Large Deviations for Intersections of Random Walks

We prove a large deviations principle for the number of intersections of two independent infinite-time ranges in dimension 5 and greater, improving upon the moment bounds of Khanin, Mazel, Shlosman, and Sinaï [9]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen, and den Hollander [15], who analyzed this question for the Wiener sausage in the finite-time horizon. The proof builds on their result (which was adapted in the discrete setting by Phetpradap [12]), and combines it with a series of tools that were developed in recent works of the authors [2, 3, 5]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order 1. © 2022 Wiley Periodicals, Inc.     

Large Deviations View Points for Heavy-Tailed Random Walks

Let X ₁, X ₂,... be a sequence of i.i.d. non-negative random variables with heavy tails. W e study logarithmic asymptotics for the distributions of the partial sums S _n = X ₁ + ··· + X _n . Our main interest is in the crude estimates P(S _n > n ^x ) n ^{–x + 1} for appropriate values of x where is a specific parameter. The related conjecture proposed by Gantert (Stat. Probab. Lett. 49, 113–118) is investigated.     

Large and Moderate Deviations for Random Walks on Nilpotent Groups

We prove large and moderate deviation estimates for products of i.i.d. r.v.'s taking values on simply connected nilpotent Lie groups as a consequence of large and moderate deviation results for stochastic processes which are solutions of O.D.E. with random coefficients.     

On Large Deviations of Multivariate Heavy-Tailed Random Walks

Let {S _n ;n=1,2,…} be a random walk in R ^d and E(S ₁)=(μ ₁,…,μ _d ). Let a _j >μ _j for j=1,…,d and A=(a ₁,∞)×⋅⋅⋅×(a _d ,∞). We are interested in the probability P(S _n /n∈A) for large n in the case where the components of S ₁ are heavy tailed. An objective is to associate an exact power with the aforementioned probability. We also derive sharper asymptotic bounds for the probability and show that in essence, the occurrence of the event {S _n /n∈A} is caused by large single increments of the components in a specific way.      

Quenched Large Deviations for Multidimensional Random Walk in Random Environment with Holding Times

We consider a random walk in random environment with random holding times, that is, the random walk jumping to one of its nearest neighbors with some transition probability after a random holding time. Both the transition probabilities and the laws of the holding times are randomly distributed over the integer lattice. Our main result is a quenched large deviation principle for the position of the random walk. The rate function is given by the Legendre transform of the so-called Lyapunov exponents for the Laplace transform of the first passage time. By using this representation, we derive some asymptotics of the rate function in some special cases.     

一维随机风景中的随机游动的中偏差

本文研究一维独立同分布随机风景中的随机游动的中偏差.通过给出一些有用的高阶矩估计并结合G(a)rtner-Ellis定理,得到主要结果.     

Moderate and Small Deviations for the Ranges of One-Dimensional Random Walks

We establish moderate and small deviations for the ranges of integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm. We establish moderate and small deviations for the ranges of integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm.     

Strong Laws of Large Numbers for Random Walks in Random Sceneries

In this paper, we study strong laws of large numbers for random walks in random sceneries. Some mild sufficient conditions for the validity of strong laws of large numbers are obtained.     

Upper Large Deviations for Mixing Random Sequence

In this article, we prove upper large deviations for the empirical measure generated by stationary mixing random sequence under some suitable assumptions and upper large deviations for the mixing random sequence.     

Large Deviations for Heavy-tailed Random Variables in Prospective-loss Process

In this paper, we study the precise large deviations for the prospectiveloss process with consistently varying tails. The obtained results improve some related known ones.     

重尾随机变量和的大偏差

This paper is a further investigation of large deviations for sums of random variables S_n=sum form i=1 to n X_i and S(t)=sum form i=1 to N(t) X_i,(t≥0), where {X_n,n≥1) are independent identically distribution and non-negative random variables, and {N(t),t≥0} is a counting process of non-negative integer-valued random variables, independent of {X_n,n≥1}. In this paper, under the suppose F∈G, which is a bigger heavy-tailed class than C, proved large deviation results for sums of random variables.     

Large Deviations Approximations to Distributions of the Total Distance of Compound Random Walks with von Mises Directions

This article considers the planar random walk where the direction taken by each consecutive step follows the von Mises distribution and where the number of steps of the random walk is determined by the class of inhomogeneous birth processes. Saddlepoint approximations to the distribution of the total distance covered by the random walk, i.e. of the length of the resultant vector of the individual steps, are proposed. Specific formulae are derived for the inhomogeneous Poisson process and for processes with linear contagion, which are the binomial and the negative binomial processes. A numerical example confirms the high accuracy of the proposed saddlepoint approximations.     

Large Deviation Probabilities for Random Walks with Semiexponential Distributions

Siberian Mathematical Journal -     

Large Deviations for Sums of Independent Heavy-Tailed Random Variables

We obtain precise large deviations for heavy-tailed random sums , of independent random variables. are nonnegative integer-valued random variables independent of r.v. (X _i )i N with distribution functions F _i. We assume that the average of right tails of distribution functions F _i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given.     

特殊重尾随机变量随机和的大偏差

文献[1]对于一些经典重尾随机变量的随机和大偏差作了有意义的讨论,本文则讨论了另外一些同样有用的重尾随机和的大偏差.     

Large Deviations in Generalized Random Graphs with Node Weights

Generalized random graphs are considered where the presence or absence of an edge depends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a generalized random graph, where the node weights are deterministic under some regularity conditions, as well as chosen i.i.d. from a finite set with positive components. When the node weights are random variables, obstacles arise because the independence among edges no longer exists, our main tools are some results of large deviations for mixtures. After calculating, our results show that the corresponding rate functions for the deterministic case and the random case are very different.     

Large and Moderate Deviations of Random Upper Semicontinuous Functions

We firstly discuss the topological properties of the space of upper semicontinuous functions, and then we obtain large deviation principles for random upper semicontinuous functions under various topologies. Finally, we prove moderate deviation principles for random sets and random upper semicontinuous functions.     

Semiexponential Distributions and Related Large Deviation Principles for Trajectories of Random Walks

Siberian Mathematical Journal - We obtain a rather simple characterization of semiexponential distributions. This allows us to relax substantially the conditions for the fulfillment of the...     

Local Precise Large and Moderate Deviations for Sums of Independent Random Variables

Let {X, X_k : k ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution F. In this paper, the authors establish some results on the local precise large and moderate deviation probabilities for partial sums S_n =sum from i=1 to n(X_i) in a unified form in which X may be a random variable of an arbitrary type,which state that under some suitable conditions, for some constants T 0, a and τ 1/2and for every fixed γ 0, the relation P(S_n- na ∈(x, x + T ]) ~nF((x + a, x + a + T ]) holds uniformly for all x ≥γn~τ as n→∞, that is, P(Sn- na ∈(x, x + T ]) lim sup- 1 = 0.n→+∞x≥γnτnF((x + a, x + a + T ])The authors also discuss the case where X has an infinite mean.     

一类随机场越界概率的大偏差

For Y₁, Y₂,…i .i .d . with Y₁～N(μ, 1) and S_n=(?)Y_i, the large deviations are obtained for theprobabilities that(?) conditionally given (i) S_m=0, and (ii) S_mi=ξ. Applied these results to the double change points model with some nuisance parameters, we developed the large deviation for the significance level of the likelihood ratio test.