 | 0.49 | 0.71 | 0.92 | 1.13 | 1.34

##  5 Conclusion

In this study, we introduced BLOOM-Net, a novel algorithm for scalable speech
denoising. We postulated that scalable implementation of a deep learning-based
speech enhancement system is critical to handle various resource-related test
conditions that a device faces. While traditional end-to-end time-domain
source separation models have shown advanced separation performance, we
claimed that such a system cannot provide the desired scalability. Our BLOOM-
Net is with a carefully designed residual learning scheme that performs
blockwise optimization to improve the model performance in an incremental way.
Since the blockwise optimization was on each individual separator module that
greedily contributes to the model performance, BLOOM-Net achieves the
scalablility—the device can freely choose from the multiple profiles based on
its resource constraint. In doing so, the enhancement quality is reasonably
associated with the model complexity. Source codes are available at
https://saige.sice.indiana.edu/research-projects/bloom-net.

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# Asymptotics of summands I: square integrable independent random variables

Aladji Babacar Niang ,  Gane Samb Lo and  Moumouni Diallo

###### Abstract.

This paper is part of series on self-contained papers in which a large part,
if not the full extent, of the asymptotic limit theory of summands of
independent random variables is exposed. Each paper of the series may be taken
as review exposition but specially as a complete exposition expect a few
exterior resources. For graduate students and for researc _h_ ers (beginners
or advanced), any paper of the series should be considered as a basis for
constructing new results. The contents are taken from advanced books but the
organization and the proofs use more recent tools, are given in more details
and do not systematically follow previous one. Sometimes, theorems are
completed and innovated.

† Aladji Babacar Niang
LERSTAD, Gaston Berger University, Saint-Louis, Sénégal.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

†† Gane Samb Lo.
LERSTAD, Gaston Berger University, Saint-Louis, Sénégal (main affiliation).
LSTA, Pierre and Marie Curie University, Paris VI, France.
AUST - African University of Sciences and Technology, Abuja, Nigeria
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Permanent address : 1178 Evanston Dr NW T3P 0J9,Calgary, Alberta, Canada.

††† Dr Moumouni Diallo
Université des Sciences Sociale et de Gestion de Bamako ( USSGB)
Faculté des Sciences Économiques et de Gestion (FSEG)
Email<EMAIL_ADDRESS>

Keywords. central limit theorem; pre-weak and weak limits; summands of arrays
of real-valued random variable; class of admissible weak limits in the clt for
independent random variables; infinitely divisible or decomposable laws; Lévy,
Lyndeberg and Lyapounov criterai; Lynderberg-type condition; product of
Poisson type characteristic functions.
AMS 2010 Mathematics Subject Classification: 60F05; 60E07

###### Contents

  1. 1 Introduction
  2. 2 The Bounded Variance Hypothesis (BVH) and the Uniformly Asymptotic Negligibility (UAN)
  3. 3 Class of infinitely divisible (or decomposable) laws on $\mathbb{R}$
    1. 3.1 Definitions and examples
    2. 3.2 Properties
    3. 3.3 The three pillars of that theory
  4. 4 The weak convergence theorem of summands under the BVH and the UAN Condition
    1. 4.1 The Central limit theorem for centered, independent and square integrable random variables
    2. 4.2 The Central limit theorem for non-centered, independent and square integrable random variables
  5. 5 Characterizations of two important examples
    1. 5.1 Gaussian limit
    2. 5.2 Poisson limit
  6. 6 Conclusion

##  1\. Introduction

The largest part of the asymptotic theory of partial sums of random variables
concentrated on independent random variables over at least two centuries.
Almost all the greatest scientists in probability Theory (Lévy, Kolmogorov,
Lyapounov, Lynderberg, Gnedenko, Feller, etc.) engaged themselves in such an
enterprise. Besides, a very large part of the current theory on dependent
sequence of random variables is based on transformations of independence
structures, for example on notions of nearness of the dependence to
independence (weak dependence, $\phi$-mixing, associated sequence, independent
increments, etc.).

So it is important to have the deepest knowledge of that past. In Lo  (2018b),
we introduced some important elements of that theory (Central limit theorems,
laws of the large numbers, law of the iterated logarithm, zero-one laws,
etc.).

We are beginning a series on self-contained papers in which a large part of
the central limit theorem, if not the full extent, of the asymptotics of
summands of independent random varianles will exposed. Each paper of the
series may be taken as review exposition but specially as a complete
exposition expect a few exteroir ressources. For graduate students and for
researchers (beginners or advanced), any paper of the series should be
considered as a basis for constructed new results. The contents are taken from
advanced books but the organization and the proofs used more recent tools, are
given in more details and do not systematically follow previous one.
Sometimes, theorem are completed and innovated.

In this first paper of the series, we focus of the full characterization of
the CLT problem of independent summands for square integrable random
variables. The main material is extracted from Loève  (1997) as a general
guide. But we use arguments from our previous works (Lo (2017b), Lo  (2018a),
Lo  (2018b), Lo et al. (2016), etc.) to have unified and a self-contained Lo
(2018b). In particular, the text on the weak convergence of bounded measures
and its exppression on $\mathbb{R}^{k}$ ($k\geq 1$) povides tools to make the
conclusions in Loève  (1997) more clear, as we expect as least.

Papers of this series focus on complete mathematical texts rather than on a
historical review of contributions of differents authors. We refer to Loève
(1997) for that aspect.

Since the set of weak limits of independent summands for square integrable
coincide with the set of infinitely decomposable laws, this paper will deal
with the most important properties of such probability laws to the extent
needed here. More developments, including the study of stable laws, will be
given in the other papers of the series.

Let us introduce the problem, after we provide some notations. below, all
sequences and all arrays of random variables have all their elements defined
on a same probability space. So, we set a generic probability space
$(\Omega,\mathcal{A},\mathbb{P})$.

Following Loève  (1997), we transform the study of sums of independent and
centered random variables

$S_{n}=X_{1}+...+X_{n},\ n\geq 1,$

(with the notation $\sigma_{i}^{2}=\mathbb{E}X_{i}^{2}$, $i\geq 1$, if they
exist), by summands

$S_{n}=\sum_{1\leq k\leq k_{n}}X_{k,n},\ n\geq 1,$

where for each $n\geq 1$, the family $\\{X_{k,n},\ 1\leq k\leq k_{n}=k(n)\\}$
is a family of independent and centered random variables such that $F_{k,n}$
stands for the cdf of $X_{k,n}$ and $\sigma_{k,n}^{2}=\mathbb{E}X_{k,n}^{2}$,
$1\leq k\leq k_{n}$. We suppose also that $k_{n}\rightarrow+\infty$ as
$n\rightarrow+\infty$.

Notations. The notation already given and completed by the notation $f_{kn}$
for the characteristic function of $X_{k,n}$, are fixed for once.

In the case of simple summands, we have for each $n\geq 1$, $k_{n}=n$ and
$X_{k,n}=X_{k}$ for $k\in[1,n]$. Here, the rows $(X_{k,n})_{1\leq k\leq
k_{n}}$ are such that each of them is obtained by adding one element to the
predecessor. But, in the general case, no relation between families
$\mathcal{E}_{n}=\\{X_{k,n},\ 1\leq k\leq k(n)\\}$ is required. Also, in the
case of the simple sequence $(X_{k})_{k\geq 1}$, the studied array for each
$n\geq 1$, is $\\{X_{1}/s_{n},\cdots,X_{n}/s_{n}\\}$ where
$s_{n}^{2}=\mathbb{V}ar(X_{1}+\cdots+X_{n})$.

Here, we are going to investigate the general problem of finding all the
possible weak limits do $S_{n}$. Without restrictions, we this may lead to
trivial results. So we have to fix a general frame in which the study will be
done. In doing, the best way seems to go back to the complete theory of
Feller-Levy-Lynderberg and there, discover the following two fundamental
hypotheses.

##  2\. The Bounded Variance Hypothesis (BVH) and the Uniformly Asymptotic
Negligibility (UAN)

Given a sequence $(X_{k})_{k\geq 1}$ of independent, centered and square-
integrable random variables, we set $\sigma_{k}^{2}=\mathbb{E}X_{k}^{2}$,
$k\geq 1$, for $n\geq 1$

$s^{2}_{n}=\sum_{k=1}^{n}\sigma_{k}^{2},\ \ t_{n}^{2}=\max\\{\sigma_{k}^{2},\
\ 1\leq k\leq n\\}\ \ and\ \ B_{n}=t_{n}^{2}/s_{n}^{2},$

$k(n)=n$ for each $n\geq 1$ and for each $n\geq 1$

$\\{X_{k,n},\ 1\leq k\leq k(n)\\}=:\left\\{\frac{X_{k}}{s_{n}},\ 1\leq k\leq
k(n)\right\\}$

and

$S_{n}=\sum_{k=1}^{k(n)}X_{k,n}.$

The Feller-Levy-Lynderberg (F2L) theorem (see Loève  (1997), or Lo  (2018b),
Chapter 7, Section 2, Part B) ensures that:

(2.1) $S_{n}\rightsquigarrow\mathcal{N}(0,1)\ \ [WC]\ \ and\ \
B_{n}\rightarrow 0\ \ [NG]$

if and only if, for any $\varepsilon>0$,

(2.2) $L_{n}(\varepsilon)=\sum_{1\leq k\leq
k(n)}\int_{|X_{k,n}|\geq\varepsilon}X_{k,n}^{2}\ d\mathbb{P}\rightarrow 0.$

Let us see how behave the two following important quantities in that frame :

(2.3) $U(n,\varepsilon)=\sup_{1\leq k\leq
n}\mathbb{P}(|X_{k,n}|\geq\varepsilon)\ \ and\ \ MV(n)=\sum_{1\leq k\leq
k(n)}\mathbb{V}ar(X_{k,n}).$

We have, by Markov inequality,

(2.4) $U(n,\varepsilon)\leq\sup_{1\leq k\leq
n}\frac{\mathbb{E}X_{k,n}^{2}}{\varepsilon^{2}s_{n}^{2}}=\varepsilon^{-2}B_{n},$

and $MV(n)=1$ for all $n\geq 1$ and for $c=1$

$\sup_{n\geq 1}MV(n)=c<+\infty.$

The theory we are going to develop in a more general case needs the properties
we just introduced with specific names.

