Theorems of Large Deviations in the Approximation by an Infinitely Divisible Law

The general lemma of large deviations has been proved in the approximation by an in finitely divisible law with a finite spectrum. The method of cumulants is used. First, vast possibilities of this nethod are surveyed when investigating large deviations in various approximations.     

Large Deviations for the Markov Binomial Distribution

Considering the Markov binomial distribution, we study large deviations for the Poisson approximation. Apart from the standard choice of parameters, we use the approach where the parameter of approximation depends on the argument of the approximated distribution function.     

Large deviations for generalized compound Poisson risk models and its bankruptcy moments

We extend the classical compound Poisson risk model to the case where the premium income process, based on a Poisson process, is no longer a linear function. For this more realistic risk model, Lundberg type limiting results on the finite time ruin probabilities are derived. Asymptotic behaviour of the tail probabilities of the claim surplus process is also investigated.     

Moderate Deviations and Large Deviations for Kernel Density Estimators

Let f _n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ^d . It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) 0 as |x| , then the moderate deviation principle and large deviation principle for hold.     

复合泊松过程的可加性

对复合泊松分布可加性的研究在许多的文献中都可以看到,本文首先应用特征函数的方法证明了复合泊松分布的可加性.以此为基础,结合对随机过程相关性质的讨论,证明了复合泊松过程也具有与复合泊松分布可加性相似的,某种意义上的可加性性质.     

Asymptotic Expansions in the Compound Poisson Limit Theorem

A triangular array of independent infinitesimal integer-valued random variables is considered. Asymptotic expansions for the probability distributions of sums of these variables are investigated in the case of the limiting compound Poisson laws.     

Compound Poisson Approximation of the Number of Exceedances in Gaussian Sequences

Consider a finite sequence of Gaussian random variables. Count the number of exceedances of some level a, i.e. the number of values exceeding the level. Let this level and the length of the sequence increase simultaneously so that the expected number of exceedances remains fixed. It is well-known that if the long-range dependence is not too strong, the number of exceeding points converges in distribution to a Poisson distribution. However, for sequences with some individual large correlations, the Poisson convergence is slow due to clumping. Using Steins method we show that, at least for m-dependent sequences, the rate of convergence is improved by using compound Poisson as approximating distribution. An explicit bound for the convergence rate is derived for the compound Poisson approximation, and also for a subclass of the compound Poisson distribution, where only clumps of size two are considered. Results from numerical calculations and simulations are also presented.     

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.     

个体风险模型的Poisson复合模型近似

本文在近乎最一般的假定下，简述了个体风险模型的Poisson复合模型近似．特别地，借助风险间停止损失保费的总差异给出了这一近似的精度．     

Finite Time Ruin Probabilities and Large Deviations for Generalized Compound Binomial Risk Models

In this paper, we extend the classical compound binomial risk model to the case where the premium income process is based on a Poisson process, and is no longer a linear function. For this more realistic risk model, Lundberg type limiting results for the finite time ruin probabilities are derived. Asymptotic behavior of the tail probabilities of the claim surplus process is also investigated.     

Large Deviations for the Brownian Motion on Loop Groups

We shall develop the method of projective limit to establish the large deviations principle in the context of loop groups. The symmetries of Lie group allow us to obtain the exponential tightness.     

Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in Cramer zones and Linnik power zones for the distribution function of sums of independent nonidentically distributed random variables (r.v.). In this scheme of summation of r.v., the results are obtained first by mainly using the general lemma on large deviations considering asymptotic expansions for an arbitrary r.v. with regular behaviour of its cumulants [11]. Asymptotic expansions in the Cramer zone for the distribution function of sums of identically distributed r.v. were investigated in the works [1,2]. Note that asymptotic expansions for large deviations were first obtained in the probability theory by J. Kubilius [3].     

Path Space Large Deviations of a Large Buffer with Gaussian Input Traffic

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having different Hurst indices.     

On Large Deviations of Vector-Valued Additive Functions of a Markov Gaussian Field

We prove a local limit theorem for large deviations of the sums , where , is a Markov Gaussian random field, is a bounded vector-valued function, and . This paper generalizes the paper [13].     

Asymptotic Expansion for the Distribution Density Function of the Compound Poisson Process in Large Deviations

The paper is devoted to obtaining the asymptotic expansion and determination of the structure of the remainder term taking into consideration large deviations in the Cramér zone for the distribution density function of the standardized compound Poisson process. Following Deltuvien? and Saulis (Acta Appl Math 78:87–97, 2003. doi: 10.1023/A:1025783905023; Lith Math J 41:620–625, 2001) and Saulis and Statulevi?ius [Limit theorems for large deviations. Mathematics and its applications (Soviet Series), vol 73, pp 154–187, Kluwer, Dordrecht, 1991], the solution to the problem is achieved by first using a general lemma presented by Saulis (see Lemma 6.1 in Saulis and Statulevi?ius 1991, p. 154) on the asymptotic expansion for the density function of an arbitrary random variable with zero mean and unit variance and combining methods for cumulants and characteristic functions. By taking into consideration the large deviations in the Cramér zone for the density function of the standardized compound Poisson process, the result for the asymptotic expansion extends the asymptotic expansions for the density function of the sums of non-random number of summands (Deltuvien? and Saulis 2003, 2001).     

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.     

Large Deviations of Empirical Measures Under Symmetric Interaction

We prove the large deviation principle for the joint empirical measure of pairs of random variables which are coupled by a totally symmetric interaction. The rate function is given by an explicit bilinear expression, which is finite only on product measures and hence is non-convex.     

Poisson Distribution in the Polynomial Semigroup

We consider the asymptotic behavior of the distributions of arithmetic functions in polynomial semigroups.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 429–442, October–December, 2004.     

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.