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.     

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)     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

ALMOST SURE CONVERGENCE OF THE STABLE TAIL EMPIRICAL DEPENDENCE FUNCTION IN MULTIVARIATE EXTREME STATISTICS

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionandTheoremsSupposethatF(x,y)isabivariatedistributionfunctionwithtwocontinuousmarginaldistributionfunctions,say,FIandF2.DefineFissaidtohaveastabletaildependencefunction(STDF)l(x,y)ifforx20andy20,whereF(x,y)~1--F(QI(x),QZ(y)).TheconceptofSTDFwasintroducedin[6].Supposethat{(Xi,K),i21}isasequenceofi.i.d.randomvectorswithdistributionF(x,y).Ifthereedestsomesequencesofconstantsan>0,on>0,b.ERandd.ER,n>1.suc…     

Almost sure limit theorems for a stationary normal sequence

We prove almost sure limit theorems for the maximum of a stationary normal sequence under some conditions.     

Almost sure invariance principles via martingale approximation

In this paper, we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, the almost sure central limit theorem, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.     

Almost sure central limit theorems on the Wiener space

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution.     

Almost sure convergence theorems in von Neumann algebras

The subadditive sequences of operators which belong to a von Neumann algebra with a faithful normal state and a given positive linear kernel are considered. We prove the almost sure convergence in Egorov’s sense for such sequences. Partially supported by the Ministry of Science and Ministry of Absorption. Partially sponsored by a grant from the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Lower bounds on the approximation of the multivariate empirical process

Summary It is well-known that ifC is the class of rectangles 0≦x ₁≦a ₁, 0≦x ₂≦a ₂ or the class of circular discs then the normalized empirical measure onC behaves like a Brownian bridge. Our main result shows that for these two classes the distances between the normalized empirical measure and the nearest Brownian measure have entirely different order of magnitudes.     

Almost sure limit theorems for random sums of multiindex random variables

In this paper, we obtain an almost sure functional limit theorem for random sums of multiindex random variables.     

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.     

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 critical paths

M. Kress proved for a special case of Location-Scale probability distributions there always exists a probability level for which the Chance Constrained Critical Path (CCCP) remains unchanged for all probabilities greater than or equal to that value. His chance constrained problem has zero-order decision rules and individual chance constraints. This paper extends his results to most of the common probability distributions.     

Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences

Let \((X_{n}^{\ast})\) be an independent identically distributed random sequence. Let \(M_{n}^{\ast}\) and \(m_{n}^{\ast}\) denote, respectively, the maximum and minimum of \(\{X_{1}^{\ast},\cdots,X_{n}^{\ast}\}\). Suppose that some of the random variables \(X_1^{\ast},X_2^{\ast},\cdots\) can be observed and let \(\widetilde{M}_n^{\ast}\) and \(\widetilde{m}_n^{\ast}\) denote, respectively, the maximum and minimum of the observed random variables from the set \(\{X_1^{\ast},\cdots,X_n^{\ast}\}\). In this paper, we consider the asymptotic joint limiting distribution and the almost sure limit theorems related to the random vector \((\widetilde{M}_n^{\ast}, \widetilde{m}_n^{\ast}, M_n^{\ast}, m_n^{\ast})\). The results are extended to weakly dependent stationary Gaussian sequences.     

Some limit theorems for the empirical process indexed by functions

Summary Let,n1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,,P). Under weak metric entropy conditions on,n1, and under growth conditions on we show that there are non-zero numerical constantsC ₁ andC ₂ such that where (n) is a non-decreasing function ofn related to the metric entropy of. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.Portions of this article were presented during the conference on Mathematical Stochastics (February 19–25, 1984) at Oberwolfach, West Germany     

Almost sure central limit theorems for functionals of absolutely regular processes with application to U-statistics

We prove an almost sure central limit theorem for functionals of absolutely regular processes and extend this result to U-statistics.