Definition. Under the notation given above, we say that :

(i) the Bounded Variance Hypothesis (BVH) holds if and only if

$\sup_{n\geq 1}MV(n)=c<+\infty;$

(ii) the Variance Convergence Hypothesis (VCH) holds if and only if

$MV(n)\rightarrow c\in]0,+\infty[;$

(iii) the Uniformly Asymptotic Negligibility (UAN) holds if and only if, for
any $\varepsilon>0$,

$U(n,\varepsilon)\rightarrow 0.$

We express the F2L theorem as follows: If

(a) the (UAN) condition holds;

(b) the (BVH) holds, then

$S_{n}\rightsquigarrow\mathcal{N}(0,1)\ \ \text{{if \ and \ only \ if}}\ \
L_{n}(\varepsilon)\rightarrow 0,\ \text{{for any}}\ \varepsilon>0.$

In this particular case, the characteristic function of the weak law
$\mathcal{N}(0,1)$

$\psi_{\infty}(u)=\exp(-u^{2}/2)$

is such that, for any $p\in\mathbb{N}$, $p>0$, $\psi_{\infty}^{1/p}$ defined
by

$\psi_{\infty}^{1/p}(u)=\exp(-p^{-1}u^{2}/2)$

is still a characteristic function, actually of a $\mathcal{N}(0,\ p^{-1})$
law. Let us denote by $\mathcal{C}_{f}$ the class of all characteristic
functions $f:\mathbb{R}\rightarrow\mathbb{C}$ on $\mathbb{R}$.

Now we may set our general task to be done: (Task). Given the UAN and the BVH
conditions, what is the class of all possible limits $Z_{\infty}$ of
characteristic function $\psi_{\infty}$. By the particular case of F2L
theorem, we may think that the searched class can be

$\mathcal{C}_{fid}=\\{\psi\in\mathcal{C}_{f}:\ \ (\forall p>1),\
\psi^{1/p}\in\mathcal{C}_{f}\\}.$

We define $\mathcal{C}_{fid}$ as the class of infinitely divisible
characteristic functions. At least, in the current version of the central
limit theorem, the Gaussian weak limit is in $\mathcal{C}_{fid}$.

We are going to see that the suggestion in the description of the task is
effectively the global solution.

In the sequel, we will devote Section 3 on infinitely divisible (or
decomposable) laws. In Section 4, we finish the task we have given to
ourselves under the UAN Condition and the BVC. Finally in 5, rediscover the
characterization of the CLT to a Gaussian law and that of the CLT to a Poisson
law. In the next element of the series, we proceed to a general theory with
non-necessarily square-integrable random variables.

##  3\. Class of infinitely divisible (or decomposable) laws on $\mathbb{R}$

###  3.1. Definitions and examples

The basic definition is the following.

######  Definition 1.

A characteristic function $\psi\in\mathcal{C}_{f}$ is infinitely decomposable
(idecomp), denoted by $\psi\in\mathcal{C}_{fid}$ if and only if for all
positive integer $p$, $\psi^{1/p}$ is still a characteristic function.

Let us explain the notion of idecomp in terms of random variables. Suppose
that $\psi\in\mathcal{C}_{f}$, $p\geq 1$ and $\psi_{p}=\psi^{1/p}$. Suppose
that $\psi_{p}$ is the characteristic function (ch.f) of a probability measure
$\mathbb{P}_{p}$. By the Kolmogorov theorem, it is possible to construct a
probability space $(\Omega_{p},\mathcal{A}_{p},\mathbb{P}_{p})$ holding
independent real-valued random variables $Z_{p}$, $Z_{1,p}$, $\cdots$,
$Z_{p,p}$ having all the cha.f $\psi_{p}$, i.e,

$\psi_{p}(u)=\mathbb{E}_{\mathbb{P}_{p}}\exp(iuZ_{p})=\int\exp(iux)\
d\mathbb{P}_{p}.$

It is clear that $\psi_{p}^{p}$ is the cha.f of
$S_{p}=Z_{1,p}+\cdots+Z_{p,p}$. As well, $\psi$ is the cha.f of a probability
measure $\mathbb{P}_{\psi}$ on $\mathbb{R}$ and let us denote by $Z_{\psi}$ a
random variable with $\mathbb{P}_{\psi}$ as probability law.

We easily see that we may extend the definition as follows. In the definition
below, we use the notion of idecomp probability law at the place of idecomp
characteristic function or idecom random variable.

######  Definition 2.

Let $Z$ be a real-valued random variable with probability law $\mathbb{P}_{Z}$
and ch.f $\psi_{Z}$. $\mathbb{P}_{Z}$ is idecom (equivalently $\psi_{Z}$ is
idecomp or $Z$ is idecomp) if and only if one of the following assertions
holds:

(i) For all $p\in\mathbb{N}\setminus\\{0\\}$, $\psi_{Z}^{1/p}$ is a cha.f.

(ii) For all $p\in\mathbb{N}\setminus\\{0\\}$, there exists a cha.f $\psi_{p}$
such that $\psi_{Z}=\psi_{p}^{p}$.

(iii) For all $p\in\mathbb{N}\setminus\\{0\\}$, there exists a probability
$\mathbb{P}_{p}$ on $\mathbb{R}$ such that

$\mathbb{P}_{Z}=\mathbb{P}^{\otimes p}_{p},$

that is, $\mathbb{P}_{Z}$ is the convolution product of $\mathbb{P}_{p}$ by
itself $p$ times.

(iv) For all $p\in\mathbb{N}\setminus\\{0\\}$, there exists a sequence
$Z_{1,p}$, $\cdots$, $Z_{p,p}$ of independent and identically distributed
real-valued random variables such that

$Z=_{d}Z_{1,p}+\cdots+Z_{p,p}.$

Examples. Let us give some quick examples.

Example 1. (Degenerate random variable). Let $Z=a$, p.s of cha.f

$\psi_{Z}(t)=e^{iat},\ t\in\mathbb{R}.$

For $p\geq 1$, $\psi_{Z}(t)^{1/p}=e^{i(a/p)t}$, which is the cha.f of the
degenerate r.v $Z_{p}=a/p$.

Example 2. (Gaussian random variables). Let $Z\sim\mathcal{N}(m,\sigma^{2})$,
$m\in\mathbb{R}$, $\sigma\in\mathbb{R}_{+}\setminus\\{0\\}$ with cha.f

$\psi_{Z}(t)=\exp(imt-\sigma^{2}t^{2}/2),\ t\in\mathbb{R}.$

For $p\geq 1$, we have

$\psi_{Z}(t)^{1/p}=\exp(i(m/p)t-(\sigma/\sqrt{p})^{2}t^{2}/),\
t\in\mathbb{R},$

which is the cha.f of a $\mathcal{N}(m/p,\sigma^{2}/p)$ r.v.

Example 3. (Translated Poisson random variables). Let
$Z\sim\mathcal{P}(a,\lambda)\equiv+\mathcal{P}(\lambda)$, $a\in\mathbb{R}$,
$\lambda\in\mathbb{R}_{+}\setminus\\{0\\}$ with cha.f

$\psi_{Z}(t)=\exp\left(iat+\lambda(e^{it}-1)\right).$

For $p\geq 1$, we have

$\psi_{Z}(t)^{1/p}=\exp\left(i(a/p)t+(\lambda/p)(e^{it}-1)\right),\
t\in\mathbb{R},$

which is the cha.f of a $\mathcal{P}(a/p,\lambda/p)$ r.v.

Example 4. (Gamma random variables). Let $Z\sim\gamma(a,b)$, $a>0$, $b>0$,
with cha.f

$\psi_{Z}(t)=\left(1-it/b\right)^{-a},\ t\in\mathbb{R}.$

For $p\geq 1$, we have

$\psi_{Z}(t)^{1/p}=\left(1-it/b\right)^{-(a/p)},\ t\in\mathbb{R},$

which is the cha.f of a $\gamma(a/p,b)$ r.v.

Example 5. (Cauchy random variables). Let $Z\sim Ca(a,b)$, $a\in\mathbb{R}$,
$b>0$ with cha.f

$\psi_{Z}(t)=\exp(iua-b|t|),\ t\in\mathbb{R}.$

For $p\geq 1$, we have

$\psi_{Z}(t)^{1/p}=\exp(iu(a/p)-(b/p)|t|),\ t\in\mathbb{R},$

which is the cha.f of a $Ca(a/p,b/p)$ r.v.

