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Limit of the environment viewed from Sinaï’s walk

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  • Comets, Francis
  • Loukianov, Oleg
  • Loukianova, Dasha

Abstract

For Sinaï’s walk (Xk) we show that the empirical measure of the environment seen from the particle (ω̄k) converges in law to some random measure S∞. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov (1984). As a consequence an “in law” ergodic theorem holds: 1n∑k=1nF(ω̄k)⟶ℒ∫ΩFdS∞.When the last limit is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic “environment’s method” dating back to Kozlov and Molchanov (1984). In particular, we show an LLN and a mixed CLT for the sums ∑k=1nf(ΔXk) where f is bounded and depending on the steps ΔXk≔Xk+1−Xk.

Suggested Citation

  • Comets, Francis & Loukianov, Oleg & Loukianova, Dasha, 2024. "Limit of the environment viewed from Sinaï’s walk," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:spapps:v:168:y:2024:i:c:s0304414923002387
    DOI: 10.1016/j.spa.2023.104266
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    References listed on IDEAS

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    1. Bertoin, Jean, 1993. "Splitting at the infimum and excursions in half-lines for random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 17-35, August.
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