Large Deviations For Random Walks Research Articles

Quenched large deviations for random walk in a random environment

AbstractWe take the point of view of a particle performing random walk with bounded jumps on ℤd in a stationary and ergodic random environment. We prove the quenched large‐deviation principle (LDP) for the pair empirical measure of the so‐called environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function. We propose an ansatz for the minimizer of this formula. When d = 1, we verify this ansatz and generalize the nearest‐neighbor result of Comets, Gantert, and Zeitouni to walks with bounded jumps. © 2009 Wiley Periodicals, Inc.

Large deviations for random walk in a space–time product environment

We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a space--time product environment $\omega\in\Omega$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}\omega)_{n\geq0}$ where $T$ denotes the shift on $\Omega$. Conditioned on the particle having asymptotic mean velocity equal to any given $\xi$, we show that the empirical process of the environment Markov chain converges to a stationary process $\mu_{\xi}^{\infty}$ under the averaged measure. When $d\geq3$ and $\xi$ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity $\xi$, the empirical process of the environment Markov chain converges to $\mu_{\xi}^{\infty}$ under the quenched measure as well. In this case, we show that $\mu_{\xi}^{\infty}$ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob $h$-transform.

Large deviations for random walks under subexponentiality: The big-jump domain

For a given one-dimensional random walk $\{S_n\}$ with a subexponential step-size distribution, we present a unifying theory to study the sequences $\{x_n\}$ for which $\mathsf{P}\{S_n>x\}\sim n\mathsf{P}\{S_1>x\}$ as $n\to\infty$ uniformly for $x\ge x_n$. We also investigate the stronger "local" analogue, $\mathsf{P}\{S_n\in(x,x+T]\}\sim n\mathsf{P}\{S_1\in(x,x+T]\}$. Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory. When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.

Large deviations for random walks with nonidentically distributed jumps having infinite variance

Let e1, e2,... be independent random variables with distributions F1, F2,... in a triangular scheme (F i may depend on some parameter), Ee i = 0, and put \( S_n = \sum\nolimits_{i = 1}^n \xi _i ,\bar S_n = \max _{k \leqslant n} S_k \). Assuming that some regularly varying functions majorize and minorize \( F = \frac{1}{n}\sum\nolimits_{i = 1}^n {F_i } \), we find upper and lower bounds for the probabilities P(S n > x) and P(\( {\bar S_n } \) > x). These bounds are precise enough to yield asymptotics. We also study the asymptotics of the probability that a trajectory {S k } crosses the remote boundary {g(k)}; i.e., the asymptotics of P(maxk≤n(S k − g(k)) > 0). The case n = ∞ is not exclude. We also estimate excluded the distribution of the first crossing time.

On Probabilities of Large Deviations for Random Walks. II. Regular Exponentially Decaying Distributions

We establish exact asymptotic behavior for the probabilities of crossing arbitrary curvilinear boundaries in the large deviations range by random walks, whose jump distribution tails differ from an exponential function by an integrable regularly varying factor. In this interesting transient case, there exists a ``lower of the zone of large deviations, where the classical exact asymptotic results hold true, and an ``upper subzone,' where only results on the crude logarithmic asymptotics were available. Now we derive exact asymptotic behavior for the latter subzone and show that it is, in a sense, close to that described in the first part of the paper [{\em Theory Probab. Appl.}, 46 (2001), pp. 193--213], where we dealt with regularly varying distribution tails. Moreover, under an additional ``asymptotic smoothness condition on the jumps distribution, we establish an asymptotic expansion for the tails of the distributions of the sums of the jumps in the large deviations range.

Large deviations for random walks in a mixing random environment and other (non‐Markov) random walks

AbstractWe extend a recent work by S. R. S. Varadhan [8] on large deviations for random walks in a product random environment to include more general random walks on the lattice. In particular, some reinforced random walks and several classes of random walks in Gibbs fields are included. © 2004 Wiley Periodicals, Inc.

Large deviations for random walk in random environment with holding times

Suppose that the integers are assigned the random variables $\{\omega_x,\mu_x\}$ (taking values in the unit interval times the space of probability measures on $\reals_+$), which serve as an environment. This environment defines a random walk $\{X_t\}$ (called a RWREH) which, when at x, waits a random time distributed according to $\mu_x$ and then, after one unit of time, moves one step to the right with probability $\omega_x$, and one step to the left with probability $1-\omega_x$. We prove large deviation principles for $X_t/t$, both quenched (i.e., conditional upon the environment), with deterministic rate function, and annealed (i.e., averaged over the environment). As an application, we show that for random walks on Galton--Watson trees, quenched and annealed rate functions along a ray differ.

Large deviations for random walks in a random environment

AbstractWe prove two large deviation principles for random walks in a random environment: one for quenched walks in general ergodic environments and the other for averaged walks in a product environment. © 2003 Wiley Periodicals, Inc.

Large deviations for random walks on Galton–Watson trees: averaging and uncertainty

In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X n } on a Galton–Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X n | the distance between the node X n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X n |/n under the “quenched measure” (conditional upon T), and the rate function for the same ratio under the “annealed measure” (averaging on T according to the Galton–Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X n } is a λ-biased walk on a Galton–Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a “ubiquity” lemma for Galton–Watson trees, due to Grimmett and Kesten (1984).

