Joint Asymptotic Distributions of Smallest and Largest Insurance Claims

Abstract

Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalized sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.

Distinguishing Log-Concavity from Heavy Tails

“…We refer to [16][17][18][19] for related work in this direction. It is shown in Lemma 1.2 of [4] that log-concavity or log-concavity of the density is closely related to the occurrence of the principle of a single big jump.…”

Asymptotic Results for Conditional Measures of Association of a Random Sum

“…ii) It holds for every m, n, k ≥ 2 that lim t→∞ Pr X (1) m > Y (1) n , X (2) m > Z (2) r |X (2) m , Y (2) n , Z (2) r > t = 13 45 .…”

“…Proof. i) It is useful to first note that Pr S m > T n , X (1) m > Z (1) r > t|X (1) m , Y (1) n , Z (1)…”

“…The results from parts a) and b) can be derived in the same manner, and thus, we will focus only on part a). a)i) It is not difficult to find the claim from this part by applying the result from N > t = 0 for all t > 0 since α > 2, which implies that SX (1) N has a finite mean. Thus, integration by parts and an obvious change of variables lead to…”

Asymptotic results for conditional measures of association of a random sum

“…The asymptotic dependence, i.e. the joint tail behaviour, among various order statistics has been recently investigated in Hashorva (2007), Albrecher et al (2014) and Peng (2014). A recent paper, namely, Asimit et al (2014), studied the asymptotic behaviour of the conditional Kendall's tau from a statistical extremes perspective.…”