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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 19728

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Prof. Dr. Alexander Tikhomirov

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Guest Editor

Institute of Physics and Mathematics, Komi Science Center of Ural Division of the Russian Academy of Sciences, 167000 Syktyvkar, Russia
Interests: random matrices; strong mixing condition; limit theorems; circular law
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Vladimir Ulyanov

E-Mail Website
Guest Editor

1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
2. Faculty of Computer Science, National Research University—Higher School of Economics, 167005 Moscow, Russia
Interests: limit theorems of probability theory; vector-valued random variables; weak limit theorems; Gaussian processes; appoximations in statistics; transforms of probability distributions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

M. Loeve wrote: “The fundamental limit theorems of Probability theory may be classified into two groups. One group deals with the problem of limit laws of sequences of some of random variables, the other deals with the problem of limits of random variables, in the sense of almost sure convergence, of such sequences. These problems will be labeled, respectively, the Central Limit Problem (CLP) and the Strong Central Limit Problem (SCLP). Like all mathematical problems, the CLP and SCLP are not static; as answers to old queries are discovered they experience the usual development and new problems arise.”

The purpose of this Special Issue is to present new directions and new advances in limit theorems in probability theory. The list of topics can be very extensive, and it includes classical models of sums of both independent and various kinds of dependent random variables, limit theorems for random processes, functional limit theorems, limit theorems in high-dimensional spaces, limit theorems in free probability, probabilities of large deviations, small and large ball probabilities, measure concentration, and more.

Prof. Dr. Alexander Tikhomirov
Prof. Dr. Vladimir Ulyanov
Guest Editors

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Keywords

Sequences of random variables
Sums of random variables
Types of convergence
Laws of large numbers
Law of iterated logarithms
Central limit theorem
Gaussian distribution
Poisson limit distribution
Large deviation
Local limit theorems
Limit distributions of extremes
Small ball probabilities
Large ball probabilities
Measure concentration
Rate of convergence
Free probability
Random matrices
High-dimensional spaces

Related Special Issue

New Trends in Stochastic Processes, Probability and Statistics in Mathematics (4 articles)

Published Papers (16 papers)

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Editorial

Jump to: Research

4 pages, 179 KiB

Open AccessEditorial

On the Special Issue “Limit Theorems of Probability Theory”

by Alexander N. Tikhomirov and Vladimir V. Ulyanov

Mathematics 2023, 11(17), 3665; https://doi.org/10.3390/math11173665 - 25 Aug 2023

Viewed by 773

Abstract

M [...] Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

Research

Jump to: Editorial

29 pages, 49985 KiB

Open AccessArticle

On Structured Random Matrices Defined by Matrix Substitutions

by Manuel L. Esquível and Nadezhda P. Krasii

Mathematics 2023, 11(11), 2505; https://doi.org/10.3390/math11112505 - 29 May 2023

Cited by 1 | Viewed by 946

Abstract

The structure of the random matrices introduced in this work is given by deterministic matrices—the skeletons of the random matrices—built with an algorithm of matrix substitutions with entries in a finite field of integers modulo some prime number, akin to the algorithm of one dimensional automatic sequences. A random matrix has the structure of a given skeleton if to the same number of an entry of the skeleton, in the finite field, it corresponds a random variable having, at least, as its expected value the correspondent value of the number in the finite field. Affine matrix substitutions are introduced and fixed point theorems are proven that allow the consideration of steady states of the structure which are essential for an efficient observation. For some more restricted classes of structured random matrices the parameter estimation of the entries is addressed, as well as the convergence in law and also some aspects of the spectral analysis of the random operators associated with the random matrix. Finally, aiming at possible applications, it is shown that there is a procedure to associate a canonical random surface to every random structured matrix of a certain class. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

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18 pages, 336 KiB

Open AccessArticle

Bound for an Approximation of Invariant Density of Diffusions via Density Formula in Malliavin Calculus

by Yoon-Tae Kim and Hyun-Suk Park

Mathematics 2023, 11(10), 2302; https://doi.org/10.3390/math11102302 - 15 May 2023

Cited by 1 | Viewed by 783

Abstract

The Kolmogorov and total variation distance between the laws of random variables have upper bounds represented by the

L^{1}

-norm of densities when random variables have densities. In this paper, we derive an upper bound, in terms of densities such as the [...] Read more.

