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.

We establish the asymptotic normality of the sample principal components of functional stochastic processes under nonrestrictive assumptions which admit nonlinear functional time series models. We show that the aforementioned asymptotic depends only on the asymptotic normality of the sample...

In the time series literature one can often find the claim that the periodogram ordinates of an iid sequence at the Fourier frequencies behave like an iid standard exponential sequence. We review some results about functions of these periodogram ordinates, including the convergence of extremes,...

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...

A maximum-likelihood-type statistic is derived for testing a sequence of observations for no change in the parameter against a possible change. We prove that the limit distribution of the suitably normalized and centralized statistic is double exponential under the null hypothesis.

We find a necessary and sufficient condition for the weak convergence of the uniform empirical and quantile processes to a Brownian bridge in weighted Lp-distances. Under the same condition, weighted Lp-functionals of the uniform empirical and quantile processes converge in distribution to the...

In this paper we study ratios of local times of a random walk in random environment. Strong and weak limit theorems are obtained.

We develop a strong approximation of renewal processes. The consequences of this approximation are laws of the iterated logarithm and a Bahadur-Kiefer representation ofthe renewal process in terms of partial sums. The Bahadur-Kiefer representation implies that the rate of the strong...

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.