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

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.