Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

TL;DR

This paper investigates the asymptotic behavior of the maximum position in a supercritical catalytic branching random walk on Z with heavy-tailed jump distributions, establishing convergence to a non-trivial limit under regular variation.

Contribution

It introduces new results on the weak convergence of the maximum in CBRW with heavy tails, contrasting with known light-tail cases, using advanced probabilistic techniques.

Findings

Maximum of CBRW converges in distribution after normalization.

Explicit normalization formula provided for heavy-tailed jumps.

New integral equations characterize the limiting distribution.

Abstract

For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization and non-linear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with "heavy" tails, in contrast to the known behavior in case of "light" tails of the random walk jumps. The new tools such as "many-to-few lemma" and spinal decomposition appear non-efficient here. The approach developed in the paper combines the techniques of renewal theory, Laplace transform, non-linear integral equations and large deviations theory for random sums of random variables. Keywords and phrases: catalytic branching random walk, heavy tails, regular varying tails, spread of population, L log L condition.