The Kolmogorov and total variation distance between the laws of random variables have upper bounds represented by the

L^{1}

-norm of densities when random variables have densities. In this paper, we derive an upper bound, in terms of densities such as the Kolmogorov and total variation distance, for several probabilistic distances (e.g., Kolmogorov distance, total variation distance, Wasserstein distance, Forter–Mourier distance, etc.) between the laws of F and G in the case where a random variable F follows the invariant measure that admits a density and a differentiable random variable G, in the sense of Malliavin calculus, and also allows a density function. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

18 pages, 378 KiB

Open AccessArticle

Second Order Chebyshev–Edgeworth-Type Approximations for Statistics Based on Random Size Samples

by Gerd Christoph and Vladimir V. Ulyanov

Mathematics 2023, 11(8), 1848; https://doi.org/10.3390/math11081848 - 13 Apr 2023

Cited by 1 | Viewed by 796

Abstract

This article completes our studies on the formal construction of asymptotic approximations for statistics based on a random number of observations. Second order Chebyshev–Edgeworth expansions of asymptotically normally or chi-squared distributed statistics from samples with negative binomial or Pareto-like distributed random sample sizes are obtained. The results can have applications for a wide spectrum of asymptotically normally or chi-square distributed statistics. Random, non-random, and mixed scaling factors for each of the studied statistics produce three different limit distributions. In addition to the expected normal or chi-squared distributions, Student’s t-, Laplace, Fisher, gamma, and weighted sums of generalized gamma distributions also occur. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

25 pages, 358 KiB

Open AccessArticle

Limit Theorem for Spectra of Laplace Matrix of Random Graphs

by Alexander N. Tikhomirov

Mathematics 2023, 11(3), 764; https://doi.org/10.3390/math11030764 - 02 Feb 2023

Cited by 1 | Viewed by 1060

Abstract

We consider the limit of the empirical spectral distribution of Laplace matrices of generalized random graphs. Applying the Stieltjes transform method, we prove under general conditions that the limit spectral distribution of Laplace matrices converges to the free convolution of the semicircular law [...] Read more.

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

15 pages, 312 KiB

Open AccessArticle

High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures

by Yasunori Fujikoshi and Tetsuro Sakurai

Mathematics 2023, 11(3), 671; https://doi.org/10.3390/math11030671 - 28 Jan 2023

Cited by 1 | Viewed by 1008

Abstract

In this paper, we consider the high-dimensional consistencies of KOO methods for selecting response variables in multivariate linear regression with covariance structures. Here, the covariance structures are considered as (1) independent covariance structure with the same variance, (2) independent covariance structure with different variances, and (3) uniform covariance structure. A sufficient condition for model selection consistency is obtained using a KOO method under a high-dimensional asymptotic framework, such that sample size n, the number p of response variables, and the number k of explanatory variables are large, as in

p / n \to c_{1} \in (0, 1)

and

k / n \to c_{2} \in [0, 1)

, where

c_{1} + c_{2} < 1

. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

37 pages, 530 KiB

Open AccessArticle

Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Laws

by Alexander Bulinski and Nikolay Slepov

Mathematics 2022, 10(24), 4747; https://doi.org/10.3390/math10244747 - 14 Dec 2022

Cited by 3 | Viewed by 2194

Abstract

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

10 pages, 247 KiB

Open AccessArticle

Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

by Shuya Kanagawa

Mathematics 2022, 10(21), 4158; https://doi.org/10.3390/math10214158 - 07 Nov 2022

Cited by 2 | Viewed by 1004

Abstract

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term

O (n^{1 - p / 2})

, for some

p \geq 4

, is shown in both cases. From the results, it is obtained [...] Read more.

