Modern Stochastics: Theory and Applications

A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.

We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the lim sup and a law of the triple logarithm for the lim inf. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424–445]. We apply our results to several queuing systems and a birth and death process.

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477–4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360–383] under the assumption that the kernel function potentially exhibits a singular behaviour at 0. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653–677] that studied the same mathematical question for the class of Brownian semi-stationary models.

We study the frequency process $f_{1}$ of the block of 1 for a Ξ-coalescent Π with dust. If Π stays infinite, $f_{1}$ is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-Λ-coalescents with $\varLambda =\delta _{p}$, $p\in [\frac{1}{2},1)$, $f_{1}$ is not Markovian, whereas its jump chain is Markovian. For simple Λ-coalescents the distribution of $f_{1}$ at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.

Our paper starts from presentation and comparison of three definitions for the self-similar field. The interconnection between these definitions has been established. Then we consider the Lamperti scaling transformation for the self-similar field and investigate the connection between the scaling transformation for such field and the shift transformation for the corresponding stationary field. It was also shown that the fractional Brownian sheet has the ergodic scaling transformation. The strong limit theorems for the anisotropic growth of the sample paths of the self-similar field at 0 and at ∞ for the upper and lower functions have been proved. It was obtained the upper bound for growth of the field with ergodic scaling transformation for slowly varying functions. We present some examples of iterated log-type limits for the Gaussian self-similar random fields.