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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 May 2021) | Viewed by 27477

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Prof. Dr. Irina Shevtsova

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1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow 119991, Russia
2. Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia
3. Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, Moscow 119333, Russia
4. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Interests: limit theorems of probability theory; estimates of the rate of convergence; random sums; extreme problems; analytical methods of probability theory
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Prof. Dr. Victor Korolev

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Guest Editor

1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow 119991, Russia
2. Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia
3. Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, Moscow 119333, Russia
4. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
Interests: limit theorems of probability theory; convergence rate estimates; random sums; statistics constructed from samples with random size; risk theory; mixture models and their applications; statistical separation of mixtures
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

As was noted in the famous book Limit Distributions for Sums of Independent Random Variables by B.V. Gnedenko and A.N. Kolmogorov, "actually, the cognitive value of probability theory is revealed only by limit theorems." The significance of limit theorems of probability theory, and in particular, the central limit theorem, cannot be overestimated. In applied probability, there is a convention, according to which a model distribution can be regarded as reasonable and/or justified enough only if it is an asymptotic approximation, that is, there exist a more or less simple settings and the corresponding limit theorem in which the model under consideration is a limit distribution. Limit theorems suggest theoretic models for many real processes, for example, arising in physics, financial mathematics, risk theory, control theory, data mining, queuing theory, and many others. In order to successfully use an approximation hinted at by a limit theorem, one has to be able to estimate its accuracy, or to dispose a convergence rate estimate. On the other hand, the proofs of limit theorems and construction of convergence rate estimates usually involve analytical methods of probability, say, Stein’s method, method of probability metrics, smoothing inequalities, characteristic functions, Laplace transforms, etc. For the sake of optimization of the error bounds in limit theorems, one may face various extreme problems.

In this Special Issue, papers are collected that produce or improve various limit theorems of probability theory and convergence rate estimates, as well as develop analytical methods of probability theory and apply stochastic models produced by limit theorems to the solution of applied and theoretical problems in various fields.

Prof. Dr. Irina Shevtsova
Prof. Dr. Victor Korolev
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

limit theorems of probability
convergence rate estimates
asymptotic approximation
analytical methods of probability
extreme problem

Published Papers (14 papers)

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8 pages, 258 KiB

Open AccessFeature PaperArticle

On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

by Irina Shevtsova and Mikhail Tselishchev

Mathematics 2021, 9(13), 1571; https://doi.org/10.3390/math9131571 - 04 Jul 2021

Cited by 4 | Viewed by 1655

Abstract

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

21 pages, 384 KiB

Open AccessArticle

On Approximation of the Tails of the Binomial Distribution with These of the Poisson Law

by Sergei Nagaev and Vladimir Chebotarev

Mathematics 2021, 9(8), 845; https://doi.org/10.3390/math9080845 - 13 Apr 2021

Viewed by 1742

Abstract

A subject of this study is the behavior of the tail of the binomial distribution in the case of the Poisson approximation. The deviation from unit of the ratio of the tail of the binomial distribution and that of the Poisson distribution, multiplied by the correction factor, is estimated. A new type of approximation is introduced when the parameter of the approximating Poisson law depends on the point at which the approximation is performed. Then the transition to the approximation by the Poisson law with the parameter equal to the mathematical expectation of the approximated binomial law is carried out. In both cases error estimates are obtained. A number of conjectures are made about the refinement of the known estimates for the Kolmogorov distance between binomial and Poisson distributions. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

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10 pages, 274 KiB

Open AccessArticle

Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

by Eunju Hwang

Mathematics 2021, 9(8), 816; https://doi.org/10.3390/math9080816 - 09 Apr 2021

Viewed by 1213

Abstract

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient

ρ = ρ_{n}

is derived uniformly over stationary values in

[0, 1)

[...] Read more.

ρ = ρ_{n}

is derived uniformly over stationary values in

[0, 1)

, focusing on

ρ_{n} \to 1

as sample size n tends to infinity. For tail index

α \in (0, 4)

of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on

1 - ρ_{n}^{2}

, but no condition on the rate of

ρ_{n}

is required. It is shown that, for the tail index

α \in (0, 2)

, the LSE is inconsistent, for

α = 2

log n / (1 - ρ_{n}^{2})

-consistent, and for

α \in (2, 4)

n^{1 - 2 / α} / (1 - ρ_{n}^{2})

-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index

α \in (0, 4)

