On the Estimation of a Changepoint in a Tail Index

We investigate the application of a new estimator for the tail index proposed in [5] and [18]. Testing hypothesis of change at unknown place and detecting change in mean allow us to provide theoretical results on estimation of the changepoint in the tail index. We demonstrate the applicability of these results in practice. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 333–348, July–September, 2005.     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

On the asymptotic form of convex hulls of Gaussian random fields

We consider a centered Gaussian random field X = {X _t : t ∈ T} with values in a Banach space $\mathbb{B}$ defined on a parametric set T equal to ? ^m or ? ^m . It is supposed that the distribution of X _t is independent of t. We consider the asymptotic behavior of closed convex hulls W _n = conv{X _t : t ∈ T _n}, where (T _n ) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b _n ) _n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W _n ), where f is some function, is also studied.     

On the rate of convergence to bivariate stable laws

In the paper, we approximate the distribution function of a sum of independent nonidentically distributed bivariate random vectors by the distribution function of a stable vector and estimate the accuracy of such an approximation. The obtained general result is only a little bit worse when compared with known estimates for the case of multivariate independent and identically distributed random vectors or univariate nonidentically distributed summands. We also apply the result obtained to a specific scheme arising when considering the so-called Increment-Ratio Statistics.     

On Beveridge-Nelson decomposition and limit theorems for linear random fields

We consider linear random fields and show how an analogue of the Beveridge-Nelson decomposition can be applied to prove limit theorems for sums of such fields.     

CLT for linear random fields with martingale increments

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, which can be regarded as a continuation of the above-mentioned paper, CLT for sums of linear random field are proved in the case where innovations form martingale differences on the plane (that can be defined in several ways). In both papers, the so-called Beveridge–Nelson decomposition is used.     

Several modifications of DPR estimator of the tail index

In this paper, we continue the investigation of an estimator proposed in [Yu. Davydov, V. Paulauskas, and A. Račkauskas, More on p-stable convex sets in Banach spaces, J. Theor. Probab., 13:39–64, 2000] and [V. Paulauskas, A new estimator for tail index, Acta Appl. Math., 79:55–67, 2003] and considered in [V. Paulauskas and M. Vaičiulis, Once more on comparison of tail index estimators, preprint, 2010]. We propose a class of modifications of the so-called DPR estimator and demonstrate that these modifications can have better asymptotic properties than the original DPR estimator.