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Second order regular variation is a reenement of the concept of regular variation which is useful for studying rates of convergence in extreme value theory and asymptotic normality of tail estimators. For a distribution tail 1 ? F which possesses second order regular variation, we discuss how this property is inherited by 1 ? F 2 and 1 ? F 2. We also(More)
Consider a sequence {Xk, k ≥ 1} of random variables on (−∞,∞). Results on the asymptotic tail probabilities of the quantities Sn = ∑nk=0 Xk ,X(n) = max0≤k≤n Xk , and S(n) = max0≤k≤n Sk , with X0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate(More)
In this paper we study the asymptotic behavior of the tail probabilities of sums of dependent and real-valued random variables whose distributions are assumed to be subexponential and not necessarily of dominated variation. We propose two general dependence assumptions under which the asymptotic behavior of the tail probabilities of the sums is the same as(More)
Let X1, X2, . . . Xn be independent and identically distributed (i.i.d.) non-negative random variables with common distribution function (d.f.) F with unbounded support and EX2 1 < ∞. We show that for a large class of heavy tailed random variables with a finite variance the renewal function U satisfies U(x)− x μ − μ2 2μ2 ∼ − 1 μx ∫ ∞
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