Tail asymptotic of Weibull-type risks

Abstract

With motivation from Arendarczyk and Dȩbicki (2011), in this paper we derive the tail asymptotics of the product of two dependent Weibull-type risks, which is of interest in various statistical and applied probability problems. Our results extend some recent findings of Schlueter and Fischer (2012) and Bose et al. (2012).

Extremal behavior of squared Bessel processes attracted by the Brown–Resnick process

“…where the second inequality follows from Lemma 2.1 in [2] (see also Corollary 2.2 in [20]) and M, M ′ are positive constants. Clearly, for C > m we further have nP(∥δ n ∥ > C) → 0, n → ∞.…”

Extremes of randomly scaled Gumbel risks

“…For g 1 , g 2 being regularly varying and ultimately monotone [29] shows that a similar result to (1.7) is valid. In statement (b) of Theorem 2.1 we remove the assumption that g 1 and g 2 are ultimately monotone.…”

Berman’s inequality under random scaling

“…The class of distribution functions satisfying (2.7) is quite large. More importantly, under (2.7) SX has also a Weibull tail behaviour if X is a N (0, 1) random variable independent of S, see e.g., [16]. We state next our second result for Weibull-type random scaling.…”

“…where ρ ij and A ij are defined in (1.1). Note that for 1 ≤ i, j ≤ n and some positive constants c 1 , c 2 , using similar arguments as in the proof of Theorem 2.1 in [16], we have…”

“…Of course, Berman's inequality alone is not enough for extending [17] to randomly scaled Gaussian triangular arrays; some additional results (see [15,16]) which show that for some tractable tail assumptions on S the scaled random vector X behaves similarly to X are also important. Specifically, we shall deal with two large classes of random scaling: a) S is a bounded random variable with a tractable tail behaviour at the right endpoint of its distribution function, including in particular the case that its survival function is regularly varying, and b) S is a…”