Extremes and limit theorems for difference of chi-type processes^∗

¹ Department of Mathematical Sciences, Chalmers University of Technology, SE-412, 96 Gothenburg, Sweden.
² Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
³ Institute for Information and Communication Technologies, HEIG-VD, University of Applied Sciences of Western Switzerland, Route de Cheseaux 1, 1401 Yverdon-les-Bains, Switzerland.
⁴ School of Mathematics and Statistics, Southwest University, Beibei District 400715 Chongqing, China.
lcx98@swu.edu.cn

Received: 16 August 2015
Revised: 27 April 2016
Accepted: 13 July 2016

Abstract

Let {ζ_m,k^(κ)(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_m,k^(κ)(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.