Large deviations for Poisson systems of independent random walks

Summary We prove large deviation theorems for occupation time functionals of independent random walks started from a Poisson field on Z ^d. In dimensions 1 and 2 the large deviation tails are larger than exponential. Exact asymptotics are derived.Partially supported by the National Science Foundation under Grant MCS 81-02131 and MCS 81-00256Alfred P. Sloan Research Fellow     

Large deviations and phase transition for random walks in random nonnegative potentials

We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on Z^d${\mathbb{Z}}^d$. We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on Z^d${\mathbb{Z}}^d$, Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.     

次线性期望下独立随机变量列的大偏差和中偏差

本文研究在次线性期望下的独立随机变量列的大偏差和中偏差原理. 利用次可加方法, 我们得 到次线性期望下的大偏差原理. 与次线性期望下的中心极限定理相应的中偏差原理也被建立.     

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

We 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 walks with nonidentically distributed jumps having infinite variance

Let = _i = 1^n _i , S_n = _k n S_k

Large deviations for trajectories of sums of independent random variables

Given a sequence of independent, but not necessarily identically distributed random variables,Y _i , letS _k denote thekth partial sum. Define a function by taking to be the piecewise linear interpolant of the points (k, S _k ), evaluated att, whereS ₀=0, andk=0, 1, 2,... Fort[0, 1], let . The are called trajectories. With regularity and moment conditions on theY _i , a large deviation principle is proved for the .     

Large deviations for Markov processes with discontinuous statistics,II: random walks

Summary Let ¹ and ² be Borel probability measures on ^d with finite moment generating functions. The main theorem in this paper proves the large deviation principle for a random walk whose transition mechanism is governed by ¹ when the walk is in the left halfspace ¹ = {x ^d :x ₁0} and whose transition mechanism is governed by ² when the walk is in the right halfspace ² = {x ^d :x ₁>0}. When the measures ¹ and ² are equal, the main theorem reduces to Cramér's Theorem.This research was supported in part by a grant from the National Science Foundation (NSF-DMS-8902333)This research was supported in part by a grant from the National Science Foundation (NSF-DMS-8901138) and in part by a Lady Davis Fellowship while visiting the Faculty of Industrial Engineering and Management at the Technion during the spring semester of 1989     

Large deviations probabilities for random walks in the absence of finite expectations of jumps

 Let be independent identically distributed random variables with regularly varying distribution tails:

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). Received: 15 November 2000 / Revised version: 27 February 2001 / Published online: 19 December 2001     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

Large deviations for sums of independent random variables with dominatedly varying tails

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.     

Maxima of branching random walks vs. independent random walks

In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random walks. A comparison of our results with the known results of branching random walk then identifies the limit behaviors which are due to the number of particles and those which are determined by the branching structure.     

Large deviations for Gibbs random fields

A large deviation principle for Gibbs random fields on Z^d is proven and a corresponding large deviations proof of the Gibbs variational formula is given. A generalization of the Lanford theory of large deviations is also obtained.This work was partially supported by NSF-DMR81-14726     

Self-normalized moderate deviations for independent random variables

Let X1,X2,...be a sequence of independent random variables(r.v.s) belonging to the domain of attraction of a normal or stable law.In this paper,we study moderate deviations for the self-normalized sum ∑ni=1 Xi/Vn,p,where Vn,p =(∑ni=1 |Xi|p)1/p(p>1).Applications to the self-normalized law of the iterated logarithm,Studentized increments of partial sums,t-statistic,and weighted sum of independent and identically distributed(i.i.d.) r.v.s are considered.