An Almost Sure Functional Limit Theorem for the Domain of Geometric Partial Attraction of Semistable Laws

An almost sure functional limit theorem is obtained for variables being in the domain of geometric partial attraction of a semistable law.     

Almost sure limit theorems of mantissa type for semistable domains of attraction

A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods lead to previously unknown versions of semistable almost sure limit theorems. This research has been carried out while the author was staying at the University of Debrecen, Hungary, with the kind support of Deutsche Forschungsgemeinschaft.     

An almost sure limit theorem for random sums of independent random variables in the domain of attraction of a semistable law

In this paper we prove an almost sure limit theorem for random sums of independent random variables in the domain of attraction of a p-semistable law and describe the limit law.     

Almost sure versions of limit theorems for random sums of multiindex random variables

In this paper we obtain an almost sure version of a limit theorem for random sums of multiindex random variables that belong to the domain of attraction of a p-stable law.     

Almost sure limit theorems for stable distributions

Consider a sequence of i.i.d. random variables in the domain of attraction of a stable distribution with an exponent in (0,2]. A universal result in almost sure limit theorem for the partial sums is established. Our results substantially extend and improve those on the almost sure central limit theorem previously obtained by Jonsson 2007, Berkes and Csáki 2001, and Hörmann 2007.     

An Improved Result in Almost Sure Central Limit Theory for Products of Partial Sums with Stable Distribution

Consider a sequence of i.i.d. positive random variables with the underlying distribution in the domain of attraction of a stable distribution with an exponent in (1, 2]. A universal result in the almost sure limit theorem for products of partial sums is established. Our results significantly generalize and improve those on the almost sure central limit theory previously obtained by Gonchigdanzan and Rempale and by Gonchigdanzan. In a sense, our results reach the optimal form.     

An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables

Let X,X₁,X₂,… be a sequence of independent and identically distributed positive random variables with EX=μ>0. In this paper we show that the almost sure central limit theorem for self-normalized products of sums holds only under the assumptions that X belongs to the domain of attraction of the normal law.     

φ-混合重尾随机向量序列的Chover型重对数律

本文讨论同分布的φ-混合随机向量序列其共同分布属于某个没有Gauss分量的广义的半稳定律的吸引场部分和的积分检验的极限结果,由此可推出相应的Chover型重对数律.     

Series representation for semistable laws and their domains of semistable attraction

If the centered and normalized partial sums of an i.i.d. sequence of random variables converge in distribution to a nondegenerate limit then we say that this sequence belongs to the domain of attraction of the necessarily stable limit. If we consider only the partial sums which terminate atk _n wherek _n+1 ck _n then the sequence belongs to the domain of semistable attraction of the necessarily semistable limit. In this paper, we consider the case where the limiting distribution is nonnormal. We obtain a series representation for the partial sums which converges almost surely. This representation is based on the order statistics, and utilizes the Poisson process. Almost sure convergence is a useful technical device, as we illustrate with a number of applications.This research was supported by a research scholarship from the Deutsche Forschungsgemeinschaft (DFG).     

Existence of Multivariate Max-Universal Laws

If suitably normalized maxima of an i.i.d. sample converge in distribution, the limiting distribution is known to be max-infinitely divisible and the common distribution of the sample is said to belong to its domain of attraction. We prove the existence of max-universal distributions belonging to the domain of attraction of every max-infinitely divisible law. The proof follows in the spirit of corresponding results for normalized sums of i.i.d. random variables originated by Doeblin and shows that necessarily the sampling size has to be rapidly increasing. Restricting the growth rate of the sampling size, we prove that one necessarily deals with max-semistable distributions and their domains of attraction. 2000 Mathematics subject classification Primary—60G70 Secondary—60E99, 60F05     

Merging of Linear Combinations to Semistable Laws

We prove merge theorems along the entire sequence of natural numbers for the distribution functions of suitably centered and normed linear combinations of independent and identically distributed random variables from the domain of geometric partial attraction of any non-normal semistable law. Surprisingly, for some sequences of linear combinations, not too far from those with equal weights, the merge theorems reduce to ordinary asymptotic distributions with semistable limits. The proofs require working out general conditions for merging in terms of characteristic functions.     

Central Limit Theorem for Linear Processes with Infinite Variance

This paper addresses the following classical question: Given a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process satisfy the central limit theorem? We study the question for several classes of dependent random variables. For independent and identically distributed random variables we show that the central limit theorem for the linear process is equivalent to the fact that the variables are in the domain of attraction of a normal law, answering in this way an open problem in the literature. The study is also motivated by models arising in economic applications where often the innovations have infinite variance, coefficients are not absolutely summable, and the innovations are dependent.     

Merging asymptotic expansions for semistable random variables

Merging asymptotic expansions are established for distribution functions from the domain of geometric partial attraction of a semistable law. The length of the expansion depends on the exponent of the semistable law and on the characteristic function of the underlying distribution. We obtain sufficient conditions for the quantile function in order to get real infinite asymptotic expansion. The results are generalizations of the existing theory in the stable case.     

Almost sure central limit theorems for random fields

A general almost sure limit theorem is presented for random fields. It is applied to obtain almost sure versions of some (functional) central limit theorems. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

随机变量序列函数的几乎处处中心极限定理

该文证明了随机元序列的一个一般的几乎处处中心极限定理, 并把这一结论应用于随机变量序列的函数.     

On sample path properties of semistable processes

This paper contains three main results: In the first result a correspondence principle between semistable measures on L_p, 1 ≤ p < ∞, and Banach space valued semistable processes is established. In the second result it is shown that the paths of a Banach space valued semistable process belong to L_p with probability zero or one, and necessary and sufficient conditions for the two alternatives to hold are given. In the third result necessary and sufficient conditions are given for almost sure path absolute continuity for certain Banach space valued semistable processes.     

Strong and conditional invariance principles for samples attracted to stable laws

Summary. We prove almost sure convergence of a representation of normalized partial sum processes of a sequence of i.i.d. random variables from the domain of attraction of an α-stable law, α<2. We obtain an explicit form of the limit in terms of the LePage series representation of stable laws. One consequence of these results is a conditional invariance principle having applications to option pricing as well as to resampling by signs and permutations. Received: 11 April 1994 / In revised form: 5 November 1996     

The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics

The Chung–Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented.     

Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/-self-similar -stable processes with stationary increments are -stable motions. The answer is yes for = 2, no for 1<2 and unknown for 0<<1. We single out the log-fractional stable processes for 1<2, different from -stable motions for ≠2. They can be regarded as the limit of fractional stable processes as the exponent in the kernel tends to 0. The paper ends with a limit theorem for partial sum processes of moving averages of iid random variables in the domain of attraction of a strictly stable law, with log-fractional stable processes as limits in law. The conditions involve de Haan's class Π of slowly varying functions.