We show that most random walks in the domain of attraction of a symmetric stable law have a non-trivial almost sure central limit theorem with the normal law as the limit.

We study the detection of a possible change in a stationary autoregressive process of order r. The test statistics are based on weighted supremum and Lp-functionals of the residual sums. Some limit theorems are proven under necessary and sufficient conditions.

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 consider the asymptotic distributions of suprema of heavily weighted quantile processes. We give representations of the limiting random variables in terms of partial sums of independent exponentially distributed random variables and Poisson process. There are three different types of limiting...

We prove the weak convergence of the empirical process of c(n)-close pairs. The limit is always Gaussian, but its form depends on c(n) and the distribution of the observations.

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.

We approximate the empirical process, based on multivariate random samples with an arbitrary distribution function, by a single Gaussian process.

The local time of a Brownian motion can be constructed from its excursions. The normalised excursion processes converge to a limiting process. We study the rates of convergence in these weak convergence results.

Theorems of approximation of Gaussian processes for the sequential empirical process of the permutations of independent random variables are established. The results are applied to simulate critical values for the functionals of sequential empirical processes used in change point analysis. The...