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A functional limit theorem for locally perturbed random walks

Iksanov A., Pilipenko A.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 2

353--368

A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed “locally perturbating set”) are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. Assuming that the particle cannot jump over the membrane, we prove that the weak scaling limit of the particle position is a skew Brownian motion with parameter γ ϵ 2 [−1, 1]. The path of a skew Brownian motion is obtained by taking each excursion of a reflected Brownian motion, independently of the others, positive with probability 2−1(1 + γ) and negative with probability 2−1(1 - γ). To prove the weak convergence result, we give a new approach which is based on the martingale characterization of a skew Brownian motion. Among others, this enables us to provide the explicit formula for the parameter γ. In the previous articles, the explicit formulae for the parameter have only been obtained under the assumption that outside the membrane the particle performs unit jumps.

J1 convergence of partial sum processes with a reduced number of jumps

Krizmanić D.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

107--128

Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of càdlàg functions D[0, 1] with one of the Skorokhod topologies have already been obtained. The mostly used Skorokhod J1 topology is inappropriate when clustering of large values of the partial sum processes occurs. When all extremes within each cluster of high-threshold excesses do not have the same sign, Skorokhod M1 topology also becomes inappropriate. In this paper we alter the definition of the partial sum process in order to shrink all extremes within each cluster to a single one, which allows us to obtain the functional J1 convergence. We also show that this result can be applied to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and m-dependent sequences.

Almost sure limit theorems for semi-selfsimilar processes

Fazekas I., Rychlik Z.

Probability and Mathematical Statistics

2005

Vol. 25, Fasc. 2

241--255

An integral analogue of the almost sure limit theorem is presented for semi-selfsimilar processes. In the theorem, instead of a sequence of random elements, a continuous time random process is involved; moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the theorem is applied to obtain almost sure limit theorems for semistable processes. Discrete versions of the above theorems are proved. In particular, the almost sure functional limit theorem is obtained for semistable random variables.