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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: [email protected]
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Abstract

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) 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
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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