Some results on increments of the partially observed empirical process

The author investigates the almost sure behaviour of the increments of the partially observed, uniform empirical process. Some functional laws of the iterated logarithm are obtained for this process. As an application, new laws of the iterated logarithm are established for kernel density estimators.

     

On almost sure noncentral limit theorems

LetX(t) be a fractional Brownian motion or Hermite process of indexH. SetX _m (t)=m ^–H X(mt), which we view as an element ofC[0, 1]. Let {x} denote a point mass at x. Then The corresponding results for certain partial sums in the domain of attraction toX(t) are shown to hold.     

Refinement of the almost sure central limit theorem for associated processes

In this paper, for the partial sumsS _n of a stationary associated random process it is proved that the logarithmic averages converge almost surely. The asymptotic normality of the normalized difference between the logarithmic averages and their limiting value is established. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 513–522, October, 2000.     

On the convergence of generalized moments in almost sure central limit theorem

Let {ζ_k} be the normalized sums corresponding to a sequence of i.i.d. variables with zero mean and unit variance. Define random measures

and let G be the normal distribution. We show that for each continuous function h satisfying ∫ hdG<∞ and a mild regularity assumption, one has

a.s.     

An extension of almost sure central limit theorem for order statistics

Let {X _n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M _n and m _n denote the first and the second largest maxima. Assume that there are normalizing sequences a _n  > 0, b _n and a nondegenerate limit distribution G, such that . Assume also that {d _k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

A tail empirical process approach to some nonstandard laws of the iterated logarithm

We show that a number of nonstandard laws of the iterated logarithm have limiting constants which may be expressed as the extreme values of functionals off, wheref varies over suitable compact sets of functions. By solving the corresponding extremal problems, we show how these constants are generated. In addition, several new laws are presented.     

Rate of convergence in the central limit theorem for empirical processes

Let _n and be an empirical process and a generalized Brownian bridge, respectively, indexed by a class of real measurable functions. From the central limit theorem for empirical processes it follows that for allr0. In this paper, assuming the class to be countably determined, under certain conditions we obtain an estimate for some constantC. Vapnik-ervonenkis class and the indicators of lower left orthants provide examples of classes considered here.     

A note on the almost sure limit theorem for U-statistic

Summary In this note we prove an almost sure limit theorem for the products of U-statistics.     

Liminf results on the increments of an (N,d)-Gaussian process

We establish some liminf theorems on the increments of a (N,d)-Gaussian process with the usual Euclidean norm, via estimating upper bounds of large deviation probabilities on the suprema of the (N,d)-Gaussian process. This revised version was published online in June 2006 with corrections to the Cover Date.     

An empirical process interpretation of a model of species survival

We study a model of species survival recently proposed by Michael and Volkov. We interpret it as a variant of empirical processes, in which the sample size is random and when decreasing, samples of smallest numerical values are removed. Micheal and Volkov proved that the empirical distributions converge to the sample distribution conditioned not to be below a certain threshold. We prove a functional central limit theorem for the fluctuations. There exists a threshold above which the limit process is Gaussian with variance bounded below by a positive constant, while at the threshold it is half-Gaussian.     

On the almost sure central limit theorem for randomly indexed sums

Let {S_n, n ≥ 1} be partial sums of independent identically distributed random variables. The almost sure version of CLT is generalized on the case of randomly indexed sums {S_Nn, n ≥ 1}, where {N_n, n ≥ 1} is a sequence of positive integer‐valued random variables independent of {S_n, n ≥ 1}. The affects of nonrandom centering and norming are considered too (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A uniform CLT for uniformly bounded families of martingale differences

This paper develops a method to get empirical central limit theorems for martingale differences that are uniformly bounded. The main idea is to generalize to martingales some ideas of E. Gine and J. Zinn [Ann. Prob. 12, 929–989 (1984)]. We consider two examples: An extension of a theorem of R. Dudley from i.i.d. to a certain type of Markov chain, and a uniform CLT for the baker's transformation.     

On long time almost sure asymptotics of renormalized branching diffusion processes

We are concerned with the long time behavior of branching diffusion processes. We give a partial answer to the following question: given a smooth density g₀, a branching rate and a spatial motion, does there exist a (nonspatially homogeneous) binary offspring distribution such that the corresponding renormalized branching process tends a.s. to g₀(y) dy as time grows to infinity?     

Bahadur-Kiefer theorems for the product-limit process

In the random censorship from the right model, strong and weak limit theorems for Bahadur-Kiefer type processes based on the product-limit estimator are established. The main theorm is sharp and may be considered as a final result as far as this type of research is concerned. As a consequence of this theorem a sharp uniform Bahadur representation for product-limit quantiles is obtained.     

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.     

Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions

Rates of convergence of the Komlos-Major-Tusnady type in the invariance principle for empirical processes indexed by functions are obtained. The conditions are given in terms of the empirical entropy and the accuracy of the Haar type approximation for the corresponding classes of functions. The recent results of Massart⁽²⁵⁾ as well as some new results for empirical characteristic functions are obtained as a corollary.     

The almost sure central limit theorems for the maxima of sums under some new weak dependence assumptions

We prove the almost sure central limit theorems for the maxima of partial sums of r.v.’s under a general condition of dependence due to Doukhan and Louhichi. We will separately consider the centered sequences and the sequences with positive expected values.     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.