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.

TWO-MACHINE FLOW-SHOP MINIMUM-LENGTH SCHEDULING WITH INTERVAL PROCESSING TIMES

2009 ◽  
Vol 26 (06) ◽  
pp. 715-734 ◽  
Author(s):  
C. T. NG ◽  
NATALJA M. MATSVEICHUK ◽  
YURI N. SOTSKOV ◽  
T. C. EDWIN CHENG
Keyword(s):  
Flow Shop ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Minimum Length ◽  
Lower And Upper Bounds ◽  
Minimal Set ◽  
Processing Times ◽  
Necessary And Sufficient ◽  
Two Machines

The flow-shop minimum-length scheduling problem with n jobs processed on two machines is addressed where processing times are uncertain: only lower and upper bounds of the random processing times are given before scheduling, but their probability distributions are unknown. For such a problem, there may not exist a dominant schedule that remains optimal for all possible realizations of the processing times and so we look for a minimal set of schedules that are dominant. We obtain necessary and sufficient conditions for the case when it is possible to fix the order of two jobs in a minimal set of dominant schedules. The necessary and sufficient conditions are proven for the case when one schedule dominates all the others. We characterize also the case where there does not exist non-trivial schedule domination. All the conditions proven may be tested in polynomial time of n.

scholarly journals Weighted Strong Law of Large Numbers for Random Variables Indexed by a Sector

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Przemysław Matuła ◽  
Michał Seweryn
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Multidimensional Indices ◽  
Necessary And Sufficient

We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector.

A theorem on the cumulative product of independent random variables

1959 ◽  
Vol 55 (4) ◽  
pp. 333-337 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Product Theorem ◽  
Image Position ◽  
Introductory Discussion ◽  
Necessary And Sufficient ◽  
Complex Valued

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under whichreferred to as the product theorem.

scholarly journals On bounds for convergence rates in combinatorial strong limit theorems and its applications

2020 ◽  
Vol 65 (4) ◽  
pp. 688-698
Author(s):  
Andrei N. Frolov ◽  
Keyword(s):  
Limit Theorems ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Independent Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Set Of Permutations ◽  
Combinatorial Sums ◽  
Necessary And Sufficient

We find necessary and sufficient conditions for convergences of series of weighted probabilities of large deviations for combinatorial sums i Xniπn(i), where Xnij is a matrix of order n of independent random variables and (πn(1), πn(2), . . . , πn(n)) is a random permutation with the uniform distribution on the set of permutations of numbers 1, 2, . . . , n, independent with Xnij. We obtain combinatorial variants of results on convergence rates in the strong law of large numbers and the law of the iterated logarithm under conditions closed to optimal ones. We discuss applications to rank statistics.

scholarly journals On a new Class of Tempered Stable Distributions: Moments and Regular Variation

2012 ◽  
Vol 49 (4) ◽  
pp. 1015-1035 ◽  
Author(s):  
Michael Grabchak
Keyword(s):  
Regular Variation ◽  
Stable Distribution ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Tempered Stable Distribution ◽  
New Class ◽  
Tempered Stable Distributions ◽  
Necessary And Sufficient

We extend the class of tempered stable distributions, which were first introduced in Rosiński (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.

Multivariate Convex Orderings, Dependence, and Stochastic Equality

1998 ◽  
Vol 35 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Marco Scarsini
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Multivariate Normal ◽  
Random Vectors ◽  
Linear Combinations ◽  
Convex Ordering ◽  
Normal Distributions ◽  
Necessary And Sufficient ◽  
Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

scholarly journals On a new Class of Tempered Stable Distributions: Moments and Regular Variation

2012 ◽  
Vol 49 (04) ◽  
pp. 1015-1035
Author(s):  
Michael Grabchak
Keyword(s):  
Regular Variation ◽  
Stable Distribution ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Tempered Stable Distribution ◽  
New Class ◽  
Tempered Stable Distributions ◽  
Necessary And Sufficient

We extend the class of tempered stable distributions, which were first introduced in Rosiński (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.

Multivariate Convex Orderings, Dependence, and Stochastic Equality

1998 ◽  
Vol 35 (01) ◽  
pp. 93-103 ◽  
Author(s):  
Marco Scarsini
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Multivariate Normal ◽  
Random Vectors ◽  
Linear Combinations ◽  
Convex Ordering ◽  
Normal Distributions ◽  
Necessary And Sufficient ◽  
Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

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