Limit theorems for random matrix products with a simple top Lyapunov exponent
Abstract
We study products of independent, identically distributed random matrices with an exponential moment and a simple top Lyapunov exponent, without assuming irreducibility. Outside an exceptional invariant subspace called the equator, we establish contraction on average for a weighted projective distance and a spectral gap for the associated Markov operator. We obtain a central limit theorem, a Berry--Esseen bound, deviation estimates near the mean, and an almost sure invariance principle both for the growth of a fixed vector outside the equator and for the operator norm. For fixed vectors, a non-arithmeticity assumption also yields a local limit theorem and a renewal theorem. Our contraction criterion applies more generally to mostly contracting random maps on locally compact spaces with a Lyapunov function, including certain random interval maps.
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BibTeX
@misc{barrientos2026limit,
title = {{Limit theorems for random matrix products with a simple top Lyapunov exponent}},
author = {Pablo G. Barrientos and Dominique Malicet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12678},
eprint = {2610.12678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12678},
}
RIS
TY - UNPB
TI - Limit theorems for random matrix products with a simple top Lyapunov exponent
AU - Pablo G. Barrientos
AU - Dominique Malicet
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12678
UR - https://arxiv.org/abs/2610.12678
AB - We study products of independent, identically distributed random matrices with an exponential moment and a simple top Lyapunov exponent, without assuming irreducibility. Outside an exceptional invariant subspace called the equator, we establish contraction on average for a weighted projective distance and a spectral gap for the associated Markov operator. We obtain a central limit theorem, a Berry--Esseen bound, deviation estimates near the mean, and an almost sure invariance principle both for the growth of a fixed vector outside the equator and for the operator norm. For fixed vectors, a non-arithmeticity assumption also yields a local limit theorem and a renewal theorem. Our contraction criterion applies more generally to mostly contracting random maps on locally compact spaces with a Lyapunov function, including certain random interval maps.
ER -