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September 5, 2025Open Access

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

Key Points

Convergence to a stable law is established for random matrix products, enhancing understanding of their behavior.
A local limit theorem is proven for the couples of log norms and directions, providing insights into convergence rates.
The analysis is based on products of nonnegative random matrices, expanding foundational knowledge in probability theory.
The results offer implications for systems modeled by random matrices, suggesting potential real-world applications.

Abstract

Abstract We consider the products Gₙ = Aₙ A₁ G n = A n ⋯ A 1 of independent and identically distributed nonnegative d d d × d matrices (Aᵢ) ₈ ₁ (A i) i ⩾ 1. For any starting point x {R}_+ᵈ x ∈ R + d with unit norm, we establish the convergence to a stable law for the norm cocycle | Gₙx | log | G n x |, jointly with its direction Gₙ x = Gₙ x / | Gₙ x | G n · x = G n x / | G n x |. We also prove a local limit theorem for the couple (|Gₙx|, Gₙ x) (log | G n x |, G n · x) and find the exact rate of its convergence.

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

Key Points

Abstract

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