The asymptotic distribution of sums of records
type: Item
instance of: scholarly article
title: The asymptotic distribution of sums of records
zbMATH Open document ID: 0931.60011
DOI: 10.1023/A:1009933902505
author: Barry C. Arnold
author: José A. Villaseñor
published in: Extremes
publication date: 1999-08-10T00:00:00Z
review text: Let \(X_1,X_2,\dots\) be a sequence of independent identically distributed random variables. Denote by \(X^{(0)}=X_1, X^{(1)}, X^{(2)},\dots \) the associated upper record sequence and take \(T_n=\sum_{j=0}^n X^{(j)}\). The authors describe three cases in which \(T_n\) can be properly normalized to have a nontrivial limit distribution. More precisely, if \(X_1\) has the standard exponential or Gumbel distribution, then the sums of records are asymptotically normal; on the other hand, sums of beta records (including uniform records) are attracted to non-normal limit law \(L_{\gamma}\) with the Laplace transform \(\psi_{\gamma}(t)=\exp\{(-1/\gamma) \int_0^1 (1-\exp(-tx))/x dx\}\). These cases are investigated by essentially different approaches, and the problem of identifying all possible limit laws for the normalized \(T_n\) remains open.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 1323316
zbMATH Keywords: associated upper records
zbMATH Keywords: sums of records
zbMATH Keywords: convergence in distribution
zbMATH Keywords: beta records
zbMATH Keywords: Gumbel records
zbMATH Keywords: exponential records
zbMATH Keywords: asymptotic normality
MaRDI profile type: Publication
full work available at URL: a:1009933902505
OpenAlex ID: W1498019195
Recommended article: On distributions of sums of record values
Recommended article: Asymptotic distributions and {B}erry-{E}sseen bounds for sums of record values
Recommended article: The asymptotic distributions of sums of record values for distributions with regularly varying tails.
Recommended article: Q3469964
Recommended article: On the classes of asymptotic distributions for the sums of record values. I
Recommended article: On the classes of asymptotic distributions for the sums of record values. II
Recommended article: Q4821671
Recommended article: The asymptotic distributions of sums of record values for distributions with lognormal-type tails
Recommended article: Asymptotic properties of sums of upper records
description: scientific article; zbMATH DE number 1323316