Now, let us focus on properties of such laws.

###  3.2. Properties

Property 1. If $\psi_{1}$ and $\psi_{2}$ are two idecomp cha.f, then
$\psi=\psi_{1}\psi_{2}$ is an idecomp cha.f.

Proof. Suppose that $\psi_{1}$ and $\psi_{2}$ are two idecomp cha.f and let
$p\geq 1$. Thus $\psi^{1/p}=\psi_{1}^{1/p}\psi_{2}^{1/p}$ is the cha.f of the
convolution product of the probability measures associated to the cha.f
$\psi_{i}^{1/p}$ ($i\in\\{1,2\\}$).

Property 2. If $\psi$ is an idecomp cha.f, the conjugate $\overline{\psi}$ is
also an idecomp cha.f and the complex square norm $\|\psi\|^{2}$ is an idecomp
cha.f.

Proof. Let $\psi$ be the cha.f of $X$, i.e.,
$\psi(t)=\psi_{X}(t)=\mathbb{E}(e^{itX})$, it is clear that

$\mathbb{E}(e^{-itX})=\mathbb{E}(\overline{e^{itX}})=\overline{\mathbb{E}(e^{itX})}=\overline{\psi}(t).$

This and $\mathbb{E}(e^{-itX})=\psi_{-X}(t)$ for $t\in\mathbb{R}$ show that
$\psi$ is a cha.f. It is also direct to see that $X$ and $-X$ are idecomp or
non-idecomp at the same time or not. Finally by Property 1,
$\|\psi\|^{2}=\psi\overline{\psi}$ is idecomp if $\psi$ is.

Property 3. If $\psi$ is an idecomp cha.f, then $\psi^{1/n}$ converges to 1
everywhere, as $n\rightarrow+\infty$.

Proof. Suppose that $\psi$ is an idecomp cha.f. Let us denote, for all $n\geq
1$, $\psi_{n}=\psi^{1/n}$, that is a cha.f. But $\|\psi\|\leq 1$ and
$\|\psi_{n}\|^{2}=\|\psi\|^{2/n}$ converges to $g$ with $g=0$ on $\psi=0$ and
$g=1$ on $\psi\neq 0$. Let us show that $\psi$ cannot take the null value.
Indeed $\psi$ is continuous (at zero in particular) and $\psi(0)=1$. So
$\psi>1/2$ on an interval $]-r,r[$, $r>0$ and next $g=1$ on $]-r,r[$. But the
function $h\equiv 1$ is the cha.f of the random variable $Z=0$. By Proposition
in Billinsgley  (1968) (see page 388), we get that $g=h$ and then $g=1$
everywhere, so $\|\psi_{n}\|^{2}\rightarrow 1$. This ensures that $\psi$ does
not take the null value. Finally, we get rid of the norm by

$\psi^{1/n}=\exp\left(\frac{1}{n}\log\psi\right)\rightarrow 1\ as\
n\rightarrow+\infty.$

Property 4. Let $(\psi_{n})_{n\geq 1}$ be a sequence of idecomp cha.f’s such
that $\psi_{n}\rightarrow\psi$ and $\psi$ is continuous at zero. Then $\psi$
is an idecomp cha.f.

Proof. Let $\mathcal{C}_{fid}\ni\psi_{p}\rightarrow\psi$ and $\psi$ is
continuous at zero. For any fixed $q\geq 1$,
$|\psi_{p}|^{2/q}\rightarrow|\psi|^{2/q}$. Since the $|\psi_{p}|^{2/q}$, are
cha.f and $|\psi^{2/q}|$ is continuous at zero, it comes that $\psi|^{2/q}$ is
a cha.f for any $q\geq 1$. So $|\psi|^{2}$ is an idecomp cha.f and by property
3, $\psi$ is nonwhere zero and next

$\psi_{p}^{1/q}=\exp\left(\frac{1}{q}\log\psi_{p}\right)\rightarrow\exp\left(\frac{1}{q}\log\psi\right)=\psi^{1/q}$

is a cha.f by the Levy continuity theorem.

Property 5. An cha.f $\psi$ is idecomp if and only if it is limit of a
sequence of products Poisson type cha.f.

Proof. If $\psi$ is a limit of a sequence of products Poisson type cha.f, it
is idecomp by Property 4, since products of Poisson type cha.f are idecomp
cha.f.

Conversely, let us be given an idecomp cha.f $\psi$. Since $\psi$ is non-where
equal to zero (Property 3), we have

$\log\psi=\lim_{p\rightarrow+\infty}p(\psi^{1/p}-1).$

For $p\geq 1$, let us denote by $F_{p}$ the cdf associated with the cha.f
$\psi_{p}=\psi^{1/p}$. So we have

$\Psi_{p}(t)=p(f^{1/p}-1)=\int p\left(e^{itx}-1\right)\ dF_{p}(x).$

Since the function $\hookrightarrow p\left(e^{itx}-1\right)$ is bounded on
$\mathbb{R}$, it is locally integrable and $\lambda_{F_{p}}$ is a finite
measure, we may apply Lebesgue Dominated theorem and we can conclude that for
any fixed $p\geq 1$,

$\Psi_{p}(t)=p(f^{1/p}-1)=\lim_{0<a\rightarrow+\infty}\int_{-a}^{a}p\left(e^{itx}-1\right)\
dF_{p}(x)=:\lim_{0<a\rightarrow+\infty}\Psi_{p,a}.$

By continuity of the integrand, the integral $\Psi_{p,a}$ is limit of Riemann-
Stieltjes, which are of the form

$\sum_{1\leq j\leq k(p,a)}pb_{j,p}\left(e^{ic_{j,p}u}-1\right),$

which are sums of logarithms of Poisson type cha.f. Hence $\exp(\Psi_{p,a})$
are cha.f and next $\exp(\Psi_{p})$ is a cha.f as limit of the sequence
$\exp(\Psi_{p,a})$.

Finally $\psi$ is limit of cha.f of the form $\exp(\Psi_{p})$, which is a
sequence of products of Poisson type cha.f.

Property 6. A cha.f is idecomp if and only if it is limit of a sequence of
products of cha.f of Poisson type laws.

Proof Let $\psi$ be a cha.f. Let $\psi_{p}$ a product of cha.f of type Poisson
laws

$\psi_{p}(t)=\prod_{j=1}^{k(p)}\exp\left(ia_{j,p}t+b_{j,p}\left(e^{ic_{j,p}t}-1\right)\right),\
t\in\mathbb{R},$

where the $a_{j,p}$’s and $c_{j,p}$’s are real numbers and the $b_{j,p}$’s
positive numbers. We have, for $p$ fixed and for $q\geq 1$

$\psi_{p}(t)^{1/q}=\exp\left(i\left\\{\frac{1}{q}\sum_{j=1}^{k(p)}a_{j,p}\right\\}t+\frac{1}{q}\sum_{j=1}^{k(p)}b_{j,p}\left(e^{ic_{j,p}t}-1\right)\right),\
\ t\in\mathbb{R}.$

This is still a product of cha.f’s of type Poisson type laws and is a cha.f.
If $\psi_{p}\rightarrow\psi$, thus by Property 4, $\psi$ is idecomp.

We will need more facts on cha.f’s that we will introduce when needed,

We begin by studying the case of bounded variances. First, we deal with three
important results that constitute the pillars of the current theory.

###  3.3. The three pillars of that theory

In this subsection, we assume that both the UAN and the BVH hold.

######  Lemma 1.

(Comparison Lemma) The complex function $\log f_{k,n}$ is well-defined and for
any $u\in\mathbb{R}$

$\sum_{k=1}^{k(n)}\left\\{\log f_{k,n}(u)-(f_{k,n}(u)-1)\right\\}\rightarrow
0,$

as $n\rightarrow+\infty$.

Proof. Let $u\in\mathbb{R}$ fixed and $n\geq 1$. Then for any
$k\in\\{1,\cdots,k(n)\\}$, we have the one order expansion

$f_{k,n}(u)=1+\theta_{k,n}u^{2}\sigma_{k,n}^{2}/2,$

with $|\theta_{k,n}|<1$ and $|\circ|$ stands for the norm in $\mathbb{C}$ or
the absolute value when applied to real numbers. In all this chapter, numbers
of the form $\theta_{\circ}$, possibly written with primes or double primes,
are only required to have norms less than one and their values are not
important. So, we get

$\max_{1\leq k\leq k(n)}|f_{k,n}(u)-1|\leq\frac{u^{2}B_{n}}{2}\rightarrow 0\
as\ \ n\rightarrow+\infty.$

Next for $v_{k,n}=\theta_{k,n}u^{2}\sigma_{k,n}^{2}/2$, we surely have that
$\max_{1\leq k\leq k(n)}|v_{k,n}|\leq(u^{2}B_{n})/2$ goes to zero. We also
have for all $u\mathbb{R}$,

