Estimates for the distribution of sums and maxima of sums of random variables without the Cramér condition
type: Item
instance of: scholarly article
title: Estimates for the distribution of sums and maxima of sums of random variables without the Cramér condition
zbMATH Open document ID: 0969.60047
DOI: 10.1007/BF02674739
author: A. A. Borovkov
published in: Siberian Mathematical Journal
publication date: 2001-06-10T00:00:00Z
full work available at URL: https://eudml.org/doc/121252
review text: Let \(X_1,X_2,\dots\) be identically distributed independent random variables with distribution function \(F(t)\), and let \[ S_n=\sum_{j=1}^n X_j,\quad \overline S_n(a)=\max_{k\leq n}(S_k-ka). \] It is assumed that the tails \(F(-t)=\mathbf{P}\{X_j<-t\}\) and \(1-F(t)=\mathbf{P}\{X_j\geq t\}\) are majorized or minorized by a regularly varying or semiexponential function. The main results concern the upper and lower bounds for the probabilities \(\mathbf{P}\{S_n>x\}\) and \(\mathbf{P}\{\overline S_n(a)>x\}\). Various cases are considered. Among them are the cases in which the mean value of \(X_1\) does not exist, the variance does not exist, etc. The main technical tools are a properly chosen truncating procedure and exponential bounds of Chebyshev type. Necessary and sufficient conditions are also presented under which the uniform relative convergence to a stable law holds. The article is concluded with the laws of iterated logarithm in the case when the second moment is infinite.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1603201
zbMATH Keywords: heavy tails
zbMATH Keywords: boundary problem
MaRDI profile type: Publication
cites work: Large deviation probabilities for random walks with semiexponential distributions
cites work: Probabilities of large deviations for random walks with regular distribution of jumps.
cites work: Some Limit Theorems for Large Deviations
cites work: Probability Inequalities for Sums of Independent Random Variables
cites work: Q5184259
cites work: Q3902241
cites work: Q3746581
cites work: Large-Deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential Distribution
cites work: Asymptotic behaviour of Wiener-Hopf factors of a random walk
cites work: Point process and partial sum convergence for weakly dependent random variables with infinite variance
cites work: A contribution to the theory of large deviations for sums of independent random variables
cites work: Q5572037
cites work: Probabilities of Large Deviations for Sums of Independent Random Variables Attracted to a Stable Law
cites work: Large deviations of heavy-tailed sums with applications in insurance
cites work: Large deviations of sums of independent random variables
cites work: On a Property of Sums of Independent Random Variables
cites work: Nonuniform estimate of the rate of convergence in the CLT with stable limit distribution
cites work: On the maximum of sums of random variables and the supremum functional for stable processes
cites work: Q3339031
cites work: Q5687297
cites work: Q4506364
description: scientific article; zbMATH DE number 1603201