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term

O (n^{1 - p / 2})

, for some

p \geq 4

, is shown in both cases. From the results, it is obtained that asymptotic expansions for the Cram

\overset{´}{e}

r–von Mises statistics of the uniform distribution

U (0, 1)

hold with the remainder term

O (n^{1 - p / 2})

for any

p \geq 4

. The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function

u (x, y)

. The key condition for the convergence is the nuclearity of a linear operator

T_{u}

defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

20 pages, 357 KiB

Open AccessArticle

Poissonization Principle for a Class of Additive Statistics

by Igor Borisov and Maman Jetpisbaev

Mathematics 2022, 10(21), 4084; https://doi.org/10.3390/math10214084 - 02 Nov 2022

Cited by 1 | Viewed by 1033

Abstract

In this paper, we consider a class of additive functionals of a finite or countable collection of the group frequencies of an empirical point process that corresponds to, at most, a countable partition of the sample space. Under broad conditions, it is shown that the asymptotic behavior of the distributions of such functionals is similar to the behavior of the distributions of the same functionals of the accompanying Poisson point process. However, the Poisson versions of the additive functionals under consideration, unlike the original ones, have the structure of sums (finite or infinite) of independent random variables that allows us to reduce the asymptotic analysis of the distributions of additive functionals of an empirical point process to classical problems of the theory of summation of independent random variables. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

17 pages, 348 KiB

Open AccessFeature PaperArticle

Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities

by Yuri Yakubovich, Oleg Rusakov and Alexander Gushchin

Mathematics 2022, 10(21), 3955; https://doi.org/10.3390/math10213955 - 25 Oct 2022

Cited by 2 | Viewed by 1049

Abstract

We consider a sequence of i.i.d. random variables,

(ξ) = {(ξ_{i})}_{i = 0, 1, 2, \dots}

E ξ_{0} = 0

E ξ_{0}^{2} = 1

, and subordinate it by [...] Read more.

We consider a sequence of i.i.d. random variables,

(ξ) = {(ξ_{i})}_{i = 0, 1, 2, \dots}

E ξ_{0} = 0

E ξ_{0}^{2} = 1

, and subordinate it by a doubly stochastic Poisson process

Π (λ t)

, where

λ \geq 0

is a random variable and

Π

is a standard Poisson process. The subordinated continuous time process

ψ (t) = ξ_{Π (λ t)}

is known as the PSI-process. Elements of the triplet

(Π, λ, (ξ))

are supposed to be independent. For sums of n, independent copies of such processes, normalized by

\sqrt{n}

, we establish a functional limit theorem in the Skorokhod space

D [0, T]

, for any

T > 0

, under the assumption

E | ξ_{0} |^{2 h} < \infty

for some

h > 1 / γ^{2}

. Here,

γ \in (0, 1]

reflects the tail behavior of the distribution of

λ

, in particular,

γ \equiv 1

when

E λ < \infty

. The limit process is a stationary Gaussian process with the covariance function

E e^{- λ u}

u \geq 0

. As a sample application, we construct a martingale from the PSI-process and establish a convergence of normalized cumulative sums of such i.i.d. martingales. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

28 pages, 4001 KiB

Open AccessArticle

Asymptotic Properties and Application of GSB Process: A Case Study of the COVID-19 Dynamics in Serbia

by Mihailo Jovanović, Vladica Stojanović, Kristijan Kuk, Brankica Popović and Petar Čisar

Mathematics 2022, 10(20), 3849; https://doi.org/10.3390/math10203849 - 17 Oct 2022

Cited by 6 | Viewed by 1213

Abstract

This paper describes one of the non-linear (and non-stationary) stochastic models, the GSB (Gaussian, or Generalized, Split-BREAK) process, which is used in the analysis of time series with pronounced and accentuated fluctuations. In the beginning, the stochastic structure of the GSB process and its important distributional and asymptotic properties are given. To that end, a method based on characteristic functions (CFs) was used. Various procedures for the estimation of model parameters, asymptotic properties, and numerical simulations of the obtained estimators are also investigated. Finally, as an illustration of the practical application of the GSB process, an analysis is presented of the dynamics and stochastic distribution of the infected and immunized population in relation to the disease COVID-19 in the territory of the Republic of Serbia. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

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38 pages, 402 KiB

Open AccessArticle

Local Laws for Sparse Sample Covariance Matrices

by Alexander N. Tikhomirov and Dmitry A. Timushev

Mathematics 2022, 10(13), 2326; https://doi.org/10.3390/math10132326 - 03 Jul 2022

Cited by 1 | Viewed by 1076

Abstract

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order

n \times m

with

n / m \to y

(where

y > 0

) and sparse probability

n p_{n} > {log}^{β} n

[...] Read more.

