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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 withnonnegative entries. In a wide range of situations, the normalized matrixproduct $P n-{\Vert P n\Vert}$ does not converge and we shall be ratherinterested in the asymptotic behavior of the normalized columns $P nU i-\VertP nU i\Vert$, where $U 1,\dots,U d$ are the canonical $d\times 1$ vectors. Ourmain result in Theorem~A gives a sufficient condition ${\bf C}$ over thesequence $A 1,A 2,\dots$ 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$J n e\emptyset$ and $J n^c$ such that each one of the sequences of normalizedcolumns, 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 $j n-\inJ n^c$. The existence of sequences of such {\it dominant columns} implies thatfor any probability vector $X$ with positive entries, the probability vector$P nX-\Vert P nX\Vert$, converges as $n$ tends to $+\infty$. Our mainapplication of Theorem~A and our initial motivation is related to an {\itErd\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 realline, known as {\it Bernoulli convolutions}.



Autor: Éric Olivier LATP, Alain Thomas LATP

Fuente: https://arxiv.org/



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