  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}.
