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

Abstract

Let $P_n$ be the $n$-step right product $A_1\cdots A_n$, where $A_1,A_2,\dots$ is a given infinite sequence of $d\times d$ matrices with nonnegative entries. In a wide range of situations, the normalized matrix product $P_n/{\Vert P_n\Vert}$ does not converge and we shall be rather interested in the asymptotic behavior of the normalized columns $P_nU_i/\Vert P_nU_i\Vert$, where $U_1,\dots,U_d$ are the canonical $d\times 1$ vectors. Our main result in Theorem~A gives a sufficient condition ${\bf (C)}$ over the sequence $A_1,A_2,\dots$ ensuring the existence of {\it dominant columns} of $P_n$, having the same projective limit $V$: more precisely, for any rank $n$, there exists a partition of $\{1,\dots,d\}$ made of two subsets $J_n\ne\emptyset$ and $J_n^c$ such that each one of the sequences of normalized columns, say $P_nU_{j_n}/\Vert P_nU_{j_n}\Vert$ with $j_n\in J_n$ tends to $V$ as $n$ tends to $+\infty$ and are {\it dominant} in the sense that the ratio $\Vert P_nU_{j_n'}/\Vert P_nU_{j_n}\Vert$ tends to $0$, as soon as $j_n'\in J_n^c$. The existence of sequences of such {\it dominant columns} implies that for any probability vector $X$ with positive entries, the probability vector $P_nX/\Vert P_nX\Vert$, converges as $n$ tends to $+\infty$. Our main application of Theorem~A (and our initial motivation) is related to an {\it Erd\H os problem} concerned with a family of probability measures $\mu_\beta$ (for $1<\beta<2$ a real parameter) fully supported by a subinterval of the real line, known as {\it Bernoulli convolutions}.