Projective convergence of columns for inhomogeneous products of matrices with nonnegative entries

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Let Pₙ be the n-step right product A₁ Aₙ, where A₁,A₂, is a given infinite sequence of d× d matrices with nonnegative entries. In a wide range of situations, the normalized matrix product Pₙ/ Pₙ does not converge and we shall be rather interested in the asymptotic behavior of the normalized columns PₙUᵢ/ PₙUᵢ, where U₁,,Ud are the canonical d× 1 vectors. Our main result in Theorem A gives a sufficient condition (C) over the sequence A₁,A₂, ensuring the existence of dominant columns of Pₙ, having the same projective limit V: more precisely, for any rank n, there exists a partition of {1,,d} made of two subsets Jₙ≠ and Jₙᶜ such that each one of the sequences of normalized columns, say PₙUⱼₙ/ PₙUⱼₙ with jₙ∈ Jₙ tends to V as n tends to +∞ and are dominant in the sense that the ratio PₙUⱼₙ'/ PₙUⱼₙ tends to 0, as soon as jₙ'∈ Jₙᶜ. The existence of sequences of such dominant columns implies that for any probability vector X with positive entries, the probability vector PₙX/ PₙX, converges as n tends to +∞. Our main application of Theorem A (and our initial motivation) is related to an Erd os problem concerned with a family of probability measures μβ (for 1<β<2 a real parameter) fully supported by a subinterval of the real line, known as Bernoulli convolutions.