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Journal of Applied Probability Journal of Applied Probability

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Weak convergence on randomly deleted sets

Part of: Stochastic processes Limit theorems

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
Corresponding
E-mail address:
rrusso@stat.uiowa.edu
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Abstract

Suppose t 1, t 2,… are the arrival times of units into a system. The kth entering unit, whose magnitude is X k and lifetime L k , is said to be ‘active’ at time t if I(t k < t k + L k ) = I k,t = 1. The size of the active population at time t is thus given by A t = ∑ k≥1 I k,t . Let V t 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 λ n is a measurable function on n-dimensional Euclidean space. Of interest is the weak limiting behaviour of the process λ*(t) whose value is λ m (V t ) or 0, according to whether A t = m > 0 or A t = 0.

Keywords

Weak convergence random 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

Baum, L. E., and Katz, M. (1963). Convergence rates in the law of large numbers. Bull. Amer. Math. Soc. 69, 771–772.CrossRefGoogle Scholar
Billingsley, P. (1986). Probability and Measure, 2nd edn. Wiley, New York.Google Scholar
Rothmann, M. D., and Russo, R. P. (1994). Persistent convergence on randomly deleted sets. Statist. Prob. Lett. 20, 367–373.CrossRefGoogle Scholar
Serfling, R. J. (1983). Approximation Theorems in Mathematical Statistics. Wiley, New York.Google Scholar
Uspensky, J. V. (1937). Introduction to Mathematical Probability. McGraw-Hill, New York.Google Scholar

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