For a skew normal random sequence, convergence rates of the distribution of its partial maximum to the Gumbel extreme value distribution are derived. The asymptotic expansion of the distribution of the normalized maximum is given under an optimal choice of norming constants. We find that the optimal convergence rate of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/ log n.

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)

Let {Xn, n ≥ 1} be an independent, identically distributed random sequence with each Xn having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.

Let {Xn(t),t∈[0,∞)},n∈ℕ, be standard stationary Gaussian processes. The limit distribution oft∈[0,T(n)]|Xn(t)| is established asrn(t), the correlation function of {Xn(t),t∈[0,∞)},n∈ℕ, which satisfies the local and long-range strong dependence conditions, extending the results obtained in Seleznjev (1991).

In an attempt to improve the ultrasonic testing capability of a conventional C-scan system, a flaw detection method using an ultrahigh gain is developed in this paper. A time-dependent threshold for image segmentation is applied to identify automatically material anomalies present in the sample. A singly scattered response model is used with extreme value statistics to calculate the confidence bounds of grain noise. The result is a time-dependent threshold associated with the grain noise that can be used for segmentation. Ultrasonic imaging experiments show that the presented method has advantages over a traditional fixed threshold approach with respect to false positives and missed flaws. The results also show that a low gain is adverse to the detection of microflaws with subwavelength dimensions. The forward model is expected to serve as an effective tool for the probability of detection of flaws and the inspection of coarse-grained materials in the future.