Barbara Szyszkowicz

Weak convergence of weighted empirical type processes under contiguous and changepoint alternatives

Let X1, X2,... be independent random variables. We study asymptotic behaviour of two-time parameter empirical type processes based on observations, ranks and sequential ranks. We introduce weight functions and derive the limiting distributions of these processes under the null hypothesis of Xi being identically distributed, as well as under a class of continguous alternatives which can accommodate the possible occurrence of a changepoint in the series of measurements.

Strong invariance principles for sequential Bahadur-Kiefer and Vervaat error processes of long-range dependent sequences

In this paper we study strong approximations (invariance principles) of the sequential uniform and general Bahadur-Kiefer processes of long-range dependent sequences. We also investigate the strong and weak asymptotic behavior of the sequential Vervaat process, i.e., the integrated sequential Bahadur-Kiefer process, properly normalized, as well as that of its deviation from its limiting process, the so-called Vervaat error process. It is well known that the Bahadur-Kiefer and the Vervaat error processes cannot converge weakly in the i.i.d. case. In contrast to this we conclude that the Bahadur-Kiefer and Vervaat error processes, as well as their sequential versions, do converge weakly to a Dehling-Taqqu type limit process for certain long-range dependent sequences.

Changepoint problems and contiguous alternatives

We consider U-statistic type processes which can be used for detecting a changepoint in a random sequence. The asymptotic results when all random variables are assumed to have the same distribution are already known. We derive the limiting distribution of such processes under contiguous alternatives.

Convergence of weighted partial sums when the limiting distribution is not necessarily Radon

Let be a non-separable Banach space of real-valued functions endowed with a weighted sup-norm. We consider partial sum processes as random functions with values in . We establish weak convergence statements for these processes via their weighted approximation in probability by an appropriate sequence of Gaussian random functions. The main result deals with convergence of distributions of certain functionals in the case when the Wiener measure is not necessarily a Radon measure on .

Lp-approximations of weighted partial sum processes

We prove an optimal asymptotic result for weighted Lp-distance of partial sum processes of independent identically distributed random variables. Using this result, we prove also the convergence in distribution of weighted Lp-functionals of U-statistic type processes which are used in change-point analysis.

The Chibisov—O'Reilly theorem for empirical processes under contiguous measures☆

We consider empirical processes based on independent observations. We prove that the weak convergence of weighted empirical processes of i.i.d. random variables for the optimal class of weight function as proven in Csorgo, Csorgo, Horvath and Mason (Ann. Probabl. 14, 1986, 31-85) continueds to hold under contiguous measures for the same class of weight functions.