A criterion of convergence of nonrandomly centered random sums of independent identically distributed random variables

1998 ◽  
Vol 89 (5) ◽  
pp. 1495-1506 ◽  
Author(s):  
V. Yu. Korolev ◽  
V. M. Kruglov
Keyword(s):  
Random Variables ◽  
Random Sums ◽  
Independent Identically Distributed

scholarly journals Large deviations for weighted random sums

2013 ◽  
Vol 18 (2) ◽  
pp. 129-142 ◽  
Author(s):  
Aurelija Kasparavičiūtė ◽  
Leonas Saulis
Keyword(s):  
Large Deviations ◽  
Normal Approximation ◽  
Random Variables ◽  
Main Idea ◽  
Random Variable ◽  
Tail Probability ◽  
Random Sums ◽  
Exponential Inequalities ◽  
The Difference ◽  
Independent Identically Distributed

In the present paper we consider weighted random sums ZN = ∑j=1NajXj, where 0 ≤ aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throughout this paper N is independent of {X, Xj , j = 1, 2,...} and, for definiteness, it is assumed Z0 = 0. The main idea of the paper is to present results on theorems of large deviations both in the Cramér and power Linnik zones for a sum ~ZN = (ZN − EZN )(DZN )−1/2 , exponential inequalities for a tail probability P(~ZN > x) in two cases: µ = 0 and µ ≠ 0 pointing out the difference between them. Only normal approximation is considered. It should be noted that large deviations when µ ≠ 0 have been already considered in our papers [1,2].

scholarly journals On the rates of convergence to symmetric stable laws for distributions of normalized geometric random sums

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3073-3084
Author(s):  
Tran Hung ◽  
Phan Kien
Keyword(s):  
Rate Of Convergence ◽  
Random Variables ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Stable Laws ◽  
Random Sums ◽  
Probability Metric ◽  
Geometric Random Variable ◽  
Independent Identically Distributed

Let X1,X2,... be a sequence of independent, identically distributed random variables. Let ?p be a geometric random variable with parameter p?(0,1), independent of all Xj, j ? 1: Assume that ? : N ? R+ is a positive normalized function such that ?(n) = o(1) when n ? +?. The paper deals with the rate of convergence for distributions of randomly normalized geometric random sums ?(?p) ??p,j=1 Xj to symmetric stable laws in term of Zolotarev?s probability metric.

scholarly journals Asymptotics for Weighted Random Sums

2012 ◽  
Vol 44 (04) ◽  
pp. 1142-1172 ◽  
Author(s):  
Mariana Olvera-Cravioto
Keyword(s):  
Random Vector ◽  
Branching Processes ◽  
Random Variables ◽  
Random Sum ◽  
Autoregressive Processes ◽  
Random Sums ◽  
Regularly Varying ◽  
Independent Identically Distributed

Let {X i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C 1, C 2,…) be a nonnegative random vector independent of the {X i } with N∈ℕ∪ {∞}. We study the weighted random sum S N =∑{i=1} N C i X i , and its maximum, M N =sup{1≤k N+1∑ i=1 k C i X i . This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M N &gt; x)∼ P(S N &gt; x)∼ E[∑ i=1 N F̄(x/C i )] as x→∞. When E[X 1]&gt;0 and the distribution of Z N =∑ i=1 N C i is also intermediate regularly varying, we obtain the asymptotics P(M N &gt; x)∼ P(S N &gt; x)∼ E[∑ i=1 N F̄}(x/C i )] +P(Z N &gt; x/E[X 1]). For completeness, when the distribution of Z N is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(M N &gt; x) ∼ P(S N &gt; x)∼ P(Z N &gt; x / E[X 1 ] hold.

scholarly journals Asymptotics for Weighted Random Sums

2012 ◽  
Vol 44 (4) ◽  
pp. 1142-1172 ◽  
Author(s):  
Mariana Olvera-Cravioto
Keyword(s):  
Random Vector ◽  
Branching Processes ◽  
Random Variables ◽  
Random Sum ◽  
Autoregressive Processes ◽  
Random Sums ◽  
Regularly Varying ◽  
Independent Identically Distributed

Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C1, C2,…) be a nonnegative random vector independent of the {Xi} with N∈ℕ∪ {∞}. We study the weighted random sum SN=∑{i=1}NCiXi, and its maximum, MN=sup{1≤kN+1∑i=1kCiXi. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄(x/Ci)] as x→∞. When E[X1]>0 and the distribution of ZN=∑ i=1NCi is also intermediate regularly varying, we obtain the asymptotics P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄}(x/Ci)] +P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x)∼ P(ZN > x / E[X1] hold.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

2021 ◽  
Vol 499 (1) ◽  
pp. 124982
Author(s):  
Benjamin Avanzi ◽  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet ◽  
Bernard Wong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Independent Identically Distributed
Sign in / Sign up

Export Citation Format

Share Document