Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T05:06:26.702Z Has data issue: false hasContentIssue false
  • English
  • Français

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: kbathreya@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Athreya, K. B. and Janicki, R. (2016).Asymptotics of powers of binomial and multinomial probabilities.Statist. Prob. Lett. 112,58–62.Google Scholar
[2]Davis, B. and McDonald, D. (1995).An elementary proof of the local central limit theorem.J. Theoret. Prob. 8,693–701.Google Scholar
[3]Farmer, J. D. and Leth, S. C. (2005).An asymptotic formula for powers of binomial coefficients.Math. Gazette 89,385–391.Google Scholar
[4]Pólya, G. and Szegö, G. (1945).Aufgaben und Lehrsätze aus der Analysis.Dover,New York (in German).Google Scholar
[5]Spitzer, F. (1976).Principles of Random Walk (Graduate Texts Math. 34),2nd edn.Springer,New York.Google Scholar