$\displaystyle\log f_{k,n}(u)$ $\displaystyle=$
$\displaystyle\log(1+(f_{k,n}(u)-1))=\log(1+v_{k,n})=v_{k,n}+\theta_{k,n}^{\prime}v_{k,n}^{2}$
$\displaystyle=$
$\displaystyle(f_{k,n}(u)-1)+\theta_{k,n}^{\prime}v_{k,n}^{2},$

which leads to, as $n\rightarrow+\infty$,

$\displaystyle\left|\sum_{k=1}^{k(n)}\left\\{\log
f_{k,n}(u)-(f_{k,n}(u)-1)\right\\}\right|$ $\displaystyle\leq$
$\displaystyle\sum_{1}^{k(n)}|\theta_{k,n}^{\prime}|v_{k,n}^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{k(n)}\frac{u^{4}}{4}|\theta_{k,n}|^{2}\sigma_{k,n}^{4}\
\ \ \ (L3)$ $\displaystyle\leq$
$\displaystyle\frac{u^{4}B_{n}}{4}\sum_{k=1}^{k(n)}\sigma_{k,n}^{2}\ \ \ \
(L4)$ $\displaystyle\leq$ $\displaystyle\frac{cu^{4}B_{n}}{4}\rightarrow 0.$

The proof of Lemma 1 is over. $\square$

Now, let us use new expressions of the results in Lemma 1. Since the variables
$X_{k,n}$ are centered, we have

$\forall n\geq 1,\ \forall 1\leq k\leq k(n),\ \int X_{k,n}\ d\mathbb{P}=\int
x\ dF_{k,n}(x)=0\ and\ \ \int x\ \ dF_{k,n}(x)=\sigma^{2}_{k,n}.$

Let us set, for $n\geq 1$,

$\psi_{n}(u)\equiv\sum_{k=1}^{k(n)}(f_{k,n}(u)-1)=\sum_{k=1}^{k(n)}\int\left(e^{iux}-1\right)\
dF_{k,n}(x),\ u\in\mathbb{R}.$

By using the remark that $\mathbb{E}X_{k,n}=0$, i.e. $\int x\ dF_{k,n}(x)=0$,
we get

$\displaystyle\psi_{n}(u)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{k(n)}\int\left(e^{iux}-1-iux\right)\ dF_{k,n}(x)$
$\displaystyle=$ $\displaystyle\int\left(e^{iux}-1-iux\right)\
\sum_{k=1}^{k(n)}dF_{k,n}(x)$ $\displaystyle=$
$\displaystyle\int\frac{1}{x^{2}}\left(e^{iux}-1-iux\right)\
x^{2}\sum_{k=1}^{k(n)}dF_{k,n}(x),\ u\in\mathbb{R}.$

But, by putting

$dK_{n}(x)=x^{2}\sum_{k=1}^{k(n)}dF_{k,n}(x),$

we get

$\psi_{n}(u)=\int\frac{1}{x^{2}}\left(e^{iux}-1-iux)\right)\ dK_{n}(x).$

Finally, Lemma 1 can be expressed as

######  Lemma 2.

$\forall
u\in\mathbb{R},\log\left(\prod_{k=1}^{k(n)}f_{k,n}(u)\right)-\psi_{n}(u)\rightarrow
0,\ as\ n\rightarrow+\infty,$

where

$dK_{n}(x)=x^{2}\sum_{k=1}^{k(n)}dF_{k,n}(x)$

and

$\psi_{n}(u)=\int\frac{1}{x^{2}}\left(e^{iux}-1-iux\right)\ dK_{n}(x).$

This lemma becomes the second pillar. The third is the following

######  Lemma 3.

For any $n\geq 1$, $\exp(\Psi_{n})$ is an idecomp cha.f and is the cha.f of a
centered random variable of variance

$\int\ dK_{n}(x)=s_{n}^{2}.$

Proof. Let $n\geq 1$ be fixed. We have

$\Psi_{n}(u)=\int\ g(u,x)\ dK_{n}(x)\ with\
g(u,x)=\frac{e^{iux}-1-iux}{x^{2}},\ x\in\mathbb{R}.$

Clearly $g$ is continuous on $\mathbb{R}\times\mathbb{R}^{\ast}$ (with
$\mathbb{R}^{\ast}=\mathbb{R}\setminus\\{0\\}$) and for $u$ fixed, $g(u,0)$ is
the extension of $g(u,x)$ by limit, since an expansion at zero gives

$g(u,x)=\frac{1+iux-u^{2}x^{2}/2-1-iux+O(x^{3})}{x^{2}}\rightarrow-u^{2}/2\
as\ x\rightarrow 0.$

So, for $u$ fixed, $x\mapsto g(u,x)$ is continuous everywhere. Moreover we
have

(3.2) $\forall u\in\mathbb{R},\ \forall x\in\mathbb{R}^{\ast},\
|g(u,x)|\leq\frac{2}{x^{2}}+\frac{|u|}{|x|}$

and

$\int x^{-2}\ dK_{n}(x)=\sum_{k=1}^{k(n)}\int dF_{k,n}(x)=k(n),$

and, by using $(|x|\leq 1+x^{2})$

$\displaystyle\int|x|^{-1}\ dK_{n}(x)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{k(n)}\int\frac{x^{2}}{|x|}\ dF_{k,n}(x)$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{k(n)}\int|x|\ dF_{k,n}(x)$
$\displaystyle\leq$ $\displaystyle\sum_{k=1}^{k(n)}\int(1+x^{2})\ dF_{k,n}(x)$
$\displaystyle=k(n)+s_{n}^{2}.$

We conclude that $g(u,x)$ is bounded by $g_{0}(x)=2x^{-2}+|ux^{-1}|$ which is
$K_{n}$-integrable. So by the dominated convergence theorem, $\Psi_{n}$ is
continuous at zero. Also, as an improper Riemann-Stieltjes integral, for
$\varepsilon>0$ fixed, we can find a number $A>0$ such that for $a\geq A$

$\left|\psi_{n}(u)-\Psi_{n,a}(x)\right|<\varepsilon\ with\
\Psi_{n,a}(u)=\int_{-a}^{a}g(u,x)\ dK_{n}(x).$

Now, since $\Psi_{n,a}(u)$ is continuous, it a limit of a sequence of of
Riemann-Stieltjes sums: there exists a partition of $[-a,a]$

$-a=x_{0,p}<\cdots<x_{j-1,p}<x_{j,p}<\cdots<x_{\ell(p),p}=a$

and a sequence of points $c_{j,p}\in(x_{j,p},x_{j+1,p})$, $0\leq
j\leq\ell(p)-1$,

$S_{p}(u)=\sum_{j=0}^{\ell(p)-1}\\{K_{n}(x_{j+1,p})-K_{n}(x_{j,p})\\}g(u,c_{j,p})\rightarrow\Psi_{n,a}(u),$

as $\max\\{x_{j+1,p}-x_{j,p},\ 1\leq j\leq\ell(p)-1\\}\rightarrow 0$ as
$p\rightarrow+\infty$. We may choose all the $c_{j,p}$ not null from the
interior of $(x_{j,p},x_{j+1,p})$ ($x_{j,p}<x_{j+1,p}$). We have

$\displaystyle S_{p}(u)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\ell(p)-1}\frac{\lambda_{K_{n}}(]x_{j,p},x_{j+1,p}])}{c_{j,p}^{2}}\biggr{(}e^{ic_{j,p}u}-1-ic_{j,p}u\biggr{)}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\ell(p)-1}-i\frac{\lambda_{K_{n}}(]x_{j,p},x_{j+1,p}])}{c_{j,p}^{2}}c_{j,p}u+\frac{\lambda_{K_{n}}(]x_{j,p},x_{j+1,p}])}{c_{j,p}^{2}}\biggr{(}e^{ic_{j,p}u}-1\biggr{)}$
$\displaystyle=:$
$\displaystyle\sum_{j=0}^{\ell(p)-1}-i\mu_{j,p}u+\lambda_{j,p}\biggr{(}e^{ic_{j,p}u}-1\biggr{)},$

with

$\lambda_{j,p}=\frac{\lambda_{K_{n}}(]x_{j,p},x_{j+1,p}])}{c_{j,p}^{2}}\ and\
\mu_{j,p}=\frac{\lambda_{K_{n}}(]x_{j,p},x_{j+1,p}])}{c_{j,p}^{2}}c_{j,p}.$

We clearly see that $\exp(S_{p})$ is the product of Poisson type cha.f
converging to $\exp(\Psi_{n})$ as $p\rightarrow+\infty$ and
$a\rightarrow+\infty$. But we also have that $\exp(\Psi_{n})$ is continuous.
So by the Lévy continuity theorem, $\exp(\Psi_{n})$ is a cha.f and it is
idecomp by Property 4 (see page 3.2).

