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 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.

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.

REFINEMENT OF CONVERGENCE RATES FOR WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES

2012 ◽  
Vol 05 (01) ◽  
pp. 1250007
Author(s):  
Si-Li Niu ◽  
Jong-Il Baek
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Counting Processes ◽  
Precise Asymptotics ◽  
Strong Law ◽  
Large Numbers ◽  
Passage Times

In this paper, we establish one general result on precise asymptotics of weighted sums for i.i.d. random variables. As a corollary, we have the results of Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368], Gut and Spătaru [Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000) 1870–1883; Precise asymptotics in the Baum–Katz and Davis laws of large numbers, J. Math. Anal. Appl. 248 (2000) 233–246], Gut and Steinebach [Convergence rates and precise asymptotics for renewal counting processes and some first passage times, Fields Inst. Comm. 44 (2004) 205–227] and Heyde [A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975) 173–175]. Meanwhile, we provide an answer for the possible conclusion pointed out by Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368].

scholarly journals Complete Moment Convergence for Sung’s Type Weighted Sums ofB-Valued Random Elements

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Wei Li ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Real Numbers ◽  
Strong Law ◽  
Moment Convergence ◽  
Large Numbers ◽  
Random Elements ◽  
Necessary And Sufficient

Letp≥1/αand1/2<α≤1.Let{X,Xn,  n≥1}be a sequence of independent and identically distributedB-valued random elements and let{ani,  1≤i≤n,  n≥1}be an array of real numbers satisfying∑i=1naniq=O(n)for someq>p.We give necessary and sufficient conditions for complete moment convergence of the form∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞,  ∀ε>0, where0<v<p.A strong law of large numbers for weighted sums of independentB-valued random elements is also obtained.

Two-parameter strong laws and maximal inequalities for U-statistics

1987 ◽  
Vol 107 (1-2) ◽  
pp. 133-151 ◽  
Author(s):  
Terry R. McConnell
Keyword(s):  
Law Of Large Numbers ◽  
Weak Type ◽  
Sufficient Conditions ◽  
Strong Law ◽  
Strong Laws ◽  
Maximal Inequalities ◽  
U Statistics ◽  
Large Numbers ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

Convergence rates in the weak law of large numbers for weighted sums of i.i.d. random variables and applications in errors-in-variables models

2019 ◽  
Vol 19 (06) ◽  
pp. 1950041 ◽  
Author(s):  
Mingyang Zhang ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Regression Models ◽  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Weighted Sums ◽  
Errors In Variables ◽  
Error In Variables ◽  
Large Numbers ◽  
Necessary And Sufficient

We prove necessary and sufficient conditions for the convergence rates in the weak law of large numbers for weighted sums of independent and identically distributed (i.i.d.) random variables. This result is applied to simple linear error-in-variables regression models.

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