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Moment convergence in conditional limit theorems

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Svante Janson
Svante Janson*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: svante.janson@math.uu.se
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Abstract

Consider a sum ∑1 N Y i of random variables conditioned on a given value of the sum ∑1 N X i of some other variables, where X i and Y i are dependent but the pairs (X i ,Y i ) form an i.i.d. sequence. We consider here the case when each X i is discrete. We prove, for a triangular array ((X ni ,Y ni )) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

Keywords

conditional distributionlimit theoremmoment convergenceoccupancyhashingrandom forestbranching process

MSC classification

Primary: 60F05: Central limit and other weak theorems

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 421 - 437
Copyright
Copyright © Applied Probability Trust 2001 

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