Moderate deviations principle for products of sums of random variables

Let(Xn)n≥1 be a sequence of independent identically distributed(i.i.d.) positive random variables with EX1 = μ,Var(X1) = σ2.In the present paper,we establish the moderate deviations principle for the products of partial sums(Πnk=1Sk/n!μn)1/(γbn√(2n))1where γ = σ/μ denotes the coefficient of variation and(bn) is the moderate deviations scale.     

Large deviations for product of sums of random variables

In this paper, we obtain sample path and scalar large deviation principles for the product of sums of positive random variables. We study the case when the positive random variables are independent and identically distributed and bounded away from zero or the left tail decays to zero sufficiently fast. The explicit formula for the rate function of a scalar large deviation principle is given in the case when random variables are exponentially distributed.     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

M-negatively associated random variables,which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its par- ticular case,are introduced and studied.Large deviation principles and moderate devi- ation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.     

关于重尾随机和大偏差的一个注记

设｛Xκ，κ≥1｝为一列独立同分布的非随机变量，且具有共同的分布函数F。记Sn为序列｛Xκ，κ≥1｝的前n项部分和。在F属于ERV分布族的假定下，文中证明了关于随机和SN(t)的随机中心化的精细大偏差结果。这里N（t）为一个与｛Xκ，κ≥1｝独立的非负整数值的随机过程。     

On sums of independent random variables without power moments

In 1952 Darling proved the limit theorem for the sums of independent identically distributed random variables without power moments under the functional normalization. This paper contains an alternative proof of Darling’s theorem, using the Laplace transform. Moreover, the asymptotic behavior of probabilities of large deviations is studied in the pattern under consideration.     

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 .     

On large deviations for sums of random vectors in Rk

The paper deals with limit theorems for probabilities of large deviations for sums of independent identically distributed random vectors. We give more detailed bounds for the remainder in von Bahr's limit theorem. New asymptotic formulas for probabilities of large deviations on the outside of balls are established.     

On replica symmetry of large deviations in random graphs

The following question is due to Chatterjee and Varadhan (2011). Fix and take , the Erd?s‐Rényi random graph with edge density p, conditioned to have at least as many triangles as the typical . Is G close in cut‐distance to a typical ? Via a beautiful new framework for large deviation principles in , Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d‐regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all such that lies on the convex minorant of where is the rate function of a binomial with parameter p. In particular, the answer for triangles involves rather than the natural guess of where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of is conditioned to exceed the typical value of the largest eigenvalue of . Building on the work of Chatterjee and Diaconis (2012) we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn (2001) and Galvin and Tetali (2004). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109–146, 2015     

Superlarge deviation probabilities for sums of independent lattice random variables with exponential decreasing tails

In the note we study large and superlarge deviation probabilities of sum of i.i.d. lattice random variables, whose distribution function has an exponentially decreasing tail at infinity.     

Superlarge deviations for sums of random variables with arithmetical super-exponential distributions

Local limit theorems are obtained for superlarge deviations of sums S(n) = ξ(1) + ... + ξ(n) of independent identically distributed random variables having an arithmetical distribution with the right-hand tail decreasing faster that that of a Gaussian law. The distribution of ξ has the form ?(ξ = k) = \(e^{ - k^\beta L(k)} \), where β > 2, k ∈ ? (? is the set of all integers), and L(t) is a slowly varying function as t → ∞ which satisfies some regularity conditions. These theorems describing an asymptotic behavior of the probabilities ?(S(n) = k) as k/n → ∞, complement the results on superlarge deviations in [4, 5].     

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.     

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.     

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums , whereX _n,n⩾1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t⩾0 is a process of non-negative integer-valued random variables, independent ofX _n,n⩾1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t⩾0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.     

Large deviations for sums of i.i.d. random compact sets

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

     

A unified approach to the large deviations for small perturbations of random evolution equations

LetX ^ɛ = {X ^ɛ (t ; 0 ⩽t ⩽ 1 } (ɛ > 0) be the processes governed by the following stochastic differential equations:

离散随机序列加权和的若干极限定理

设{Xn,n≥0}是任意离散随机变量序列,{ank,0≤k≤n,n≥0)是一常数阵列,我们引入随机序列渐近对数似然比的概念,作为表征随机序列的真实概率测度P与参考测度Q之间的差异的度量,用分析方法,得到了随机序列Jamison型加权和的若干随机偏差定理.     

Small deviations of series of independent positive random variables with weights close to exponential

Let ξ, ξ₀, ξ₁, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ _j=0 ^∞ a(j)ξ _j was studied under different assumptions on the rate of decrease of the probability ?(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ?(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ?(ξ < x) ～ bx ^α as x → 0, b > 0, a > 0, the asymptotics ?(S < x) is obtained in an explicit form up to the factor x ^o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].     

长尾宽上限相依的不同分布的随机变量和的精确大偏差

研究了服从长尾分布族上的随机变量和的精确大偏差问题,其中假设代表索赔额的随机变量序列是一列宽上限相依的、不同分布的随机变量序列。在给定一些假设条件下,得到了部分和与随机和的两种一致渐近结论。     

Precise large deviations for widely orthant dependent random variables with dominatedly varying tails

For the widely orthant dependent (WOD) structure, this paper mainly investigates the precise large deviations for the partial sums ofWOD and non-identically distributed random variables with dominatedly varying tails. The obtained results extend some corresponding results.