Convergence to Stable Laws and a Local Limit Theorem for Products of Positive Random Matrices

Jianzhang Mei () and Quansheng Liu ()
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Jianzhang Mei: Tsinghua University
Quansheng Liu: CNRS UMR 6205, LMBA

Journal of Theoretical Probability, 2025, vol. 38, issue 4, 1-31

Abstract: Abstract We consider the products $$G_n = A_n \cdots A_1$$ G n = A n ⋯ A 1 of independent and identically distributed nonnegative $$d \times d$$ d × d matrices $$(A_i)_{i \geqslant 1}$$ ( A i ) i ⩾ 1 . For any starting point $$x \in {\mathbb {R}}_+^d$$ x ∈ R + d with unit norm, we establish the convergence to a stable law for the norm cocycle $$\log | G_nx |$$ log | G n x | , jointly with its direction $$G_n \cdot x = G_n x / | G_n x |$$ G n · x = G n x / | G n x | . We also prove a local limit theorem for the couple $$ (\log |G_nx|, G_n \cdot x)$$ ( log | G n x | , G n · x ) and find the exact rate of its convergence.

Keywords: Products of random matrices; Stable laws; Weak convergence; Rate of convergence; Local limit theorem; Primary 60B10; 60G50; 60E07; Secondary 60B20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01434-9

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