Let us study the differentiability of $\Psi_{n}$. We have

$\left|\frac{\partial g(u,x)}{\partial
x}\right|=\left|\frac{ix(e^{iux}-1}{x^{2}})\right|\leq\frac{2}{|x|}\in\mathcal{L}^{1}({K_{n}}),$

and hence

$\Psi_{n}^{\prime}(u)=\int\frac{ix(e^{iux}-1}{x^{2}})\ dK_{n}(x)\ and\
\Psi_{n}^{\prime}(0)=0.$

Also

$\left|\frac{\partial^{2}g(u,x)}{\partial^{2}x}\right|=\left|\frac{-x^{2}e^{iux}}{x^{2}}\right|=1\in\mathcal{L}^{1}(K_{n}),$

and hence

$\Psi_{n}^{\prime\prime}(u)=-\int e^{iux}\ dK_{n}(x)\ and\
\Psi_{n}^{\prime\prime}(0)=-s_{n}^{2}.$

Finally, let $Z_{n}$ be a r.v. with cha.f $\exp(\Psi_{n})$. The first and
second derivatives of $\exp(\Psi_{n})$ are

$\Psi_{n}^{\prime}(u)\exp(\Psi_{n}(u))\ and\
\\{\Psi_{n}^{\prime\prime}(u)\exp(\Psi_{n}(u))+\left(\Psi_{n}^{\prime}(u)\right)^{2}\exp(\Psi_{n}(u))\\}$

taking the values

$\Psi_{n}^{\prime}(0)\exp(\Psi_{n}(0))=0\ and\
\\{\Psi_{n}^{\prime\prime}(0)\exp(\Psi_{n}(0))+\left(\Psi_{n}^{\prime}(0)\right)^{2}\exp(\Psi_{n}(0))\\}=-s_{n}^{2}.$

We conclude that $\mathbb{E}Z_{n}=0$ and $\mathbb{V}ar(Z_{n})=s_{n}^{2}$. The
relation

$\Psi_{n}^{\prime\prime}(u)=-\int e^{iux}\ dK_{n}(x)\ \ \ (C)$

shows that $\Psi_{n}^{\prime\prime}(u)$ characterizes $K_{n}$ and vice-versa.
Now, for two functions $\Psi_{n}$ and $\Phi_{n}$ such that
$\Psi_{n}^{\prime\prime}=\Phi_{n}^{\prime\prime}$ with
$\Psi_{n}(0)=\Phi_{n}(0)=0$ and $\Psi_{n}^{\prime}(0)=\Phi_{n}^{\prime}(0)=0$,
we have

$\forall u\in\mathbb{R},\ \ \Psi_{n}^{\prime}(u)=\Phi_{n}^{\prime}(u)+d_{1},$

and by applying this for $u=0$, we get $d_{1}=0$. Next, we have

$\forall u\in\mathbb{R},\ \ \Psi_{n}(u)=\Phi_{n}(u)+d_{2},$

and by applying this for $u=0$, we get $d_{2}=0$. So $\Psi=\Phi$ and we have
the following fact.

######  Fact 1.

$K_{n}$ characterizes $\Psi_{n}$ and vice-versa. $\blacksquare$

##  4\. The weak convergence theorem of summands under the BVH and the UAN
Condition

###  4.1. The Central limit theorem for centered, independent and square
integrable random variables

We are going to conclude the discussion above to find solutions of the CLT
problem under the BVH and the UAN Condition. We will have two studies from
which of them we draw a final conclusion.

Study (A). From Lemma 2 and from the notations above, we have

$\forall\ t\in\mathbb{R},\ \Psi_{S_{n}}(t)-\exp(\Psi_{n}(t))\rightarrow 0\ as\
n\rightarrow+\infty.$

But $\exp(\Psi_{n}(\circ))$ is an idecomp cha.f for any $n\geq 1$ and is
linked to

(4.1) $\Psi_{n}(u)=\int\ g(u,x)\ d\lambda_{K_{n}}(x)\ with\
g(u,x)=\frac{e^{iux}-1-iux}{x^{2}},\ x\in\mathbb{R},$

where $\lambda_{K_{n}}$ is the Lebesgue-Stieltjes measure associated with the
df $K_{n}$. Now, we are using the weak convergence theory of bounded measures
on $\mathbb{R}$ as exposed Chapter 6 in Lo et al. (2016).

Direct part. Let us suppose that $\lambda_{K_{n}}$ pre-weakly converges to
some df $\lambda_{K}$, i.e., (for $C(K)$ standing for the set continuity
points of $K$),

$\forall x\in C(K),\ K_{n}(x)\rightarrow K(x)\ as\ n\rightarrow+\infty.$

By Part (i) of Proposition 37 in Chapter 6 in Lo et al. (2016), we have

$\lambda_{K}(\mathbb{R})\leq\liminf_{n\rightarrow+\infty}\lambda_{K_{n}}(\mathbb{R})\leq
c,$

since, for any $n\geq 1$,

$\lambda_{K_{n}}(\mathbb{R})=\sum_{k=1}^{k(n)}\int
x^{2}dF_{k,n}(x)=\sum_{k=1}^{k(n)}\mathbb{V}ar(X_{k,n})\leq c$

from the BVH. Hence the pre-weak limit $\lambda_{K}$ is a bounded measure. Now
we apply the integral Helly-Bray theorem as in Theorem 30 in Chapter 6 in Lo
et al. (2016) to (4.1) (See above). By (3.2), for any fixed real number $u$,
the function $g(u,x)$ (in $s$) in (4.1) is continuous and satisfies
$g(\pm\infty)=0$. So by the cited Helly-Bray integral theorem, we have

$\forall u\in\mathbb{R},\ \Psi_{n}(u)\rightarrow\Psi_{K}(u)=\int\ g(u,x)\
d\lambda_{K}(x)=:\int\frac{e^{ix}-1-iux}{x^{2}}\ d\lambda_{K}(x).$

Now, from the expression of $\Psi_{K}(u)$ and from (3.2), we see that
$\Psi_{K}(u)$ is a parametrized (in $u$) integral and by the dominated
convergence theorem, $\Psi_{K}(u)$ is continuous and $\Psi_{K}(0)=0$.
Therefore,

$\forall u\in\mathbb{R},\
\exp(\Psi_{n}(u))\rightarrow\exp(\Psi_{K}(u))=:f_{K}(u).$

Since $f_{K}(\circ)$ is continuous at $zero$ and $f_{K}(0)=1$, we get by the
Lévy continuity theorem (See Theorem 11 in Chapter 3 in Lo et al. (2016)), we
conclude that $f_{K}$ is cha.f and by designating by $\mathcal{K}_{K}$ the
probability law associated to the cha.f $f_{K}$, we have

$S_{n}\rightsquigarrow\mathcal{K}_{K}.$

By Property 4 (see page 3.2 above), $\mathcal{K}$ is an idecomp probability
law, following the fact that each $\exp(\Psi_{n}(\circ))$, $n\geq 1$, is an
idecomp cha.f.

Indirect Part. Suppose that for some df $K_{0}$,

$S_{n}\rightsquigarrow\mathcal{K}_{K_{0}},$

where $\mathcal{K}_{K_{0}}$ is the probability law associated to $K_{0}$. We
are going to use a Prohorov’s type argument. By the asymptotic tightness
theorem (See Theorem 29 in Lo et al. (2016)), any sub-sequence
$(\lambda_{K_{n_{j}}})_{j\geq 1}$ of $(\lambda_{K_{n}})_{n\geq 1}$ contains a
sub-sequence $(\lambda_{K_{n_{j_{\ell}}}})_{\ell\geq 1}$ pre-weakly converging
to some $\lambda_{K^{\ast}}$. By the direct part,

$S_{n_{j_{\ell}}}\rightsquigarrow\mathcal{K}_{K^{\ast}},$

where $\mathcal{K}_{K^{\ast}}$ is associated to a cha.f
$f_{K^{\ast}}=\exp(\Psi_{K^{\ast}})$, with

$\forall u\in\mathbb{R},\
\exp(\Psi_{n_{j_{\ell}}}(u))\rightarrow\exp(\Psi_{K^{\ast}}(u))=:f_{K}(u)$

and

$\forall u\in\mathbb{R},\ \Psi_{K^{\ast}}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\
d\lambda_{K^{\ast}}(x).$

By uniqueness of the weak limit,
$\mathcal{K}_{K^{\ast}}=_{d}\mathcal{K}_{K_{0}}$. Then each sub-sequence of
$(K_{n})_{n\geq 1}$ contains a sub-sequence converging to $K_{0}$. We conclude
that by Prohorov theorem

$K_{n}\ \rightsquigarrow_{pre}K_{0}.$

In both parts, $\mathbb{V}ar(K)<+\infty$ and by Fact 1 applied to $K$, we may
conclude that $K$ and $\Psi_{K}$ characterizes one the other.

We conclude as follows.

######  Theorem 1.

Under the BVH and the UAN Condition for summands of independent, centered and
square integrable real valued random variables, we have:

(a) If

$S_{n}\rightsquigarrow\mathcal{K},$

where $\mathcal{K}$ is a probability law, then is $\mathcal{K}$ is idecomp.

(b) For any idecomp probability law $\mathcal{K}$ of a centered and square
integrable random variable $Z$, for which for any $n\geq 1$, there exists
$X_{1,n}$, $\cdots$, $X_{n,n}$ independent and of same law (they are
necessarily centered and square integrable) such that

$Z=X_{1,n}+\cdots+X_{n,n}=:S_{n}.$

Then clearly, $\mathcal{K}$ is a weak limit of summands of independent,
centered and square integrable real valued random variables under the BVH and
the UAN Condition.

(3) We have, under the BVH and the UAN Condition,

$S_{n}\rightsquigarrow\mathcal{K}_{K},$

for some df $K$, if and only if (using the notation stated above)

$K_{n}\rightsquigarrow_{pre}K.$

Moreover

$\Psi_{K}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\ dK$

and $K$ characterize each other, and $\exp(\Psi_{K}(\circ))$ is the
characteristic function of $\mathcal{K}_{K}$.