; and no restriction on the rate of

ρ_{n}

is necessary. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

28 pages, 476 KiB

Open AccessArticle

Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

by Gerd Christoph and Vladimir V. Ulyanov

Mathematics 2021, 9(7), 775; https://doi.org/10.3390/math9070775 - 02 Apr 2021

Cited by 2 | Viewed by 2241

Abstract

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized

T_{0}^{2}

statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

4 pages, 217 KiB

Open AccessArticle

On Small Deviation Asymptotics in the L₂-Norm for Certain Gaussian Processes

by Leonid Rozovsky

Mathematics 2021, 9(6), 655; https://doi.org/10.3390/math9060655 - 19 Mar 2021

Cited by 1 | Viewed by 1060

Abstract

The results obtained allow finding sharp small deviations in a Hilbert norm for centered Gaussian processes in the case where their covariances have a special form of the eigenvalues and allow us to describe small deviation asymptotics for certain Gaussian processes. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

36 pages, 544 KiB

Open AccessArticle

Statistical Estimation of the Kullback–Leibler Divergence

by Alexander Bulinski and Denis Dimitrov

Mathematics 2021, 9(5), 544; https://doi.org/10.3390/math9050544 - 04 Mar 2021

Cited by 9 | Viewed by 3459

Abstract

Asymptotic unbiasedness and

L^{2}

-consistency are established, under mild conditions, for the estimates of the Kullback–Leibler divergence between two probability measures in

R^{d}

, absolutely continuous with respect to (w.r.t.) the Lebesgue measure. These estimates are based on certain k-nearest [...] Read more.

Asymptotic unbiasedness and

L^{2}

-consistency are established, under mild conditions, for the estimates of the Kullback–Leibler divergence between two probability measures in

R^{d}

, absolutely continuous with respect to (w.r.t.) the Lebesgue measure. These estimates are based on certain k-nearest neighbor statistics for pair of independent identically distributed (i.i.d.) due vector samples. The novelty of results is also in treating mixture models. In particular, they cover mixtures of nondegenerate Gaussian measures. The mentioned asymptotic properties of related estimators for the Shannon entropy and cross-entropy are strengthened. Some applications are indicated. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

32 pages, 481 KiB

Open AccessArticle

Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

by Ruslan Gabdullin, Vladimir Makarenko and Irina Shevtsova

Mathematics 2021, 9(5), 501; https://doi.org/10.3390/math9050501 - 01 Mar 2021

Viewed by 1195

Abstract

Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

13 pages, 338 KiB

Open AccessArticle

Asymptotically Normal Estimators for the Parameters of the Gamma-Exponential Distribution

by Alexey Kudryavtsev and Oleg Shestakov

Mathematics 2021, 9(3), 273; https://doi.org/10.3390/math9030273 - 30 Jan 2021

Cited by 7 | Viewed by 2317

Abstract

Currently, much research attention has focused on generalizations of known mathematical objects in order to obtain adequate models describing real phenomena. An important role in the applied theory of probability and mathematical statistics is the gamma class of distributions, which has proven to be a convenient and effective tool for modeling many real processes. The gamma class is quite wide and includes distributions that have useful properties such as, for example, infinite divisibility and stability, which makes it possible to use distributions from this class as asymptotic approximations in various limit theorems. One of the most important tasks of applied statistics is to obtain estimates of the parameters of the model distribution from the available real data. In this paper, we consider the gamma-exponential distribution, which is a generalization of the distributions from the gamma class. Estimators for some parameters of this distribution are given, and the asymptotic normality of these estimators is proven. When obtaining the estimates, a modified method of moments was used, based on logarithmic moments calculated on the basis of the Mellin transform for the generalized gamma distribution. On the basis of the results obtained, asymptotic confidence intervals for the estimated parameters are constructed. The results of this work can be used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

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17 pages, 1510 KiB

Open AccessArticle

A Random Walk Model for Spatial Galaxy Distribution

by Vladimir V. Uchaikin, Vladimir A. Litvinov, Elena V. Kozhemyakina and Ilya I. Kozhemyakin

Mathematics 2021, 9(1), 98; https://doi.org/10.3390/math9010098 - 05 Jan 2021

Cited by 1 | Viewed by 2171

Abstract

A new statistical model of spatial distribution of observed galaxies is described. Statistical correlations are involved by means of Markov chain ensembles, whose parameters are extracted from the observable power spectrum by adopting of the Uchaikin–Zolotarev ansatz. Markov chain trajectories with the Lévy–Feldheim distributed step lengths form the set of nodes imitating the positions of galaxy. The model plausibly reproduces the two-point correlation functions, cell-count data and some other important properties. It can effectively be used in the post-processing of astronomical data for cosmological studies. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

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11 pages, 276 KiB

Open AccessArticle

On the Accuracy of the Exponential Approximation to Random Sums of Alternating Random Variables

by Irina Shevtsova and Mikhail Tselishchev

Mathematics 2020, 8(11), 1917; https://doi.org/10.3390/math8111917 - 01 Nov 2020

Cited by 2 | Viewed by 1315

Abstract

Using the generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with a finite non-zero first moment, we prove moment-type error-bounds in the Kantorovich distance for the exponential approximation to random sums of possibly dependent random variables with positive finite expectations, in particular, to geometric random sums, generalizing the previous results to alternating and dependent random summands. We also extend the notions of new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) distributions to alternating random variables in terms of the corresponding distribution functions and provide a criteria in terms of conditional expectations similar to the classical one. As corollary, we provide simplified error-bounds in the case of NBUE/NWUE conditional distributions of random summands. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