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order

n \times m

with

n / m \to y

(where

y > 0

) and sparse probability

n p_{n} > {log}^{β} n

(where

β > 0

). The bounds of the distance between the empirical spectral distribution function of the sparse sample covariance matrices and the Marchenko–Pastur law distribution function that was obtained in the complex domain

z \in D

with

Im z > v_{0} > 0

(where

v_{0}

) were of order

{log}^{4} n / n

and the domain bounds did not depend on

p_{n}

while

n p_{n} > {log}^{β} n

. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

13 pages, 288 KiB

Open AccessArticle

On De la Peña Type Inequalities for Point Processes

by Naiqi Liu, Vladimir V. Ulyanov and Hanchao Wang

Mathematics 2022, 10(12), 2114; https://doi.org/10.3390/math10122114 - 17 Jun 2022

Cited by 1 | Viewed by 1177

Abstract

There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain de la Peña’s inequalities for a stochastic integral of multivariate point processes. The proof is primarily based on Doléans–Dade exponential formula and the optional stopping theorem. As an application, we obtain an exponential inequality for block counting process in

Λ -

coalescent. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

6 pages, 251 KiB

Open AccessFeature PaperArticle

A New Bound in the Littlewood–Offord Problem

by Friedrich Götze and Andrei Yu. Zaitsev

Mathematics 2022, 10(10), 1740; https://doi.org/10.3390/math10101740 - 19 May 2022

Cited by 1 | Viewed by 1027

Abstract

The paper deals with studying a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the concentration function of a weighted sum of independent identically distributed random variables is estimated in terms of the concentration function of a symmetric infinitely divisible distribution whose spectral measure is concentrated on the set of plus-minus weights. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

16 pages, 386 KiB

Open AccessArticle

On the M-Estimator under Third Moment Condition

by Rundong Luo, Yiming Chen and Shuai Song

Mathematics 2022, 10(10), 1713; https://doi.org/10.3390/math10101713 - 17 May 2022

Cited by 2 | Viewed by 1708

Abstract

Estimating the expected value of a random variable by data-driven methods is one of the most fundamental problems in statistics. In this study, we present an extension of Olivier Catoni’s classical M-estimators of the empirical mean, which focus on the heavy-tailed data by imposing more precise inequalities on exponential moments of Catoni’s estimator. We show that our works behave better than Catoni‘s both in practice and theory. The performances are illustrated in the simulation and real data. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

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11 pages, 263 KiB

Open AccessArticle

Large Deviations for the Maximum of the Absolute Value of Partial Sums of Random Variable Sequences

by Xia Wang and Miaomiao Zhang

Mathematics 2022, 10(5), 758; https://doi.org/10.3390/math10050758 - 27 Feb 2022

Cited by 1 | Viewed by 1191

Abstract

Let

{ξ_{i} : i \geq 1}

be a sequence of independent, identically distributed (i.i.d. for short) centered random variables. Let

S_{n} = ξ_{1} + \dots + ξ_{n}

denote the partial sums of

{ξ_{i}} .

[...] Read more.

Let

{ξ_{i} : i \geq 1}

be a sequence of independent, identically distributed (i.i.d. for short) centered random variables. Let

S_{n} = ξ_{1} + \dots + ξ_{n}

denote the partial sums of

{ξ_{i}} .

We show that sequence

{\frac{1}{n} max_{1 \leq k \leq n} | S_{k} | : n \geq 1}

satisfies the large deviation principle (LDP, for short) with a good rate function under the assumption that

P (ξ_{1} \geq x)

and

P (ξ_{1} \leq - x)

have the same exponential decrease. Full article

(This article belongs to the Special Issue Limit Theorems of Probability Theory)

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