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Harmonic Functions of Random Walks in a Semigroup via Ladder Heights

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  • Irina Ignatiouk-Robert

    (Université de Cergy-Pontoise)

Abstract

We investigate harmonic functions and the convergence of the sequence of ratios $$({\mathbb {P}}_x(\tau _\vartheta> n)/{\mathbb {P}}_e(\tau _\vartheta > n))$$ ( P x ( τ ϑ > n ) / P e ( τ ϑ > n ) ) for a random walk on a countable group killed upon the time $$\tau _\vartheta $$ τ ϑ of the first exit from some semigroup with an identity element e. Several results of classical renewal theory for one-dimensional random walk killed at the first exit from the positive half-line are extended to a multi-dimensional setting. For this purpose, an analogue of the ladder height process and the corresponding renewal function V are introduced. The results are applied to multi-dimensional random walks (X(t)) killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet–Deny theory.

Suggested Citation

  • Irina Ignatiouk-Robert, 2021. "Harmonic Functions of Random Walks in a Semigroup via Ladder Heights," Journal of Theoretical Probability, Springer, vol. 34(1), pages 34-80, March.
  • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-019-00974-1
    DOI: 10.1007/s10959-019-00974-1
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    1. Duraj, Jetlir, 2014. "Random walks in cones: The case of nonzero drift," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1503-1518.
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