Showing papers by "Amir Dembo published in 1994"

Limit Distribution of Maximal Non-Aligned Two-Sequence Segmental Score

Abstract: Consider two independent sequences $X_1,\ldots, X_n$ and $Y_1,\ldots, Y_n$. Suppose that $X_1,\ldots, X_n$ are i.i.d. $\mu_X$ and $Y_1,\ldots, Y_n$ are i.i.d. $\mu_Y$, where $\mu_X$ and $\mu_Y$ are distributions on finite alphabets $\sigma_X$ and $\sigma_Y$, respectively. A score $F: \sigma_X \times \sigma_Y\rightarrow \mathbb{R}$ is assigned to each pair $(X_i, Y_j)$ and the maximal nonaligned segment score is $M_n = \max_{0\leq i, j\leq n - \Delta, \Delta \geq 0} \{\sum^\Delta_{k=1} F(X_{i+k}, Y_{j+k})\}$. The limit distribution of $M_n$ is derived here when $\mu_X$ and $\mu_Y$ are not too far apart and $F$ is slightly constrained.

The rate-distortion dimension of sets and measures

Abstract: Data compression of independent samples drawn from a fractal set is considered. The asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution. This quantity is shown to be identical to Renyi's (1959) information dimension. For self-similar fractal sets this dimension is distribution dependent-in sharp contrast with the behavior of absolutely continuous measures. The rate-distortion dimension of a set is defined as the maximal rate-distortion dimension for distributions supported on this set. Kolmogorov's metric dimension is an upper bound on the rate-distortion dimension, while the Hausdorff dimension is a lower bound. Examples of sets for which the rate-distortion dimension differs from these bounds are provided. >

Critical Phenomena for Sequence Matching with Scoring

Abstract: Consider two independent sequences $X_1,\ldots, X_n$ and $Y_1,\ldots, Y_n$. Suppose that $X_1,\ldots, X_n$ are i.i.d. $\mu X$ and $Y_1,\ldots, Y_n$ are i.i.d. $\mu_Y$, where $\mu_X$ and $\mu_Y$ are distributions on finite alphabets $\sum_X$ and $\sum_Y$, respectively. A score $F: \sum_X \times \sum_Y \rightarrow \mathbb{R}$ is assigned to each pair $(X_i, Y_j)$ and the maximal nonaligned segment score is $M_n = \max_{0\leq i, j \leq n - \Delta, \Delta \geq 0}\{\sum^\Delta_{l=1}F(X_{i+l}, Y_{j+l})\}$. Our result is that $M_n/\log n \rightarrow \gamma^\ast(\mu_X, \mu_Y)$ a.s. with $\gamma^\ast$ determined by a tractable variational formula. Moreover, the pair empirical measure of $(X_{i+l}, Y_{j+l})$ during the segment where $M_n$ is achieved converges to a probability measure $ u^\ast$, which is accessible by the same formula. These results generalize to $X_i, Y_j$ taking values in any Polish space, to intrasequence scores under shifts, to long quality segments and to more than two sequences.

A Topological Criterion for Hypothesis Testing

Abstract: A simple topological criterion is given for the existence of a sequence of tests for composite hypothesis testing problems, such that almost surely only finitely many errors are made.

Large exceedances for multidimensional levy processes

Abstract: Three results on hitting a rare set by the increments of an $\mathbb{R}^d$-valued random process with stationary independent increments are presented: the first time that it occurs, the duration of such a segment and the typical trajectory during the segment.

Sanov's theorem for sub-sampling from individual sequences

Abstract: Let x = (X1, 2,-...,m,. .. ) be a Z-valued deterministic sequence such that Lm = m-1 =1 S, converge to Px weakly. Consider the following random sub-sampling scheme. Fix 6 E (0, 1), and m = m(n) such that n/m --, 3, generating the random variables Xl', .. ., Xnm by sampling n values out of (xl,..., xm) without replacement, i.e. X? = xj; for i = 1,...,n where each choice of jl 7 j2 ' ... # jn E { 1, ... , m} is equally likely (and independent of the sequence x). The next proposition shows that perhaps somewhat surprisingly (see Remark 1 immediately following its statement), the large deviations of the empirical measure of the resulting sample admits a rate function which is independent of the particular sequence x but different from the

A large deviations analysis of range tracking loops

Abstract: Large deviations theory is applied to the analysis of a discrete time range tracking loop. It is shown that the resulting asymptotics differ from those of the continuous time diffusion limit. >