Stable limits for sums of dependent infinite variance random variables

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.     

Asymptotic behavior for sums of non-identically distributed random variables

For any given positive integer m, let X_i, 1 ≤ i ≤ m be m independent random variables with distributions F_i, 1 ≤ i ≤ m. When all the summands are nonnegative and at least one of them is heavy-tailed, we prove that the lower limit of the ratio ■equals 1 as x →∞. When the summands are real-valued, we also obtain some asymptotic results for the tail probability of the sums. Besides, a local version as well as a density version of the above results is also presented.     

Limit behavior of sums of a random number of independent random variables

Lithuanian Mathematical Journal -     

Limit behavior of centered random sums of independent identically distributed random variables

Under different assumptions, necessary and sufficient conditions are given for the weak convergence of sums of a random number of independent identically distributed random variables. Supported by the Russian Foundation for Fundamental Researches (grant No. 93-011-1446). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

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.     

Moments of sums of independent random variables

Upper bounds are obtained for the absolute moments of order p>1 of sums of independent random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 120–121, 1989.     

Large deviations of sums of random variables

Lithuanian Mathematical Journal - In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some results...     

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…     

A central limit theorem for random sums of random variables

Anscombe (1952) (also see Chung (1974)) has developed a central limit theoremof random sums of independent and identically distributed random variables. Applicability of this theorem in practice, however, is limited since the normalization requires random factors. In this paper we establish sufficient conditions under which the central limit theorem holds when such random factors are replaced by the underlying asymptotic mean and standard ddeviation. An application of this result in the context of shock models is also given.     

On the growth of sums of independent random variables

Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 158–164.This research was partially supported by the Russian Foundation for Basic Research, grant 02-01-00779, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by V. V. Petrov.