Integral analogues of almost sure limit theorems

Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.     

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.     

Strong Laws and Summability for φ-Mixing Sequences of Random Variables

In this note the almost sure convergence of stationary, -mixing sequences of random variables according to summability methods is linked to the fulfillment of a certain integrability condition generalizing and extending the results for i.i.d. sequences. Furthermore we give via Baum-Katz type results an estimate for the rate of convergence in these laws.     

独立随机序列最大值的几乎处处极限定理

本文研究了独立随机序列最大值分布的几乎必然收敛性.利用有关协方差的不等式和加权平均,获得独立随机序列最大值的几乎处处极限.将独立同分布随机序列的结论,推广了独立但不同分布的情形.     

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.     

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 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)     

CONVERGENCE RATES IN THE STRONG LAWS OF NONSTATIONARY ρ-MIXING RANDOM FIELDS

1 IntroductionLet d be a positive integer, Nd be the d-dimensional lattice equipped with the coordinatewise partial order, 5. For any A C N', set SA = Z.,. X., IAI =the cardinal number of A.For anyn E Nd, let (n) = {m E N',m 5 n}, S. = s(.), Inl = I(n)l = nlnZ' nd and hull' denote the Euclidean norm. Occasionally, n,k,n, etc. will also denote positive integers, thereader will not be confused from their context. A d-dimensional discrete field of real randomvariables {Xk; k E N'} wil…     

On almost sure limiting behavior of weighted sums of random fields

We study necessary and sufficient conditions for the almost sure convergence of averages of independent random variables with multidimensional indices obtained by certain summability methods.     

Stable and Semistable Hemigroups: Domains of Attraction and Self-Decomposability

A hemigroup of probability measures builds the set of distributions of the increments of an independent increment process. Decomposability properties of the hemigroup lead to stable respectively semistable hemigroups and enable us to show that such hemigroups appear as limits of certain functional limit theorems for operator-normed independent (non-identically) distributed random vectors. Regular variation properties of the norming operators show that the functional limit theorems are closely related to limits of infinitesimal triangular arrays of independent random vectors, i.e., to operator-self-decomposable respectively operator-semi-self-decomposable laws.     

Multivariate time series analysis

Some key theorems in the asymptotic theory for multivariate time series, using spectrl methods, are established. These theorems relate to various estimation situations including multiple systems of regressions, the determination of the frequency of a periodic signal and the determination of the velocity of a signal propagated through a dispersive medium and received with noise at a number of recorders. The theorems are of a general kind and relate to the almost sure convergence of averages of the periodogram and to the limiting covariance properties and the central limit theorem for such averages. Some brief indications are given concerning extensions of the results to cases where processes are observed that are stationary in time and homogenous with respect to spatial translation.     

A universal result in almost sure central limit theory

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561–574; Schatte, Math. Nachr. 137 (1988) 249–256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular, a.s. central limit theorems and various related ‘logarithmic’ limit theorems have been obtained for several classes of independent and dependent random variables. In this paper we extend this theory and show that not only the central limit theorem, but every weak limit theorem for independent random variables, subject to minor technical conditions, has an analogous almost sure version. For many classical limit theorems this involves logarithmic averaging, as in the case of the CLT, but we need radically different averaging processes for ‘more sensitive’ limit theorems. Several examples of such a.s. limit theorems are discussed.     

On the Very Strong Summability of Orthogonal Series

We prove two theorems pertaining to the very strong summability of orthogonal series for general summability methods and present four consequences of them for classical summability methods.     

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).     

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.     

On the invariant mean and statistical convergence

Two concepts - one of almost convergence and the other of statistical convergence - play a very active role in recent research on summability theory. The definition of almost convergence introduced by Lorentz [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190] originated from the concept of the Banach limit, while the statistical convergence introduced by Fast [H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244] was defined through the concept of density. Both involve non-matrix methods of summability and they are incompatible. In this work we define two new kinds of summability methods by using these two mutually incompatible concepts of the Banach limit and of density to deal with those sequences which are statistically convergent but not almost convergent or vice versa.     

渐近负相关随机域强定律的收敛率

In this paper,a notion of negative side ρ-mixing (ρ--mixing) which can be regarded as asymptotic negative association is defined,and some Rosenthal type inequalities for ρ--mixing random fields are established. The complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are also discussed for ρ--mixing random fields. The results obtained extend those for negatively associated sequences and ρ*-mixing random fields.     

Some Limit Theorems for Weighted Sums of Arrays of Nod Random Variables

In this paper the authors study the complete, weak and almost sure convergence for weighted sums of NOD random variables and obtain some new limit theorems for weighted sums of NOD random variables, which extend the corresponding theorems of Stout [1], Thrum [2] and Hu et al. [3].     

Random walks on diestel-leader graphs

We investigate various features of a quite new family of graphs, introduced as a possible example of vertex-transitive graph not roughly isometric with a Cayley graph of some finitely generated group. We exhibit a natural compactification and study a large class of random walks, proving theorems concerning almost sure convergence to the boundary, a strong law of large numbers and a central limit theorem. The asymptotic type of then-step transition probabilities of the simple random walk is determined.     

Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma.