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Article Dans Une Revue Probability Theory and Related Fields Année : 2020

Conditioned local limit theorems for random walks defined on finite Markov chains

Résumé

Let (X n) n 0 be a Markov chain with values in a finite state space X starting at X 0 = x ∈ X and let f be a real function defined on X. Set S n = n k=1 f (X k), n 1. For any y ∈ R denote by τ y the first time when y + S n becomes non-positive. We study the asymptotic behaviour of the probability P x (y + S n ∈ [z, z + a] , τ y > n) as n → +∞. We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order n 3/2 and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability P x (τ y = n) as n → +∞.
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Dates et versions

hal-02878963 , version 1 (23-06-2020)

Identifiants

Citer

Ion Grama, Ronan Lauvergnat, Emile Le Page. Conditioned local limit theorems for random walks defined on finite Markov chains. Probability Theory and Related Fields, 2020, 176 (1-2), pp.669-735. ⟨10.1007/s00440-019-00948-8⟩. ⟨hal-02878963⟩
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