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

TL;DR

This paper investigates the asymptotic behavior of normalized columns in products of nonnegative matrices, establishing conditions for convergence to dominant projective limits, with applications to Bernoulli convolutions.

Contribution

It provides a sufficient condition for the existence of dominant columns with the same projective limit in inhomogeneous matrix products, advancing understanding of their asymptotic behavior.

Findings

Existence of dominant columns with the same projective limit under condition (C)

Convergence of normalized probability vectors in matrix products

Application to Bernoulli convolutions and Erdős problem

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}.