Estimate of the rate of convergence of distributions of additive functionals of a sequence of sums of independent random variables

1978 ◽  
Vol 19 (3) ◽  
pp. 371-383 ◽  
Author(s):  
I. S. Borisov
Keyword(s):  
Rate Of Convergence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Additive Functionals

scholarly journals Complete convergence for weighted sums of pairwise independent random variables

2017 ◽  
Vol 15 (1) ◽  
pp. 467-476
Author(s):  
Li Ge ◽  
Sanyang Liu ◽  
Yu Miao
Keyword(s):  
Rate Of Convergence ◽  
Complete Convergence ◽  
Moving Average ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Pairwise Independent Random Variables ◽  
Moving Average Processes

Abstract In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

A Central Limit Theorem for Cumulative Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

On Fibonacci (or lagged Bienaymé-Galton-Watson) branching processes

1981 ◽  
Vol 18 (03) ◽  
pp. 583-591
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Discrete Time ◽  
Population Model ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
First Time

This paper is concerned with a discrete-time population model in which a new individual entering the population at time t can produce offspring for the first time at time t + 2 and then subsequently at times t + 3, t + 4, ···. The numbers of offspring produced on each occasion are independent random variables each with the distribution of Z for which EZ = m <∞, and individuals have independent lines of descent. This model is contrasted with the corresponding Bienaymé-Galton-Watson one. If Xn denotes the number of individuals in the population at time n, it is shown that z–nXn almost surely converges to a random variable W, as n→∞, where Various properties of W are obtained, in particular W > 0 a.s. if and only if EZ | log Z | < ∞ Results are also given on the rate of convergence of to z when Var Z < ∞ and these display a surprising dependence on the size of z.

On some rate of convergence questions

2010 ◽  
Vol 47 (3) ◽  
pp. 373-387
Author(s):  
Dao Tuyen
Keyword(s):  
Rate Of Convergence ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Convergence In Probability ◽  
Weighted Sums

This paper gives answers to some questions posed in Hanson and Wright, Z. Wahrscheinlichkeitstheor. Verw. Geb. , 19 (1971), on rates of convergence in probability to zero for weighted sums of independent random variables.

scholarly journals On the constant in the nonuniform version of the Berry-Esseen theorem

2005 ◽  
Vol 2005 (12) ◽  
pp. 1951-1967 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Rate Of Convergence ◽  
Random Variables ◽  
Stein’S Method ◽  
Independent Random Variables ◽  
Stein's Method ◽  
Concentration Inequality

In 2001, Chen and Shao gave the nonuniform estimation of the rate of convergence in Berry-Esseen theorem for independent random variables via Stein-Chen-Shao method. The aim of this paper is to obtain a constant in Chen-Shao theorem, where the random variables are not necessarily identically distributed and the existence of their third moments are not assumed. The bound is given in terms of truncated moments and the constant obtained is21.44for most values. We use a technique called Stein's method, in particular the Chen-Shao concentration inequality.

On Fibonacci (or lagged Bienaymé-Galton-Watson) branching processes

1981 ◽  
Vol 18 (3) ◽  
pp. 583-591 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Discrete Time ◽  
Population Model ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
First Time

This paper is concerned with a discrete-time population model in which a new individual entering the population at time t can produce offspring for the first time at time t + 2 and then subsequently at times t + 3, t + 4, ···. The numbers of offspring produced on each occasion are independent random variables each with the distribution of Z for which EZ = m <∞, and individuals have independent lines of descent. This model is contrasted with the corresponding Bienaymé-Galton-Watson one. If Xn denotes the number of individuals in the population at time n, it is shown that z–nXn almost surely converges to a random variable W, as n→∞, where Various properties of W are obtained, in particular W > 0 a.s. if and only if EZ | log Z | < ∞ Results are also given on the rate of convergence of to z when Var Z < ∞ and these display a surprising dependence on the size of z.

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