A conditioned local limit theorem for non-negative random matrices

M. Peigné (1) , Thi da Cam Pham (2, 3)

1 IDP - Institut Denis Poisson
2 CERADE ESAIP - ESAIP Centre d’Etudes et de Recherche pour l’Aide à la Décision
3 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

Let $(S_n)_n$ be the random process on $\mathbb R$ driven by the product of i.i.d. non-negative random matrices and $\tau$ its exit time from $]0, +\infty[$. By using the adapted strategy initiated by D. Denisov and V. Wachtel, we obtain an asymptotic estimate and bounds of the probability that the process $(S_k)_k$ remains non negative up to time $n$ and simultaneously belongs to some compact set $[b, b+\ell ]\subset \mathbb R^{*+}$ at time $n$.

Mots clés: product of random matrices; Markov chains; first exit time; local limit theorem; random walk

Domaines: Probabilités [math.PR]

M. Peigné, Thi da Cam Pham. A conditioned local limit theorem for non-negative random matrices. 2023. ⟨hal-03939554⟩