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Weak convergence on randomly deleted sets

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mark D. Rothmann  and
Ralph P. Russo
Mark D. Rothmann*
Affiliation:
University of Iowa
Ralph P. Russo*
Affiliation:
University of Iowa
*
∗Postal address: Department of Statistics, University of Iowa, Iowa City, IA 52242, USA
∗Postal address: Department of Statistics, University of Iowa, Iowa City, IA 52242, USA
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Abstract

Suppose t1, t2,… are the arrival times of units into a system. The kth entering unit, whose magnitude is Xk and lifetime Lk, is said to be ‘active’ at time t if I(tk < tk + Lk) = Ik,t = 1. The size of the active population at time t is thus given by At = ∑k≥1Ik,t. Let Vt denote the vector whose coordinates are the magnitudes of the active units at time t, in their order of appearance in the system. For n ≥ 1, suppose λn is a measurable function on n-dimensional Euclidean space. Of interest is the weak limiting behaviour of the process λ*(t) whose value is λm(Vt) or 0, according to whether At = m > 0 or At = 0.

Keywords

Weak convergencerandom deletion

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G17: Sample path properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 893 - 902
Copyright
Copyright © Applied Probability Trust 1998 

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References

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