On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails

We establish first-order approximations and asymptotic expansions for probabilities of crossing arbitrary curvilinear boundaries in the large deviations range by random walks with regularly varying distribution tails. In particular, we study the large deviations probabilities for the sums and maxima of partial sums of independent and identically distributed random variables, including the asymptotic behavior of the densities when they exist. Extensions to the regular case (when the distribution tail differs from the exponential one by a regularly varying factor) are considered in part II of the paper.

Probabilities of Large Deviations on the Whole Axis

Probabilities of Large Deviations on the Whole Axis

Dominating Points and the Asymptotics of Large Deviations for Random Walk on $\mathbb{R}^d$

Let $\mu(\cdot)$ be a probability measure on $\mathbb{R}^d$ and $B$ be a convex set with nonempty interior. It is shown that there exists a unique "dominating" point associated with $(\mu, B)$. This fact leads (via conjugate distributions) to a representation formula from which sharp asymptotic estimates of the large deviation probabilities $\mu^{\ast n}(nB)$ can be derived.

A Problem on Large Deviations in a Space of Trajectories

Previous article Next article A Problem on Large Deviations in a Space of TrajectoriesI. F. PinelisI. F. Pinelishttps://doi.org/10.1137/1126006PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Godovanchuk, Probabilities of large deviations for sums of independent random variables attracted to a stable law, Theory Prob. Appl., 23 (1978), 602–608 0422.60020 LinkGoogle Scholar[2] A. A. Borovkov, Analysis of large deviations in boundary-value problems with arbitrary boundaries, Sibirsk. Mat. Ž., 5 (1964), 253–289 29:645 0166.14303 A. A. Borovkov, Analysis of large deviations in boundary-value problems with arbitrary boundaries. II, Sibirsk. Mat. Ž., 5 (1964), 750–767 29:4117 0202.48502 Google Scholar[3] A. A. Borovkov, Boundary-value problems for random walks and large deviations in function spaces, Theory Prob. Appl., 12 (1967), 575–595 0178.20004 LinkGoogle Scholar[4] A. A. Mogul'skii, Large deviations for trajectories of multi-dimensional random walks, Theory Prob. Appl., 21 (1976), 300–315 0366.60031 LinkGoogle Scholar[5] A. V. Nagaev, Limit theorems that take into account large deviations when Cramér's condition is violated, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, 13 (1969), 17–22, (In Russian.) 43:8108 0226.60043 Google Scholar[6] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– 48:1287 0219.60027 Google Scholar[7] D. Kh. Fuk and , S. V. Nagaev, Probability inequalities for sums of independent random variables, Theory Prob. Appl., 16 (1971), 643–660 0259.60024 LinkGoogle Scholar[8] A. V. Nagaev, doctoral dissertation, Tashkent, 1970. (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Hidden regular variation for point processes and the single/multiple large point heuristicThe Annals of Applied Probability, Vol. 32, No. 1 Cross Ref On the Nonuniform Berry–Esseen Bound Cross Ref Tail asymptotics for delay in a half-loaded GI/GI/2 queue with heavy-tailed job sizes21 June 2015 | Queueing Systems, Vol. 81, No. 4 Cross Ref An asymptotically Gaussian bound on the Rademacher tailsElectronic Journal of Probability, Vol. 17, No. none Cross Ref Large deviations for random walks under subexponentiality: The big-jump domainThe Annals of Probability, Vol. 36, No. 5 Cross Ref An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions11 September 2008 | Siberian Mathematical Journal, Vol. 49, No. 4 Cross Ref Integro-local and integral theorems for sums of random variables with semiexponential distributionsSiberian Mathematical Journal, Vol. 47, No. 6 Cross Ref Large deviations of the first passage time for a random walk with semiexponentially distributed jumpsSiberian Mathematical Journal, Vol. 47, No. 6 Cross Ref On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution TailsA. A. Borovkov and K. A. Borovkov25 July 2006 | Theory of Probability & Its Applications, Vol. 46, No. 2AbstractPDF (215 KB)Large-Deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential DistributionD. A. Korshunov25 July 2006 | Theory of Probability & Its Applications, Vol. 46, No. 2AbstractPDF (155 KB)Exact Asymptotics for Large Deviation Probabilities, with Applications Cross Ref Sample path large deviations in finer topologies4 April 2007 | Stochastics and Stochastic Reports, Vol. 67, No. 3-4 Cross Ref The time until the final zero crossing of random sums with application to nonparametric bandit theoryApplied Mathematics and Computation, Vol. 63, No. 2-3 Cross Ref Probabilities of Large Deviations on the Whole AxisL. V. Rozovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 1AbstractPDF (1937 KB)Sharp Exponential Inequalities for the Martingales in the 2-Smooth Banach Spaces and Applications to “Scalarizing” Decoupling Cross Ref Large Deviations of Sums of Independent Random Variables without Several Maximal SummandsV. V. Vinogradov and V. V. Godovan’chuk17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 3AbstractPDF (383 KB)Probabilities of Large Deviations of Sums of Independent Random Variables with Common Distribution Function in the Domain of Attraction of the Normal LawL. V. Rozovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 4AbstractPDF (1377 KB) Volume 26, Issue 1| 1981Theory of Probability & Its Applications History Submitted:05 February 1979Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126006Article page range:pp. 69-84ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Probability Inequalities for Sums of Independent Random Variables

Probability Inequalities for Sums of Independent Random Variables

Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces

Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces