加权随机 $d$-正则图的极值谱行为

Extremal behavior of stochastic integrals driven by regularly varying Lévy processes

We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regularly varying with index $α>0$. For predictable integrands with a finite $(α+δ)$-moment, for some $δ>0$, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy process and we determine its limit measure associated with regular variation on the space of càdlàg functions.

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random…

Weighted maxima and sums of non-stationary random length sequences in heavy-tailed models

The sums and maxima of weighted non-stationary random length sequences of regularly varying random variables may have the same tail and extremal indices, Markovich and Rodionov (2020). The main constraints are that there exists a unique series in a scheme of series with the minimum tail index, the tail of the term number is lighter than the tail of the terms and the weights are positive constants. These assumptions are changed here: a bounded random number of series is allowed to have the minimum tail index, the tail of the term number may be heavier than the tail of the terms and the weights …

Concave majorant of stochastic processes and Burgers turbulence

The asymptotic solution of the inviscid Burgers equations with initial potential $ψ$ is closely related to the convex hull of the graph of $ψ$. In this paper, we study this convex hull, and more precisely its extremal points, if $ψ$ is a stochastic process. The times where those extremal points are reached, called extremal times, form a negligible set for Lévy processes, their integrated processes, and Itô processes. We examine more closely the case of a Lévy process with bounded variation. Its extremal points are almost surely countable, with accumulation only around the extremal values. Thes…

正则树上简单随机游走局部时的极值过程

Extremal process of the local time of simple random walk on a regular tree

本文研究正则有根树上连续时间简单随机游走在两种起止条件下叶节点局部时的极值极限，证明其收敛为随机强度测度下的装饰 Poisson 点过程，并借助同构定理通过 Lindeberg 型交换确定装饰普适性。

Regularly Varying Random Fields

We study the extremes of multivariate regularly varying random fields. The crucial tools in our study are the tail field and the spectral field, notions that extend the tail and spectral processes of Basrak and Segers (2009). The spatial context requires multiple notions of extremal index, and the tail and spectral fields are applied to clarify these notions and other aspects of extremal clusters. An important application of the techniques we develop is to the Brown-Resnick random fields.