Showing papers by "Amir Dembo published in 1996"
TL;DR: In this article, it was shown that for certain environment distributions where the drifts 2ωx-1 can take both positive and negative values, the chance of the RWRE deviating below this speed has a polynomial rate of decay, and the exponent in this power law was determined.
Abstract: Suppose that the integers are assigned i.i.d. random variables {ωx} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {Xk} (called a RWRE) which, when atx, moves one step to the right with probability ωx, and one step to the left with probability 1-ωx. Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2ωx-1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(−Cn1/3). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment.
102 citations
Journal Article•
TL;DR: In this article, the authors prouvons le principe de grandes deviations (LDP) for les mesures empiriques en τ-topologie, dans les cases of suites stationnaires sous les conditions de melange fort α(n) 0, ou Φ(n), << exp(-nl(n)) avec l(n).
Abstract: Nous prouvons le principe de grandes deviations (LDP) pour les mesures empiriques en τ-topologie, dans les cas de suites stationnaires sous les conditions de melange fort α(n) 0, ou Φ(n) << exp(-nl(n)) avec l(n) → ∞. Les exemples de chaines de Markov recurrentes au sens de Doeblin montrent que ces conditions ne permettent pas d'amelioration substantielle, et que l'existence meme du principe de grandes deviations depend du choix de la mesure initiale.
97 citations
TL;DR: The moderate deviation principle holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, converges in probability at an exponential rate as discussed by the authors.
Abstract: We prove that the Moderate Deviation Principle (MDP) holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, converges in probability at an exponential rate. A consequence of this MDP is the tightness of the method of bounded martingale differences in the regime of moderate deviations.
68 citations
TL;DR: Refinements of Sanov's large deviations theorem lead via Csiszár's information theoretic identity to refinements of the Gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence.
Abstract: Refinements of Sanov's large deviations theorem lead via Csiszar's information theoretic identity to refinements of the Gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence. Sharp bounds on the growth of the block length with the length of the conditioning sequence are derived.
53 citations
TL;DR: Using a transportation approach, this article showed that for every probability measure (P,Q,Q_1 and Q_2) on product spaces, there exist r.c.d. inequalities in product spaces.
Abstract: Using a transportation approach we prove that for every probability measures $P,Q_1,Q_2$ on $\Omega^N$ with $P$ a product measure there exist r.c.p.d. $
u_j$ such that $\int
u_j (\cdot|x) dP(x) = Q_j(\cdot)$ and $$ \int dP (x) \int \frac{dP}{dQ_1} (y)^\beta \frac{dP}{dQ_2} (z)^\beta (1+\beta (1-2\beta))^{f_N(x,y,z)} d
u_1 (y|x) d
u_2 (z|x) \le 1 \;, $$ for every $\beta \in (0,1/2)$. Here $f_N$ counts the number of coordinates $k$ for which $x_k
eq y_k$ and $x_k
eq z_k$. In case $Q_1=Q_2$ one may take $
u_1=
u_2$. In the special case of $Q_j(\cdot)=P(\cdot|A)$ we recover some of Talagrand's sharper concentration inequalities in product spaces.
40 citations
01 Jan 1996
TL;DR: In this paper, Stein's method is applied to study the convergence rate of the normal approximation for sums of non-linear functionals of correlated Gaussian random variables, for the exceedances of r-scans of i.i.d.
Abstract: Stein’s method is applied to study the rate of convergence in the normal approximation for sums of non-linear functionals of correlated Gaussian random variables, for the exceedances of r-scans of i.i.d. random variables, and in a multinomial setting.
32 citations
TL;DR: In this paper, the authors prove the large deviation principle and compute the resulting rate function for the latter empirical measure under the assumptions that the empirical measure of the m-sequence converges and that n/m tends to some 0 < β < 1.
9 citations
01 Jan 1996
TL;DR: In this paper, the large deviations for the model of random mass distribution proposed by Aldous were studied, based on a suitable approximation argument, and they proved Aldous' conjecture concerning the large deviation behavior.
Abstract: We study the large deviations for the model of random mass distribution proposed by Aldous. Based on a suitable approximation argument, we prove Aldous’ conjecture concerning the large deviations behavior.
7 citations
01 Jan 1996
TL;DR: For every probability measure P;Q1;Q2 on N with P a product measure there exist r.c.p.d. j such that R j(jx)dP (x )= Qj() and Z as mentioned in this paper.
Abstract: Using a transportation approach we prove that for every probability measures P;Q1;Q2 on N with P a product measure there exist r.c.p.d. j such that R j(jx)dP (x )= Qj() and Z