On maxima of chi-processes over threshold dependent grids

Abstract

In this paper, with motivation from [30] and the considerable interest in stationary chi-processes, we derive asymptotic joint distributions of maxima of stationary strongly dependent chi-processes on a continuous time and an uniform grid on the real axis. Our findings extend those for Gaussian cases and give three involved dependence structures via the strongly dependence condition and the sparse, Pickands and dense grids.The impetus for this investigation comes from numerical simulations of high extremes of continuous time random processes, see e.g., [15,30,37] for Gaussian processes, [16] for the storage process with fractional Brownian motion, [13,38,39] for stationary vector Gaussian processes and standardized stationary Gaussian processes, and [41] for stationary processes. It is shown in the aforementioned contributions that the dependence between continuous time extremes and discrete time extremes is determined strongly by the sampling frequency δ and the normalization constants, see also for related discussions [5,20,31,32,41] in the financial and time series literature. Another motivation is that since the chi-processes appear naturally as limiting processes which have

Extremes and limit theorems for difference of chi-type processes

“…which by an application of (20) implies that the mixing Condition D in [2] holds for the random process ζ (20), respectively. It follows then that (21) holds.…”

“…Since chi-type processes appear naturally as limiting processes (see, e.g., [3,4,20]), when one considers two independent asymptotic models, the study of the supremum of the difference of the two chi-type processes is also of some interest in mathematical statistics and its applications. Although for k ≥ 1 the random process ζ (κ) m,k is not Gaussian and the analysis of the supremum cannot be directly transformed into the study of the supremum of a related Gaussian random field (which is the case for chi-type processes; see, e.g., [10, 17-19, 21, 22, 25]), it turns out that it is possible to apply the techniques for dealing with extremes of stationary processes developed mainly in [2,5,6].…”

“…with D 0 , K 0 defined in Theorem 2.3. We split the sum in (20) at T β κ , where β is a constant such that 0 < β < (1 − δ)/(1 + δ) and δ = sup{ r(t) : t ≥ ǫ} < 1 (see, e.g., Lemma 8.1.1 (i) in [15]). Below K is again a positive constant which value might change from line to line.…”

On the maxima of continuous and discrete time Gaussian order statistics processes

“…Now, to complete the proof of the third assertion, we only need to make some transform. Using (12) and ( 13), we get…”

“…Turkman (2012) considered this problem by adopting the model from Albin (1990), but there are some mistakes in the paper, which have been corrected by Ling et al (2017). Ling and Tan (2016) dealt with the problems for chi-processes. The goal of this paper is to study the Piterbarg's max-discretisation theorems for order statistics processes.…”