11 pages, 269 KiB

Open AccessArticle

Asymptotic Properties of MSE Estimate for the False Discovery Rate Controlling Procedures in Multiple Hypothesis Testing

by Sofia Palionnaya and Oleg Shestakov

Mathematics 2020, 8(11), 1913; https://doi.org/10.3390/math8111913 - 01 Nov 2020

Cited by 1 | Viewed by 1491

Abstract

Problems with analyzing and processing high-dimensional random vectors arise in a wide variety of areas. Important practical tasks are economical representation, searching for significant features, and removal of insignificant (noise) features. These tasks are fundamentally important for a wide class of practical applications, such as genetic chain analysis, encephalography, spectrography, video and audio processing, and a number of others. Current research in this area includes a wide range of papers devoted to various filtering methods based on the sparse representation of the obtained experimental data and statistical procedures for their processing. One of the most popular approaches to constructing statistical estimates of regularities in experimental data is the procedure of multiple testing of hypotheses about the significance of observations. In this paper, we consider a procedure based on the false discovery rate (FDR) measure that controls the expected percentage of false rejections of the null hypothesis. We analyze the asymptotic properties of the mean-square error estimate for this procedure and prove the statements about the asymptotic normality of this estimate. The obtained results make it possible to construct asymptotic confidence intervals for the mean-square error of the FDR method using only the observed data. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

14 pages, 249 KiB

Open AccessArticle

Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

by Young Sik Kim

Mathematics 2020, 8(10), 1666; https://doi.org/10.3390/math8101666 - 28 Sep 2020

Cited by 2 | Viewed by 1439

Abstract

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of

F (x) = exp {\int_{0}^{T} θ (t, x (t)) d t}

successfully [...] Read more.

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of

F (x) = exp {\int_{0}^{T} θ (t, x (t)) d t}

successfully exist under the certain condition, where

θ (t, u) = \int_{R} exp {i u v} d σ_{t} (v)

is a Fourier–Stieltjes transform of a complex Borel measure

σ_{t} \in M (R)

and

M (R)

is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of

F (x)

sucessfully holds on the Wiener space. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

24 pages, 4011 KiB

Open AccessArticle

Modeling Particle Size Distribution in Lunar Regolith via a Central Limit Theorem for Random Sums

by Andrey Gorshenin, Victor Korolev and Alexander Zeifman

Mathematics 2020, 8(9), 1409; https://doi.org/10.3390/math8091409 - 23 Aug 2020

Cited by 5 | Viewed by 2633

Abstract

A version of the central limit theorem is proved for sums with a random number of independent and not necessarily identically distributed random variables in the double array limit scheme. It is demonstrated that arbitrary normal mixtures appear as the limit distribution. This result is used to substantiate the log-normal finite mixture approximations for the particle size distributions of the lunar regolith. This model is used as the theoretical background of the two different statistical procedures for processing real data based on bootstrap and minimum

χ^{2}

estimates. It is shown that the cluster analysis of the parameters of the proposed models can be a promising tool for revealing the structure of such real data, taking into account the physico-chemical interpretation of the results. Similar methods can be successfully used for solving problems from other subject fields with grouped observations, and only some characteristic points of the empirical distribution function are given. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

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27 pages, 377 KiB

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Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics

by Victor Korolev

Mathematics 2020, 8(11), 1918; https://doi.org/10.3390/math8111918 - 01 Nov 2020

Cited by 8 | Viewed by 2054

Abstract

In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are proved. Unlike the conventional analytical approach, here the presentation follows the lines of a kind of arithmetical approach in the space of random variables or vectors. Here the operation of scale mixing in the space of distributions is replaced with the operation of multiplication in the space of random vectors/variables under the assumption that the multipliers are independent. By doing so, the reasoning becomes much simpler, the proofs become shorter and some general features of the distributions under consideration become more vivid. The first part of the paper concerns the univariate case. Some known results are discussed and simple alternative proofs for some of them are presented as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions. The results obtained here open the way to deal with natural multivariate analogs of EP distributions. In the second part of the paper, we discuss the conventionally defined multivariate EP distributions and introduce the notion of projective EP (PEP) distributions. The properties of multivariate EP and PEP distributions are considered as well as limit theorems establishing the conditions for the convergence of multivariate statistics constructed from samples with random sizes (including random sums of random vectors) to multivariate elliptically contoured EP and projective EP laws. The results obtained here give additional theoretical grounds for the applicability of EP and PEP distributions as asymptotic approximations for the statistical regularities observed in data in many fields. Full article

(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)

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