Showing papers by "Amir Dembo published in 1999"
TL;DR: In this paper, the authors studied the waiting time of a D-close version of the initial string (X, X, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 47
Abstract: Given two independent realizations of the stationary processes $\mathbf{X} = {X_n;n \geq 1}$ and $\mathbf{Y} = {Y_n;n \geq 1}$, our main quantity of interest is the waiting time $W_n(D)$ until a D-close version of the initial string $(X_1, X_2,\dots, X_n)$ first appears as a contiguous substring in $(Y_1, Y_2, Y_3,\dots)$, where closeness is measured with respect to some "average distortion" criterion. We study the asymptotics of $W_n(D)$ for large n under various mixing conditions on X and Y. We first prove a strong approximation theorem between $\logW_n(D)$ and the logarithm of the probability of a D-ball around $(X_1, X_2,\dots, X_n)$. Using large deviations techniques, we show that this probability can, in turn, be strongly approximated by an associated random walk, and we conclude that: (i) $n^{-1} \log W_n(D)$ converges almost surely to a constant R determined byan explicit variational problem; (ii) $[\log W_n(D) - R]$, properly normalized, satisfies a central limit theorem, a law of the iterated logarithm and, more generally, an almost sure invariance principle.
47 citations
TL;DR: In this article, it was shown that for any ε > 0, the Hausdorff dimension of the set of "thick points" for which T(x,r)/(r^b|\log r|) is the correct scaling to obtain a nondegenerate multifractal spectrum for transient stable occupation measure is ε ≥ 0.
Abstract: Let $T(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for a transient symmetric stable processes of index $b
14 citations