Study (B). Here, we suppose that the VCH and the UAN Condition hold. We begin
by remarking that the Comparison Lemma 1 holds since formula (3.3) (page 3.3)
holds with the use of the VCH in Line (L4).

Direct part. Let $K_{n}\rightsquigarrow K$. In particular $K_{n}\
\rightsquigarrow_{pre}K$. By the direct part of Study (A), we still have

(4.2)
$\lambda_{K}(\mathbb{R})\leq\liminf_{n\rightarrow+\infty}\lambda_{K_{n}}(\mathbb{R})=c,$

and

$S_{n}\rightsquigarrow\mathcal{K}_{K}.$

Actually, by weak convergence, we exactly have

$\lambda_{K}(\mathbb{R})=\lim_{n\rightarrow+\infty}\lambda_{K_{n}}(\mathbb{R})=c,$

but this not play any role for the direct part.

Indirect part. Let

$S_{n}\rightsquigarrow\mathcal{K}_{K},$

for some df $K$. By the indirect part of Study (A), we still have

$K_{n}\ \rightarrow_{pre}K.$

Now if, for $Z\sim\mathcal{K}_{K}$ such that $\mathbb{V}ar(Z)=c$, we have that
$\lambda_{K}(\mathbb{R})=\mathbb{V}ar(Z)$ and then

$\lim_{n\rightarrow+\infty}\lambda_{K_{n}}(\mathbb{R})=\lambda_{K}(\mathbb{R})\
\ and\ \ K_{n}\ \rightarrow_{pre}K.$

By Proposition 37 in Chapter 6 in Lo et al. (2016), we conclude that
$K_{n}\rightsquigarrow K$ as $\rightarrow+\infty$.

We conclude as follows.

######  Theorem 2.

Under the VCH and the UAN Condition for summands of independent, centered and
square integrable real valued random variables, we have the following
characterization. If $\mathcal{K}_{K}$ is associated with a random variable
$Z$ such that $\mathbb{V}ar(Z)=c$, where $c$ is the limit in the VCH, then we
have

$S_{n}\rightsquigarrow\mathcal{K}_{K},$

if and only if

$K_{n}\rightsquigarrow K.$

###  4.2. The Central limit theorem for non-centered, independent and square
integrable random variables

Let us re-conduct all the notations in Subsection 4.1. Let us denote

$\biggr{(}\forall n\geq 1,\ \forall 1\leq k\leq k(n),\
\mathbb{E}X_{k,n}=a_{k,n}\biggr{)}\ and\ \biggr{(}\forall n\geq 1,\
\sum_{k=1}^{k(n)}a_{k,n}=a_{n}\biggr{)}$

Let us write

$\forall n\geq 1,\ \
S_{n}=(S_{n}-a_{n})+a_{n}=\sum_{k=1}^{k(n)}(X_{k,n}-a_{k,n})+a_{n}=:S_{n}^{\ast}+a_{n}.$

Let us denote by $F_{k,n}^{\ast}$ the cdf of $(X_{k,n}-a_{k,n})$ for $n\geq 1$
and $1\leq k\leq k(n)$,

$\forall u\in\mathbb{R},\Psi_{K^{\ast}}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\
dK^{\ast}(x)$

and

$\forall n\geq 1,\ \forall u\in\mathbb{R},\ \
\Psi_{K_{n}^{\ast}}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\ dK_{n}^{\ast}(x)$

with

$\forall n\geq 1,\ \forall x\in\mathbb{R},\ \
K^{\ast}_{n}(x)=\int_{-\infty}^{x}y^{2}\sum_{k=1}^{k(n)}\ dF_{k,n}^{\ast}(y).$

Direct part. If $K^{\ast}_{n}(x)\rightsquigarrow_{pre}K^{\ast}$ and
$a_{n}\rightarrow a$, then

$S_{n}\rightsquigarrow\mathcal{K}_{K^{\ast}}+a=\mathcal{K}_{0}.$

Moreover, the cha.f of $\mathcal{K}_{K^{\ast}}$ is
$\exp(\Psi_{K^{\ast}}(\circ))$ and next the cha.f of $\mathcal{K}_{0}$ is

$\forall u\in\mathbb{R},\
\Psi(u)=\exp\biggr{(}iau+\Psi_{K^{\ast}}(u)\biggr{)}.$

Indirect part. Suppose that

$S_{n}\rightsquigarrow\mathcal{K}_{0},$

where $\mathcal{K}_{0}$ is associated with an a.s finite random variable $Z$.
Then $b=\limsup_{n\rightarrow+\infty}a_{n}$ is finite. Otherwise consider a
sub-sequence $a_{n_{\ell}}\rightarrow+\infty$ as $\ell\rightarrow+\infty$. So
$S_{n_{\ell}}=S_{n_{\ell}}^{\ast}+a_{n_{\ell}}$ necessarily weakly converges
to $Z$, where by Theorem 1, $S_{n_{\ell}}^{\ast}\rightsquigarrow Z^{\ast}$, of
law $\mathcal{K}_{K^{\ast}}$ and $Z^{\ast}$ finite a.s, and hence $Z$ is a.s
infinite. Hence $b=\limsup_{n\rightarrow+\infty}a_{n}$ if finite. Now, each
sub-sequence of $(a_{n})_{n\geq 1}$ contains a sub-sequence
$(a_{n^{\prime}})_{n^{\prime}\geq 1}$ is converging to $a$ finite. By the
argument given above, $S_{n^{\prime}}^{\ast}$ weakly converges to some
$\mathcal{K}_{K^{\ast}}$. By prohorov’s criteria, $S_{n}^{\ast}$ weakly
converges to $\mathcal{K}_{K^{\ast}}$ and $S_{n}$ weakly converges to
$\mathcal{K}_{K^{\ast}}+a=_{d}\mathcal{K}_{0}$. The later inequality shows
that all converging subsequences of $(a_{n})_{n\geq 1}$ converge to the same
number $a$. Finally

$S_{n}\rightsquigarrow\mathcal{K}_{K^{\ast}}+a,$

with $a_{n}\rightarrow a$. Let us summarize the discussions as follows.

######  Theorem 3.

Under the BVH and the UAN Condition for summands of independent and square
integrable real valued random variables, we have the following
characterization. Let us denote

$\biggr{(}\forall n\geq 1,\ \forall 1\leq k\leq k(n),\
\mathbb{E}X_{k,n}=a_{k,n}\biggr{)}\ and\ \biggr{(}\forall n\geq 1,\
\sum_{k=1}^{k(n)}a_{k,n}=a_{n}\biggr{)};$

$\forall n\geq 1,\ \
S_{n}=(S_{n}-a_{n})+a_{n}=\sum_{k=1}^{k(n)}(X_{k,n}-a_{k,n})+a_{n}=:S_{n}^{\ast}+a_{n};$

$F_{k,n}^{\ast}$ the cdf of $(X_{k,n}-a_{k,n})$ for $n\geq 1$ and $1\leq k\leq
k(n)$;

$\forall u\in\mathbb{R},\Psi_{K^{\ast}}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\
dK^{\ast}(x)$

and, finally,

$\forall n\geq 1,\ \forall u\in\mathbb{R},\ \
\Psi_{K^{\ast}_{n}}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\ dK_{n}^{\ast}(x)$

with

$\forall n\geq 1,\ \forall x\in\mathbb{R},\ \
K^{\ast}_{n}(x)=\int_{-\infty}^{x}y^{2}\sum_{k=1}^{k(n)}\ dF_{n}^{\ast}(y).$

We have the following facts.

(i) If $K_{n}^{\ast}\rightsquigarrow_{pre}K^{\ast}$ and $a_{n}\rightarrow a$,
then

$S_{n}\rightsquigarrow\mathcal{K}_{K^{\ast}}+a.$

(ii) If

$S_{n}\rightsquigarrow\mathcal{K}_{0}$

where $\mathcal{K}_{0}$ is associated to an a.s finite random variable $Z$,
then the sequence $(a_{n})_{n\geq 1}$ converges to a real number $a$ and

$\mathcal{K}_{0}=_{d}\mathcal{K}_{K^{\ast}}+a$

and

$K_{n}^{\ast}\rightsquigarrow_{pre}K^{\ast}.$

Moreover if the VCH holds at the place of the BVH and the variance of
$\mathcal{K}_{K^{\ast}}$ is equal to $c$, we have

$K_{n}^{\ast}\rightsquigarrow K^{\ast}$

in both parts (i) and (ii).

##  5\. Characterizations of two important examples

The two important limits of Gaussian law and Poisson law are very important.
In stochastic analysis, these laws allow to represent some stochastic process
into a discontinuous process (Poisson component) and a continuous process
(Gaussian part).

###  5.1. Gaussian limit

Let us suppose that the weak limit of the summands $(S_{n})_{n\geq 1}$ is the
standard Gaussian law

$\exp(\Psi_{K}(u))=\exp(-u^{2}/2),\ u\in\mathbb{R},$

i.e.

(5.1) $\Psi_{K}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\ dK(x)=-\frac{u^{2}}{2},\
u\in\mathbb{R}.$

But, for $\lambda_{K}=\delta_{0}$, that is, $K=1_{[0,+\infty[}$, we have

$\int\frac{e^{iux}-1-iux}{x^{2}}\
d\delta_{0}(x)=\left[\frac{e^{iux}-1-iux}{x^{2}}\right]_{x=0}=-\frac{u^{2}}{2}.$

We are going to rediscover the Lévy-Lynderberg-Feller (L2F) theorem as stated
in Lo  (2018b) (See Theorem 20 in page …).

