Extremes and limit theorems for difference of chi-type processes

Abstract

Abstract. Let {ζ (κ) m,k (t), t ≥ 0}, κ > 0 be random processes defined as the differences of two independent stationary chi-type processes with m and k degrees of freedom. In this paper we derive the asymptotics of P sup t∈[0,T ] ζ (κ) m,k (t) > u , u → ∞ under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.

On Extremal Index of max-stable stationary processes

“…In view of the aforementioned paper, several results derived here for maxstable processes are extendable to semi-min-stable processes. This latter condition is a rephrasing of the so-called B condition, see e.g., [1,12,2], which was formulated by discretising the original Berman's condition, see [6]. Condition 3.1 is weaker than the D ′ (xn) condition of Leadbetter as discussed in [22][Section 5.3.2].…”

“…with E a unit exponential random variable independent of W . Expression (3.17) of the classical Pickands constant was initially derived in [1] for some general W , see also recent contribution [2]. In [28], Proposition 3 or the formula in [24][p.44] the classical Pickands constant is the limit of a cluster index.…”

Extremes of locally stationary Gaussian and chi fields on manifolds

“…To our best knowledge, the study in the existing literature has only focused on χ-processes and fields indexed by intervals or hyper cubes, but not low-dimensional manifolds. See, for example, Albin et al [2], Bai [4], Hashorva and Ji [19], Ji et al [20], Konstantinides et al [22], Lindgren [23], Ling and Tan [24], Liu and Ji [25,26], Piterbarg [30,31], Tan and Hashorva [38,39], Tan and Wu [40]. Also it is worth mentioning that it is often assumed that X 1 , • • • , X r are independent copies of a Gaussian process or field X in the literature, while the cross-dependence among X 1 , • • • , X r is allowed under certain constraints in this work.…”