scholarly journals On a Multivariate Analog of the Zolotarev Problem

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1728
Author(s):  
Yury Khokhlov ◽  
Victor Korolev
Keyword(s):  
Geometric Distribution ◽  
Stable Distributions ◽  
Random Sums ◽  
Random Vectors ◽  
Random Summation ◽  
Mixing Distributions ◽  
Random Index ◽  
Zolotarev Problem ◽  
Elliptically Contoured ◽  
Multivariate Geometric Distribution

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.

scholarly journals Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 749 ◽  
Author(s):  
Yury Khokhlov ◽  
Victor Korolev ◽  
Alexander Zeifman
Keyword(s):  
Limit Theorems ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Random Vectors ◽  
Scale Mixtures ◽  
Elliptically Contoured ◽  
Necessary And Sufficient

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

scholarly journals Multivariate subexponential distributions and random sums of random vectors

2006 ◽  
Vol 38 (4) ◽  
pp. 1028-1046 ◽  
Author(s):  
E. Omey ◽  
F. Mallor ◽  
J. Santos
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Regular Variation ◽  
Subexponential Distributions ◽  
Random Sums ◽  
Multivariate Regular Variation ◽  
Random Vectors ◽  
Convolution Power ◽  
Random Index

Let F(x) denote a distribution function in Rd and let F*n(x) denote the nth convolution power of F(x). In this paper we discuss the asymptotic behaviour of 1 - F*n(x) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F*n(x) ∼ n(1 - F(x)). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N.

scholarly journals A formula for hidden regular variation behavior for symmetric stable distributions

Extremes ◽  
2020 ◽  
Vol 23 (4) ◽  
pp. 667-691
Author(s):  
Malin Palö Forsström ◽  
Jeffrey E. Steif
Keyword(s):  
Power Law ◽  
Spectral Measure ◽  
Regular Variation ◽  
Stable Distributions ◽  
Decay Rates ◽  
Random Vectors ◽  
Power Law Decay ◽  
Exact Power ◽  
Hidden Regular Variation ◽  
Stable Random Vectors

Abstract We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.

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.

On the distance between the distributions of random sums

2003 ◽  
Vol 40 (01) ◽  
pp. 87-106 ◽  
Author(s):  
Bero Roos ◽  
Dietmar Pfeifer
Keyword(s):  
Total Variation ◽  
Approximation Problem ◽  
Upper Bounds ◽  
Total Variation Distance ◽  
Random Sum ◽  
Random Sums ◽  
Random Summation ◽  
Variation Distance ◽  
Stop Loss

In this paper, we consider the total variation distance between the distributions of two random sums S M and S N with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum S N if M and N have the same first moments.

On the distance between the distributions of random sums

2003 ◽  
Vol 40 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Bero Roos ◽  
Dietmar Pfeifer
Keyword(s):  
Total Variation ◽  
Approximation Problem ◽  
Upper Bounds ◽  
Total Variation Distance ◽  
Random Sum ◽  
Random Sums ◽  
Random Summation ◽  
Variation Distance ◽  
Stop Loss

In this paper, we consider the total variation distance between the distributions of two random sums SM and SN with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum SN if M and N have the same first moments.

scholarly journals Multivariate geometric autoregressive and autoregressive moving average models

2021 ◽  
Vol 12 (2) ◽  
pp. 12-18
Author(s):  
S.M. Umar ◽  
S. Bala
Keyword(s):  
Autoregressive Model ◽  
Geometric Distribution ◽  
Moving Average ◽  
Autoregressive Moving Average ◽  
Positive Dependence ◽  
Associated Random Variables ◽  
Moving Average Models ◽  
Multivariate Geometric Distribution ◽  
Special Case ◽  
Autoregressive Moving Average Models

We present Autoregressive (AR) and autoregressive moving average (ARMA) processes with multivariate geometric (MG) distribution. The theory of positive dependence is used to show that in many cases, multivariate geometric autoregressive (MGAR) and multivariate autoregressive moving average (MGARMA) models consist of associated random variables. We also provide a special case of the multivariate geometric autoregressive model in which it is stationary and has multivariate geometric distribution.

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