From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

Bivariate Limit Theorems for Record Values Based on Random Sample Sizes

Sankhya A ◽

10.1007/s13171-019-00167-2 ◽

2019 ◽

Vol 82 (1) ◽

pp. 50-67

Author(s):

M. A. Abd Elgawad ◽

H. M. Barakat ◽

Ting Yan

Keyword(s):

Random Sample ◽

Limit Theorems ◽

Record Values ◽

Sample Sizes

Random Sample Sizes: Limit Theorems and Characterizations

Probability Theory and Applications ◽

10.1007/978-94-011-2817-9_7 ◽

1992 ◽

pp. 107-123 ◽

Cited By ~ 3

Author(s):

J. Galambos

Keyword(s):

Random Sample ◽

Limit Theorems ◽

Sample Sizes

Limit theorems for statistics with random sample sizes

10.1063/1.4912514 ◽

2015 ◽

Author(s):

M. E. Grigoryeva ◽

Victor Yu. Korolev ◽

Alexander I. Zeifman

Keyword(s):

Random Sample ◽

Limit Theorems ◽

Sample Sizes

Limit theorems for extremes with random sample size

Advances in Applied Probability ◽

10.1017/s0001867800008600 ◽

1998 ◽

Vol 30 (03) ◽

pp. 777-806 ◽

Cited By ~ 3

Author(s):

Dmitrii S. Silvestrov ◽

Jozef L. Teugels

Keyword(s):

Weak Convergence ◽

Sample Size ◽

Random Sample ◽

Limit Theorems ◽

Functional Limit ◽

Functional Limit Theorems ◽

Random Sample Size ◽

Extremal Processes ◽

Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

Limit theorems for the diameter of a random sample in the unit ball

Extremes ◽

10.1007/s10687-007-0038-y ◽

2007 ◽

Vol 10 (3) ◽

pp. 129-150 ◽

Cited By ~ 10

Author(s):

Michael Mayer ◽

Ilya Molchanov

Keyword(s):

Unit Ball ◽

Random Sample ◽

Limit Theorems

Limit theorems for random sums in the double array scheme

Random Summation ◽

10.1201/9781003067894-4 ◽

2020 ◽

pp. 93-152

Author(s):

Boris V. Gnedenko ◽

Victor Yu. Korolev

Keyword(s):

Limit Theorems ◽

Random Sums ◽

Double Array ◽

Array Scheme

Limit theorems for numbers satisfying a class of triangular arrays

Glasnik Matematicki ◽

10.3336/gm.56.2.01 ◽

2021 ◽

Vol 56 (2) ◽

pp. 195-223

Author(s):

Igoris Belovas ◽

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Convergence Rate ◽

Asymptotic Normality ◽

Limit Theorems ◽

Limiting Distribution ◽

Linear Recurrence ◽

Triangular Arrays ◽

Exponential Generating Function ◽

Analytical Expressions

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian.

Limit Theorems with ϑ-Rates for Random Sums of Dependent Banach-Valued Random Variables

Mathematische Nachrichten ◽

10.1002/mana.19841190106 ◽

1984 ◽

Vol 119 (1) ◽

pp. 59-75 ◽

Cited By ~ 4

Author(s):

Paul L. Butzer ◽

Dietmar Schulz

Keyword(s):

Limit Theorems ◽

Random Variables ◽

Random Sums

Limit Theorems for an Extended Coupon Collector's Problem and for Successive Subsampling with Varying Probabilites*

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319800301 ◽

1980 ◽

Vol 29 (3-4) ◽

pp. 113-132 ◽

Cited By ~ 4

Author(s):

Pranab Kumar Sen

Keyword(s):

Asymptotic Normality ◽

Limit Theorems ◽

Finite Population ◽

Waiting Times ◽

Invariance Principles ◽

Weak Invariance ◽

Coupon Collector’S Problem ◽

Asymptotic Distribution Theory ◽

Multistage Sampling ◽

Coupon Collector's Problem

Asymptotic normality as well as some weak invariance principles for bonus sums and waiting times in an extended coupon collector's problem are considered and incorporated in the study of the asymptotic distribution theory of estimators of (finite) population totals in successive sub-sampling (or multistage sampling) with varying probabilities (without replacement). Some applications of these theorems are also considered.

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

Mathematics ◽

10.3390/math8111918 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1918

Author(s):

Victor Korolev

Keyword(s):

Limit Theorems ◽

Random Sums ◽

Random Vectors ◽

Asymptotic Results ◽

Space Of Distributions ◽

Gamma Distributions ◽

Statistical Regularities ◽

Exponential Power ◽

The One ◽

Multivariate Exponential

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.