######  Theorem 4.

Let $S_{n}=X_{1,n}+\cdots+X_{k(n),n}$ summands of centered and square
integrable random variables as denoted above such that

(5.2) $\sum_{1\leq k\leq k(n)}\mathbb{V}ar(X_{k,n})=\sum_{1\leq k\leq
k(n)}\sigma^{2}_{k,n}=1.$

For $\varepsilon>0$ and $n\geq 1$, let us denote the Lynderberg function as

(5.3) $g_{n}(\varepsilon)=\sum_{k=1}^{k(n)}\int_{(|x|\geq\varepsilon)}x^{2}\
dF_{k,n}(x).$

We have the following characterization:

(5.4) $S_{n}\rightsquigarrow\mathcal{N}(0,1)\ as\ n\rightarrow+\infty$

and

(5.5) $\max_{1\leq k\leq k(n)}\mathbb{V}ar(X_{k,n})\rightarrow 0\ as\
n\rightarrow+\infty$

if and only if, for any $\varepsilon>0$, the following Lynderberg criterion
holds:

(5.6) $g_{n}(\varepsilon)\rightarrow 0\ as\ n\rightarrow+\infty.$

Proof. Let us begin by linking the Lynderberg function as (5.3) with (5.5). We
have

(5.7) $\displaystyle\max_{1\leq k\leq k(n)}\mathbb{V}ar(X_{k,n})$
$\displaystyle=$ $\displaystyle\max_{1\leq k\leq k(n)}\int x^{2}dF_{k,n}(x)$
$\displaystyle=$ $\displaystyle\sum_{1\leq k\leq k(n)}\int x^{2}dF_{k,n}(x)$
$\displaystyle=$ $\displaystyle\sum_{1\leq k\leq
k(n)}\int_{(|x|\leq\varepsilon)}x^{2}dF_{k,n}(x)$ $\displaystyle+$
$\displaystyle\sum_{1\leq k\leq k(n)}\int_{(|x|>\varepsilon)}x^{2}dF_{k,n}(x)\
\ (L3)$ $\displaystyle=$ $\displaystyle\varepsilon^{2}+g_{n}(\varepsilon),$

where we used (5.2) in the first summation in Line (L3). By letting
$n\rightarrow+\infty$ first and next, by letting $\varepsilon\rightarrow 0$,
we get that the Lynderberg criterion implies (5.5). We have:

######  Fact 2.

The Lynderberg criterion (5.6) implies (5.5), which in turn implies the UAN
hypothesis.

Now, Let us prove both implications.

Direct implication. Suppose that (5.4) and (5.5) hold. So the BVH (by (5.2))
and the UAN Condition holds (by Fact 2). Actually the BVH (5.2) is also a VCH
Conditions. So may apply both Theorems 1 and 2. By applying Theorem 1, we have

$\forall x\in C(K),\ K_{n}(x)=\sum_{1\leq k\leq k(n)}\int_{-\infty}^{x}y^{2}\
dF_{k,n}(y)\rightarrow 1_{(x\geq 0)}$

since $1_{(x\geq 0)}$ is the df associated with $\delta_{0}$. Any $x>0$ is in
$C(K)$ and then

$\displaystyle K_{n}(x)=\sum_{1\leq k\leq k(n)}\int_{-\infty}^{x}y^{2}\
dF_{k,n}(y)\rightarrow 1\ $ $\displaystyle\Leftrightarrow\sum_{1\leq k\leq
k(n)}\int y^{2}\ dF_{k,n}(y)-\sum_{1\leq k\leq k(n)}\int_{(y>x)}y^{2}\
dF_{k,n}(y)\rightarrow 1$ $\displaystyle\Leftrightarrow 1-\sum_{1\leq k\leq
k(n)}\int_{(y>x)}y^{2}\ dF_{k,n}(y)\rightarrow 1.$

Hence

$\forall x>0,\ g_{n,1}(x):=\sum_{1\leq k\leq k(n)}\int_{(y>x)}y^{2}\
dF_{k,n}(y)\rightarrow 0.$

Next, any $x<0$ is in $C(K)$ and then

(5.8) $\displaystyle K_{n}(x)$ $\displaystyle=$ $\displaystyle\sum_{1\leq
k\leq k(n)}\int_{-\infty}^{x}y^{2}\ dF_{k,n}(y)\rightarrow 0\ $
$\displaystyle\Leftrightarrow$ $\displaystyle\sum_{1\leq k\leq
k(n)}x^{2}\lambda_{F_{k,n}}(\\{x\\})+\sum_{1\leq k\leq k(n)}\int_{(y<x)}y^{2}\
dF_{k,n}(y)\rightarrow 0$ $\displaystyle\Leftrightarrow$
$\displaystyle\lambda_{K_{n}}(\\{x\\})+\sum_{1\leq k\leq
k(n)}\int_{(y<x)}y^{2}\ dF_{k,n}(y)\rightarrow 0.$

But, by Portmanteau Theorem (see Criterion (vi) of Theorem 2, page 47 in Lo et
al. (2016)), $\lambda_{K_{n}}(\\{x\\})\rightarrow\lambda_{K}(\\{x\\})$ since
$\partial\\{x\\}=\\{x\\}$ and hence $\lambda_{K}(\\{x\\})=K(x)-K(x-0)=0$ since
$x\in C(K)$.

Hence

$\forall x<0,\ g_{n,2}(x):=\sum_{1\leq k\leq k(n)}\int_{(y<x)}y^{2}\
dF_{k,n}(y)\rightarrow 0.$

By putting together the two last results, for any $\varepsilon>0$

$\displaystyle g_{n}(\varepsilon)$ $\displaystyle=$ $\displaystyle\sum_{1\leq
k\leq k(n)}\int_{(|y|>\varepsilon)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$
$\displaystyle\sum_{1\leq k\leq k(n)}\int_{(y>\varepsilon)}y^{2}\
dF_{k,n}(y)+\sum_{1\leq k\leq k(n)}\int_{(y<-\varepsilon)}y^{2}\ dF_{k,n}(y)$
$\displaystyle=$ $\displaystyle g_{n,1}(\varepsilon)+g_{n,2}(-\varepsilon)$
$\displaystyle\rightarrow$ $\displaystyle 0\ as\ n\rightarrow+\infty.\ \
\square$

Proof of the indirect implication. Let (5.6) holds. So, by Fact 2, (5.5) holds
and then the UAN is satisfied and the BVH is already satisfied as an
hypothesis of the theorem. Still by Theorem 1, (5.4) holds whenever

(5.9) $\forall x\in C(K),\ K_{n}(x)=\sum_{1\leq k\leq
k(n)}\int_{-\infty}^{x}y^{2}\ dF_{k,n}(y)\rightarrow 1_{(x\geq 0)}.$

Let us prove (5.9), by exploiting (5.6). We have $C(F)=(x<0)+(x>0)$. For
$x>0$, we have

$\displaystyle K_{n}(x)$ $\displaystyle=$ $\displaystyle\sum_{1\leq k\leq
k(n)}\int_{(y\leq x)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$ $\displaystyle
1-\sum_{1\leq k\leq k(n)}\int_{(y>x)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$
$\displaystyle 1-\sum_{1\leq k\leq k(n)}\int_{(|y|>x)}y^{2}\ dF_{k,n}(y)$
$\displaystyle=$ $\displaystyle 1-g_{n}(x)$ $\displaystyle\rightarrow$
$\displaystyle 1\ as\ n\rightarrow+\infty.\ \ \square$

For $x<0$, we have

$\displaystyle K_{n}(x)$ $\displaystyle=$ $\displaystyle\sum_{1\leq k\leq
k(n)}\int_{(y\leq x)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$
$\displaystyle\sum_{1\leq k\leq
k(n)}x^{2}\lambda_{F_{k,n}}(\\{x\\})+\sum_{1\leq k\leq k(n)}\int_{(y<x)}y^{2}\
dF_{k,n}(y)$ $\displaystyle=$ $\displaystyle\sum_{1\leq k\leq
k(n)}x^{2}\lambda_{F_{k,n}}(\\{x\\})+\sum_{1\leq k\leq
k(n)}\int_{(-y>-x)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$
$\displaystyle\sum_{1\leq k\leq
k(n)}x^{2}\lambda_{F_{k,n}}(\\{x\\})+\sum_{1\leq k\leq
k(n)}\int_{(|y|>-x)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$
$\displaystyle\lambda_{K_{n}}(\\{x\\})+\sum_{1\leq k\leq
k(n)}\int_{(|y|>-x)}y^{2}\ dF_{k,n}(y)$ $\displaystyle=$
$\displaystyle\lambda_{K_{n}}(\\{x\\})+g_{n}(-x).$

Now, by (5.6), $g_{n}(-x)\rightarrow 0$ and by using a similar technical in
line (5.8), $\lambda_{K_{n}}(\\{x\\})\rightarrow 0$.

