Second-order properties of tail probabilities of sums and randomly weighted sums

Abstract

Let X 1 , . . . , X n be independent nonnegative random variables with respective survival functions F 1 , . . . , F n , and let 1 , . . . , n be (not necessarily independent) nonnegative random variables, independent of X 1 , . . . , X n , satisfying certain moment conditions. This paper consists of two parts. In the first part, we investigate second-order expansions of P n i=1 X i > t as t → ∞ under the assumption that the F i are of second-order regular variation (2RV) with the same first-order index but with different second-order indexes. In the second part, under the assumption that the F 1 = · · · = F n have 2RV tails, second-order expansions of tail probabilities of the randomly weighted sum n i=1 i X i are studied. The closure property of 2RV under randomly weighted sum is also discussed. The main results in this paper generalize and strengthen several known results in the literature.

Asymptotic analysis of portfolio diversification

“…The uniform convergence of (3.4) follows immediately from checking that for the limit relations in Proposition 3.7 and Theorems 4.7 of Mao and Ng (2015). The details are omitted here but are available upon request.…”

“…, X * d are i.i.d. with common distribution function G. For I(t), it follows from Theorems 4.7 of Mao and Ng (2015) that,…”

A Scaling Limit for Limit Order Books Driven by Hawkes Processes

“…For instance, Geluk et al [22] first proved that the second-order regular variation of two i.i.d random variables carries over to their maxima and sum. Their result has been extended in [17,38,39,42,44] to finite and random sums. They also showed the equivalence between the second-order regular variation and asymptotic normality of the Hill estimator.…”

“…The subject of second-order regular variation has found numerous applications in statistics and probability theory. It provides an efficient tool for measuring the rate of convergence of the distribution of extreme order statistics [13,14,15], for characterizing the asymptotic normality of Hill estimators [22,51], and for establishing the second-order expansion of tail probabilities of sums of random variables [22,42,44]. Second-order regular variation has also been successfully used to model extreme environmental events [11], to assess tail risks in financial markets [9,19], to estimate losses from rare catastrophic events [47] and to establish second-order approximations for risk and concentration measures [17,32].…”