A Conditioned Local Limit Theorem for Nonnegative Random Matrices

Marc Peigné () and Da Cam Pham ()
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Marc Peigné: Université de Tours, Université d’Orléans, CNRS
Da Cam Pham: ESAIP

Journal of Theoretical Probability, 2024, vol. 37, issue 2, 1882-1901

Abstract: Abstract For any fixed real $$a > 0$$ a > 0 and $$x \in {\mathbb {R}}^d, d \ge 1$$ x ∈ R d , d ≥ 1 , we consider the real-valued random process $$(S_n)_{n \ge 0}$$ ( S n ) n ≥ 0 defined by $$ S_0= a, S_n= a+\ln \vert g_n\cdots g_1x\vert , n \ge 1$$ S 0 = a , S n = a + ln | g n ⋯ g 1 x | , n ≥ 1 , where the $$g_k, k \ge 1, $$ g k , k ≥ 1 , are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of d-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process $$(S_n)_{n \ge 0}$$ ( S n ) n ≥ 0 remains nonnegative up to time n and simultaneously belongs to some compact set $$[b, b+\ell ]\subset {\mathbb {R}}^+_*$$ [ b , b + ℓ ] ⊂ R ∗ + at time n.

Keywords: Local limit theorem; Random walk; Product of random matrices; Markov chains; First exit time; 60B15; 60F15 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01336-2

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