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Rafal Kulik

Publications and source records attributed to Rafal Kulik.

22 records · Page 2Linked to original sources
arXiv2007

Empirical process of long-range dependent sequences when parameters are estimated

In this paper we study the asymptotic behaviour of empirical processes when parameters are estimated, assuming that the underlying sequence of random variables is long-range dependent. We show completely different phenomena compared to i.i.d. situation, as well as compared to ordinary empirical processes of long range dependent sequences. Applications include Kolmogorov-Smirnov and Cramer-Smirnov-von Mises goodness-of-fit statistics.

math.ST↗
arXiv2006

Sums of extreme values of subordinated long-range dependent sequences: moving averages with finite variance

In this paper we characterize the limiting behavior of sums of extreme values of long range dependent sequences defined as functionals of linear processes with finite variance. The extremal sums behave completely different by compared to the i.i.d case. In particular, though we still have asymptotic normality, the scaling factor is relatively bigger than in the i.i.d case, meaning that the maximal terms have relatively smaller contribution to the whole sum. Also, the scaling need not depend on the tail index of the underlying marginal distribution, as it is well-known to be so in the i.i.d. situation. Furthermore, subordination may completely change the asymptotic properties of sums of extremes.

math.PR↗
arXiv2006

Poisson limits for empirical point processes

Define the scaled empirical point process on an independent and identically distributed sequence $\{Y_i: i\le n\}$ as the random point measure with masses at $a_n^{-1} Y_i$. For suitable $a_n$ we obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. We obtain limits at points where the density of $Y_i$ may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. These results also hold for multivariate $Y_i$ with little additional effort. Applications are provided both to nearest-neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas.

math.PR↗