So (5.9) holds and we have proved (5.4) and (5.5). $\blacksquare$

###  5.2. Poisson limit

The searched limit here is a translated Poisson law
$\mathcal{P}(b,\lambda)=b+\mathbb{P}(\lambda)$, with $b\in\mathbb{R}$ and
$\lambda>0$ of characteristic function

$\exp(\Psi_{K}(u))=\exp(ibu+\lambda(e^{iu}-1))=\exp(i(b+\lambda)u+\lambda(e^{iu}-1-iu)),\
u\in\mathbb{R},$

with

$\Psi_{K}(u)=i(b+\lambda)u+\Psi_{K^{\ast}}(u),\
\Psi_{K^{\ast}}(u)=\lambda(e^{iu}-1-iu),$

where $\exp(\Psi_{K^{\ast}}(\circ))$ is the cha.f of the centered Poisson law
$\mathcal{P}^{\ast}(\lambda)=(\mathcal{P}(\lambda)-\lambda)$.

Let us state the characterization theorem.

######  Theorem 5.

Let $S_{n}=X_{1,n}+\cdots+X_{k(n),n}$ be summands of independent and square
integrable random variables. As above, let $a_{k,n}=\mathbb{E}X_{k,n}$ and let
$F^{\ast}_{k,n}$ be the cdf of $X_{k,n}-a_{k,n}$. Let us introduce Lynderberg-
type functions, for $\varepsilon>0$ and $n\geq 1$, as

(5.10)
$g_{n,pois}(\varepsilon)=\sum_{k=1}^{k(n)}\int_{|x-1|>\varepsilon}x^{2}\
dF^{\ast}_{k,n}(x).$

Suppose that, as $n\rightarrow+\infty$ with
$\biggr{(}\text{{MVP(n)}}=\sum_{1\leq k\leq k(n)}\sigma^{2}_{k,n}\biggr{)}$,

(5.11) $B_{n}=\max_{1\leq k\leq k(n)}\sigma^{2}_{k,n}\rightarrow 0\ and\
\text{{MVP(n)}}\rightarrow\lambda.$

Let $b\in\mathbb{R}$. We have the following characterization.

(5.12) $S_{n}\rightsquigarrow\mathcal{P}(b,\lambda)\ as\ n\rightarrow+\infty$

if and only if,

(5.13) $\sum_{k=1}^{k(n)}\mathbb{E}(X_{k,n})=a_{n}\rightarrow a=b+\lambda\ as\
n\rightarrow+\infty$

and for any $\varepsilon>0$, the following Lynderberg Poisson-type criterion
holds:

(5.14) $g_{n,pois}(\varepsilon)\rightarrow 0\ as\ n\rightarrow+\infty.$

Proof. Based (5.11), the CVH and the UAN condition hold. We can apply Theorem
3. We study the limit of $\Psi_{K_{n}^{\ast}}(u)$, for any $u\in\mathbb{R}$ to

$\Psi_{K^{\ast}}(u)=\int\frac{e^{iux}-1-iux}{x^{2}}\
dK^{\ast}(x)=\lambda(e^{iu}-1-iu).$

Let $\lambda_{K^{\ast}}=\lambda\delta_{1}$, i.e., $K^{\ast}(x)=\lambda
1_{(x\geq 1)}$. Thus

$\int\frac{e^{iux}-1-iux}{x^{2}}\
dK^{\ast}(x)=\lambda\left[\frac{e^{iux}-1-iux}{x^{2}}\right]_{x=1}=\lambda(e^{iu}-1-iu),\
u\in\mathbb{R}.$

Proof of the direct part. Suppose that (5.12) holds. Applying Theorem 3, where
the probability law limit is associated with an a.s finite random variable,
leads to

$a_{n}\rightarrow b+\lambda\ \ and\ \ K_{n}^{\ast}\rightsquigarrow K^{\ast}.$

So (5.13) holds. We also have that $K_{n}\rightsquigarrow K$ means:

$\forall x\in C(K^{\ast}),\ \ K_{n}^{\ast}(x)\rightarrow\lambda 1_{(x\geq
1)},\ as\ n\rightarrow+\infty,$

since $C(K^{\ast})=(x<1)+(x>1)$ and
$\lambda_{K_{n}^{\ast}}(\mathbb{R})\rightarrow\lambda_{K^{\ast}}(\mathbb{R})=\lambda$.
For $x>1$, we have

$\displaystyle\sum_{k=1}^{k(n)}\int_{y\leq x}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow\lambda$
$\displaystyle\Leftrightarrow\sum_{k=1}^{k(n)}\int y^{2}\ \
dF_{k,n}^{\ast}(y)+\sum_{k=1}^{k(n)}\int_{(y>x)}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow\lambda$
$\displaystyle\Leftrightarrow\sum_{k=1}^{k(n)}\sigma^{2}_{k,n}+\sum_{k=1}^{k(n)}\int_{y>x}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow\lambda,$

where we use that $\int y^{2}\
dF_{k,n}^{\ast}(y)=\mathbb{V}ar(X_{k,n}-a_{k,n})=\sigma^{2}_{k,n}$ in the last
line. Hence

$\displaystyle\sum_{k=1}^{k(n)}\int_{y\leq x}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow\lambda$
$\displaystyle\Leftrightarrow\sum_{k=1}^{k(n)}\int_{y>x}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow 0.\ \ (L22)$

Let $\varepsilon=x-1>0$, (L22) is equivalent to

$\sum_{k=1}^{k(n)}\int_{(y-1>x-1)}y^{2}\ dF_{k,n}^{\ast}(y)\rightarrow 0,$

which is

$\sum_{k=1}^{k(n)}\int_{(|y-1|>x-1)}y^{2}\ dF_{k,n}^{\ast}(y)\rightarrow 0,$

that is

$g_{n,pois}(x-1)\rightarrow 0.$

Next, For $x<1$, we have

$\displaystyle\sum_{k=1}^{k(n)}\int_{y\leq x}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow 0$ $\displaystyle\Leftrightarrow\sum_{1\leq
k\leq k(n)}x^{2}\lambda_{F_{k,n}}(\\{x\\})+\sum_{k=1}^{k(n)}\int_{(y<x)}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow 0$
$\displaystyle\Leftrightarrow\lambda_{K_{n}^{\ast}}(\\{x\\})+\sum_{k=1}^{k(n)}\int_{(1-y>1-x)}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow 0$
$\displaystyle\Leftrightarrow\sum_{k=1}^{k(n)}\int_{(|1-y|>1-x)}y^{2}\
dF_{k,n}^{\ast}(y)\rightarrow 0,$

where we use that $\lambda_{K_{n}^{\ast}}(\\{x\\})\rightarrow 0$ (as shown in
line (5.8) above) in the last line. Hence

$g_{n,pois}(1-x)\rightarrow 0.$

By combining these results, we have for any $\varepsilon>0$, by taking either
$x-1=\varepsilon$ (for $x>1$) or $1-x=\varepsilon$ (for $x<1$), we arrive at
(5.14).

Proof of the indirect implication. Suppose that (5.13) and (5.14) are
satisfied. Let us exploit (5.14). For $x>1$,

$\displaystyle K_{n}(x)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{k(n)}\int_{(y\leq x)}y^{2}\ dF_{k,n}^{\ast}(y)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{k(n)}\sigma^{2}_{k,n}-\sum_{k=1}^{k(n)}\int_{(y-1>x-1)}y^{2}\
dF_{k,n}^{\ast}(y)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{k(n)}\sigma^{2}_{k,n}-g_{n,pois}(x-1)\rightarrow\lambda.$

For $x<1$

$\displaystyle K_{n}(x)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{k(n)}\int_{(y\leq x)}y^{2}\ dF_{k,n}^{\ast}(y)$
$\displaystyle=$
$\displaystyle\lambda_{K_{n}}(\\{x\\})+\sum_{k=1}^{k(n)}\int_{(|y-1|>1-x)}y^{2}\
dF_{k,n}^{\ast}(y)$ $\displaystyle=$
$\displaystyle\lambda_{K_{n}}(\\{x\\})+g_{n,pois}(1-x).$

So, $\lambda_{K_{n}}(\\{x\\})\rightarrow 0$ is shown exactly as in the lines
(5.8) above. Next $g_{n,pois}(1-x)\rightarrow 0$ is Assumption . Hence
$K_{n}\rightsquigarrow_{pre}\lambda 1_{(\circ\geq 1)}$. To complete the proof,
we remark that for any $n\geq 1$, $K_{n}(-\infty)=0$ and
$K_{n}(+\infty)=\sum_{k=1}^{k(n)}\sigma^{2}_{k,n}$. So

$\lambda_{K_{n}}(\mathbb{R})=K_{n}(+\infty)=\sum_{k=1}^{k(n)}\sigma^{2}_{k,n}\rightarrow\lambda=\lambda_{K}(\mathbb{R}).$

By Theorem 3, we conclude that (5.12) holds. $\blacksquare$

##  6\. Conclusion

We hope that we have given a complete exposition of the theory of the weak
limits of independent summands of square integrable random variables on
$\mathbb{R}$. The next step will be a re-do of the same theory when the
existence of the moments, even the first moment, is not required.

## References

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  * Lo et al. (2016) Lo, G.S.(2016). Weak Convergence (IA). Sequences of random vectors. SPAS Books Series. Saint-Louis, Senegal - Calgary, Canada. Doi : 10.16929/sbs/2016.0001. Arxiv : 1610.05415. ISBN : 978-2-9559183-1-9
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