Persistent convergence on randomly deleted sets
type: Item
instance of: scholarly article
title: Persistent convergence on randomly deleted sets
zbMATH Open document ID: 0809.60029
DOI: 10.1016/0167-7152(94)90128-7
author: Mark D. Rothmann
author: Ralph P. Russo
published in: Statistics \& Probability Letters
publication date: 1994-11-06T00:00:00Z
review text: Let \(0 \leq t_ 1 < t_ 2 < \dots\) be a fixed sequence of times with \(t_ n \to \infty\) as \(n \to \infty\). At time \(t_ k\) a unit with magnitude \(X_ k\) and lifetime \(L_ k\) enters a population or is placed into a system. An observer, inspecting the system at time \(t\), sees only the units with indices in \(S(t) = \{k : t_ k \leq t < t_ k + L_ k\}\). Let \(B_ n\) denote the sequence \(\{X_ k\}_{1 \leq k \leq n}\) and \(B^*_ t\) the sequence \(\{X_ k\}_{k \in S(t)}\). Suppose \(\lambda\) is a real-valued measurable function defined on the collection of finite sequences of real numbers (including the empty sequence \(\emptyset)\), with \(\lambda (\emptyset) = 0\), and symmetric in the sense that \(\lambda (x_ 1, \dots, x_ n) = \lambda (x_{r_ 1}, \dots, x_{r_ n})\) for all permutations \((x_{r_ 1}, \dots, x_{r_ n})\) of \((x_ 1, \dots, x_ n)\). In many cases \(\lambda (B_ n)\) is convergent in probability, or with probability one, to a constant \(\varphi\), or weakly convergent to a random variable \(\varphi\). The authors present conditions under which \(\lambda (B_ t^*)\) converges or fails to converge to \(\varphi\) in the same sense. The authors have been working under the assumption that \(X_ n\), \(n \geq 1\), are i.i.d. random variables, and \(L_ n\), \(n \geq 1\), are i.i.d. nonnegative random variables, jointly independent of the \(X_ k\)'s.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 681907
zbMATH Keywords: strong and weak convergence
zbMATH Keywords: persistent convergence
zbMATH Keywords: random deletion
reviewed by: Zdzisław Rychlik
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)90128-7
OpenAlex ID: W2047112825
cites work: Q3794956
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cites work: Q4085018
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description: scientific article; zbMATH DE number 681907