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On Seneta–Heyde scaling for a stable branching random walk

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  26 July 2018

Hui He ,
Jingning Liu  and
Mei Zhang
Hui He*
Affiliation:
Beijing Normal University
Jingning Liu*
Affiliation:
Beijing Normal University
Mei Zhang*
Affiliation:
Beijing Normal University
*
* Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
* Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
* Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
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Abstract

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 < α < 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

Keywords

Branching random walkdomain of attractionSeneta–Hedye scalingstable distributionderivative martingaleadditive martingale

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 565 - 599
Copyright
Copyright © Applied Probability Trust 2018 

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