Let {X i (t), t ≥ 0}, 1 ≤ i ≤ n be independent copies of a stationary process {X(t), t ≥ 0}. For given positive constants u, T , define the set of rth conjunctions C r,T,u := {t ∈ [0, T ] : X r:n (t) > u} with X r:n (t) the rth largest order statistics of X i (t), t ≥ 0, 1 ≤ i ≤ n. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions C r,T,u is not empty. Imposing the Albin's conditions on X, in this paper we obtain an exact asymptotic expansion of this probability as u tends to infinity. Furthermore, we establish the tail asymptotics of the supremum of the order statistics processes of skew-Gaussian processes and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.

Random deflated risk models have been considered in recent literatures. In this paper, we investigate second-order tail behavior of the deflated risk X = RS under the assumptions of second-order regular variation on the survival functions of the risk R and the deflator S. Our findings are applied to approximation of Value at Risk, estimation of small tail probability under random deflation and tail asymptotics of aggregated deflated risk.The main contributions of this paper concern the second-order expansions of the tail probability of the deflated risk X = RS which are then illustrated by several examples. Our main findings are utilized for the formulations of three applications, namely approximation of Value-at-Risk, estimation of small tail probability of the deflated risk, and the derivation of the tail asymptotics of aggregated risk under deflation.The rest of this paper is organized as follows. Section 2 gives our main results under second-order regular variation conditions. Section 3 shows the efficiency of our second-order asymptotics through some illustrating examples. Section 4 is dedicated to three applications. The proofs of all results are relegated to Section 5. We conclude the paper with a short Appendix. Main resultsWe start with the definitions and some properties of regular variation followed by our principal findings. A measurable function f : [0, ∞) → IR with constant sign near infinity is said to be of second-order regular variation with parameters α ∈ IR and ρ ≤ 0, denoted by f ∈ 2RV α,ρ , if there exists some function A with constant sign near infinity satisfying lim t→∞ A(t) = 0 such that for all x > 0 (cf. Bingham et al. (1987) and Resnick (2007))Here, A is referred to as the auxiliary function of f . Noting that (2.1) implies lim t→∞ f (tx)/f (t) = x α , i.e., f is regularly varying at infinity with index α ∈ IR, denoted by f ∈ RV α ; RV 0 is the class of slowly varying functions. When f is eventually positive, it is of second-order Π-variation with the second-order parameter ρ ≤ 0, denoted by f ∈ 2ERV 0,ρ , if there exist some functions a and A with constant sign near infinity and lim t→∞ A(t) = 0 such that for all x positive lim t→∞ f (tx)−f (t) a(t)

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

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