Conditioned limit theorems for products
of positive random matrices

Pham, Thi Da Cam

doi:10.30757/alea.v15-04

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Cited by 13 publications

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“…The following condition ensures that the harmonic function V is strictly positive (V (x, y) > 0 for any x ∈ X and y 0), see [19,33].…”

Section: A4mentioning

confidence: 99%

Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

Xiao,

Grama,

Liu

2023

Sci. China Math.

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Consider the random matrix products Gn := gn . . . g1, where (gn) n 1 is a sequence of independent and identically distributed positive random d × d matrices for any integer d 2. For any starting point x ∈ R d + with |x| = 1 and y 0, consider the exit time τx,y = inf{k 1 : y + log |G k x| < 0}. In this paper, we study the conditioned local probability P(y + log |Gnx| ∈ [0, ∆] + z, τx,y > n) under various assumptions on y and z. For y = o(√ n), we prove precise upper and lower bounds when z is in a compact interval and give exact asymptotics when z → ∞. We also study the case when y ≍ √ n and establish the corresponding asymptotics in function of the behaviour of z.

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“…The following condition ensures that the harmonic function V is strictly positive (V (x, y) > 0 for any x ∈ X and y 0), see [19,33].…”

Section: A4mentioning

confidence: 99%