Maxima with random indexes
type: Item
instance of: scholarly article
title: Maxima with random indexes
zbMATH Open document ID: 0911.60001
DOI: 10.1007/BF02882638
author: Shengrong Xie
published in: Chinese Science Bulletin
publication date: 1999-04-26T00:00:00Z
review text: Let \(\{\xi_i\}\) and \(\{X_i\}\) be two sequences of r.v.'s, where \(\{X_i\}\) is i.i.d. Put \(M_n=\bigvee_{i=1}^n \xi_i\) and \(\widehat M_n=\bigvee_{i=1}^n X_i.\) Let \(U\) be a regularly varying function (at \(+\infty\)) with index \(\rho\) (write \(U\in \text{RV}_\rho\)). Let \(\eta\) be a positive r.v. and \(U\in\text{RV}_\rho\) \((\rho>0).\) Suppose that for any real numbers \(u_n\) \[ \big| P\big( M_n\leq u_n\mid \eta=y\big)-P\big( \widehat M_{[U(n)]}\leq u_n\mid\eta=y\big) \big|\longrightarrow 0,\qquad n\to\infty, \] for \(y>0.\) Let \(\{N_n\}\) be a positive integer-valued r.v. such that \(N_n/n {\buildrel P\over\rightarrow}\eta\). If there exist \(a_n>0,\) \(b_n\in{\mathbb R},\) \(n\geq 1,\) such that \[ P\big( M_n\leq a_nx+b_n \big) {\buildrel W\over\longrightarrow} G(x),\quad n\uparrow\infty, \] where \(G\) is nondegenerate distribution function, then \[ P\big( M_{N_n}\leq a_{[U(n)]}x+b_{[U(n)]} \big) {\buildrel W\over\longrightarrow} \int_0^{+\infty}\big( G(x) \big)^{y^\rho}dP(\eta\leq y),\quad n\uparrow\infty. \]
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 1159090
zbMATH Keywords: regularly varying function
zbMATH Keywords: maximum
zbMATH Keywords: random index
reviewed by: K. Kubilius
MaRDI profile type: Publication
cites work: Q4162318
cites work: Q5633348
cites work: Regularly varying functions
cites work: Sur la distribution limite du terme maximum d'une série aléatoire
cites work: Q3771297
cites work: Q3757056
description: scientific article; zbMATH DE number 1159090