Limit theorems for random sums of dependent d-dimensional random vectors

1982 ◽  
Vol 61 (1) ◽  
pp. 43-57 ◽  
Author(s):  
Marek Beśka ◽  
Andrzej Kłopotowski ◽  
Leszek Słomiński
Keyword(s):  
Limit Theorems ◽  
Random Sums ◽  
Random Vectors

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

Mathematics ◽  
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.

scholarly journals Random sums of random vectors and multitype families of productive individuals

2004 ◽  
Vol 2004 (19) ◽  
pp. 975-990
Author(s):  
I. Rahimov ◽  
H. Muttlak
Keyword(s):  
Stochastic Processes ◽  
Special Form ◽  
Limit Theorems ◽  
Random Variables ◽  
Random Sums ◽  
Random Vectors ◽  
Bernoulli Random Variables

We prove limit theorems for a family of random vectors whose coordinates are a special form of random sums of Bernoulli random variables. Applying these limit theorems, we study the number of productive individuals inn-type indecomposable critical branching stochastic processes with types of individualsT1,…,Tn.

Limit theorems for random sums in the double array scheme

2020 ◽  
pp. 93-152
Author(s):  
Boris V. Gnedenko ◽  
Victor Yu. Korolev
Keyword(s):  
Limit Theorems ◽  
Random Sums ◽  
Double Array ◽  
Array Scheme

scholarly journals Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Zhicheng Chen ◽  
Xinsheng Liu
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Random Vectors ◽  
Standard Normal ◽  
Almost Sure Limit Theorems ◽  
General Standard

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.

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