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

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)i 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 log | 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 (log |Gₙx|, Gₙ · x) ( log | G n x | , G n · x ) and find the exact rate of its convergence.