Mathematics – Probability
Scientific paper
2009-08-28
Mathematics
Probability
13 pages
Scientific paper
We consider the sequence of column vectors $A_1\dots A_nV$ associated to a sequence $(A_n)_{n\in\mathbb N}$ of nonnegative $d\times d$ matrices and to a positive $d$-dimensional column vector $V$. We give some sufficient conditions on $(A_n)_{n\in\mathbb N}$ for the sequence of normalized column vectors $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$ to converge (do not confuse with the convergence of $\displaystyle{A_1\dots A_n\over\left\Vert A_1\dots A_n\right\Vert}$, that can fail even in case there exists two positive matrices $M_1$ and $M_2$ such that $A_n\in\{M_1,M_2\}$ for any $n$). These conditions seem to be strong but, given a sequence $(A_n)_{n\in\mathbb N}$ of nonnegative $d\times d$ matrices, it is often possible to find an increasing sequence $(n_k)_{k\in\mathbb N}$ such that the sequence of the matrices $A'_k=A_{n_k+1}\dots A_{n_{k+1}}$ satisfies them. This is what we do in a second paper, by considering a set of three $7\times7$ sparse $(0,1)$-matrices and all the sequences $(A_n)_{n\in\mathbb N}$ with terms in this set, in order to apply the results of the present to the multifractal analysis of some Bernoulli convolution.
Olivier Eric
Thomas Alain
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