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Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

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  • Ekaterina Vl. Bulinskaya

    (Novosibirsk State University)

Abstract

For a continuous-time catalytic branching random walk (CBRW) on $${\mathbb {Z}}$$ 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 regular variation of the jump distribution tail for underlying random walk and the well-known $$L\log L$$ L log L condition for the offspring numbers. In our classification, given in Bulinskaya (Theory Probab Appl 59(4):545–566, 2015), the analysis refers to supercritical CBRW. The principal 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 nonlinear 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 ineffective here. The approach developed in this paper combines the techniques of renewal theory, Laplace transform, nonlinear integral equations and large deviations theory for random sums of random variables.

Suggested Citation

  • Ekaterina Vl. Bulinskaya, 2021. "Maximum of Catalytic Branching Random Walk with Regularly Varying Tails," Journal of Theoretical Probability, Springer, vol. 34(1), pages 141-161, March.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-020-01009-w
    DOI: 10.1007/s10959-020-01009-w
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    References listed on IDEAS

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    1. Bhattacharya, Ayan & Hazra, Rajat Subhra & Roy, Parthanil, 2018. "Branching random walks, stable point processes and regular variation," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 182-210.
    2. Ekaterina Vladimirovna Bulinskaya, 2014. "Local Particles Numbers in Critical Branching Random Walk," Journal of Theoretical Probability, Springer, vol. 27(3), pages 878-898, September.
    3. E. Yarovaya, 2017. "Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1151-1167, December.
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