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This article is a survey of the results on asymptotic behavior of small ball probabilities in $L_2$-norm. Recent progress in this field is mainly based on the methods of spectral theory of differential and integral operators.
Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the…
Random fields play a central role in the analysis of spatially correlated data and, as a result, have a significant impact on a broad array of scientific applications. This paper studies the cepstral random field model, providing recursive…
We study the problem of empirical minimization for variance-type functionals over functional classes. Sharp non-asymptotic bounds for the excess variance are derived under mild conditions. In particular, it is shown that under some…
Extreme value theory (EVT) is a statistical tool for analysis of extreme events. It has a strong theoretical background, however, we need to choose hyper-parameters to apply EVT. In recent studies of machine learning, techniques of choosing…
A parameter estimation problem is considered, in which dispersed sensors transmit to the statistician partial information regarding their observations. The sensors observe the paths of continuous semimartingales, whose drifts are linear…
In this paper we present some limit theorems for power variation of L\'evy semi-stationary processes in the setting of infill asymptotics. L\'evy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving…
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…
The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several…
Long-range dependent random fields with spectral densities which are unbounded at some frequencies are investigated. We demonstrate new examples of covariance functions which do not exhibit regular varying asymptotic behaviour at infinity.…
We determine the asymptotic behavior of the realized power variations, or more generally of sums of a given test function evaluated at the successive increments of a L\'{e}vy process. One can completely elucidate the first order behavior…
We derive a higher-order asymptotic expansion of the conditional characteristic function of the increment of an It\^o semimartingale over a shrinking time interval. The spot characteristics of the It\^o semimartingale are allowed to have…
A theorem about asymptotic estimation of multiple integral of a special type is proved for the case when the integrand peaks at the integration domain bound, but not at a point of extremum. Using this theorem the asymptotic expansion of the…
A class of stochastic processes strongly related to random sums plays an important role in network and in finance. In this paper we study this kind of stochastic process discuss an overtime unchanged parameter and reveal its asymptotic…
In this paper we first provide several conditional limit theorems for L\'evy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of…
We use the martingale convergence method to get the weak convergence theorem on general functionals of partial sums of independent heavy-tailed random variables. The limiting process is the stochastic integral driven by $\alpha-$stable…
In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem…
There has been a growing interest in providing models for multivariate spatial processes. A majority of these models specify a parametric matrix covariance function. Based on observations, the parameters are estimated by maximum likelihood…
We obtain results on both weak and almost sure asymptotic behaviour of power variations of a linear combination of independent Wiener process and fractional Brownian motion. These results are used to construct strongly consistent parameter…
We consider a class of doubly weighted rank-based estimating methods for the transformation (or accelerated failure time) model with missing data as arise, for example, in case-cohort studies. The weights considered may not be predictable…