Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure.

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete...

We obtain a strong approximation for the logarithmic average of sample extremes. The central limit theorem and laws of the iterated logarithm are immediate consequences.

We study the asymptotic behavior of the empirical distribution function and the empirical process of squared residuals. We prove the Glivenko-Cantelli theorem for the empirical distribution function. We show that the two-parameter empirical process converges to a Gaussian process.

In this paper we consider the most visited site, Xt, of a Poisson process up to time t. Our point of departure from the literature on maximal spacings is our asymptotic analysis of where the maximal spacing occurs (i.e., the size of Xt) and not the size of the spacings.

We prove a law of the iterated logarithm for stable processes in a random scenery. The proof relies on the analysis of a new class of stochastic processes which exhibit long-range dependence.

The asymptotics for the number of times the empirical distribution function crosses the true distribution function are well-known (see Dwass, 1961; or Shorack and Wellner, 1986). We give a process version of this limit theorem and we identify the limiting process to be the local time of Brownian...

We prove that the number Z(N) of level crossings of a two-parameter simple random walk in its first NxN steps is almost surely N3/2+o(1) as N--[infinity]. The main ingredient is a strong approximation of Z(N) by the crossing local time of a Brownian sheet. Our result provides a useful algorithm...

This paper is concerned with weak convergence together with convergence rates in weighted almost sure local central limit theorems for random walks. The main tools are stochastic calculus and strong approximations.

We consider a broad class of continuous martingales whose local modulus of continuity is in some sense deterministic. We show that such martingales have Gaussian probability tails, provided we appropriately normalize them by their quadratic variation. As other applications of our methods, we...