Analysis finds convergence behavior in products of positive random matrices, indicating key properties for high-dimensional systems.
We consider the products Gₙ = Aₙ ⋯ A₁ of independent and identical distributed nonnegative d × d matrices (Aᵢ)i ≥ 1. For any starting point x ∈ R₊ᵈ with unit norm, we establish the convergence to a stable law for the norm cocycle log | Gₙx |, jointly with its direction Gₙ · x = Gₙ x / | Gₙ x |. We also prove a local limit theorem for the couple (log |Gₙx|, Gₙ · x), and find the exact rate of its convergence.
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Mei et al. (2025) studied this question.
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