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On powers of likelihood functions of random walks on ℤͩ

Published online by Cambridge University Press:  01 February 2019

Krishna B. Athreya
Krishna B. Athreya*
Affiliation:
Iowa State University
*
Department of Mathematics and Department of Statistics, Iowa State University, Ames, IA, USA. Email address: [email protected]
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Abstract

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Let {Xi}i≥1 be independent, identically distributed random vectors in ℤd,d≥1. Let LLn(x)≡ℙ(Sn=x),n≥1,x∈ℤd, be the likelihood function for Sn=∑i=1nXi. For integers j≥2 and n≥1, let an(j)≡∑x∈ℤd(Ln(x))j. We show that if X1-X2 has a nondegenerate aperiodic distribution in ℤd and 𝔼(∥X1∥2)>∞, then limn→∞n(j-1)d∕2an(j)≡a(j,d) exists and 0<a(j,d)<∞. Some extensions and open problems are also outlined.

Keywords

Likelihood functionrandom walkpowers
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 63 - 66
Copyright
Copyright © Applied Probability Trust 2018 

References

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