Page 337 - A First Course In Stochastic Models

332 ADVANCED RENEWAL THEORY

Heavy-tailed distributions
The probability distribution function B(x) of the claim sizes (service times) is said
to be heavy-tailed when B(x) does not satisfy Assumption 8.4.1. An important sub-
class of heavy-tailed distributions is the class of subexponential distributions. Let
X 1 , X 2 , . . . be a sequence of non-negative independent random variables which are
distributed according to the probability distribution function B(x). The distribution
function B(x) is said to be subexponential if B(x) < 1 for all x > 0 and
P {X 1 + · · · + X n > x} ∼ nP {X 1 > x} as x → ∞ (8.4.10)

for all n ≥ 2. It can be shown that (8.4.10) holds for all n ≥ 2 if it holds for n = 2.
A physical interpretation of subexponentiality follows by noting that condition
(8.4.10) is equivalent to
P {X 1 + · · · + X n > x} ∼ P {max (X 1 , . . . , X n ) > x} as x → ∞ (8.4.11)

for all n ≥ 2. In other words, subexponentiality means that a very large value
of a finite sum of independent subexponential random variables is most likely
caused by a very large value of one of the random variables. This property makes
subexponentiality a commonly used paradigm in insurance mathematics, especially
in modelling catastrophes. The class of subexponential distributions is a natural
subclass of heavy-tailed distributions. This subclass includes the lognormal distri-
bution, the Pareto distribution and the Weibull distribution with a shape parameter
less than 1. The equivalence of (8.4.10) and (8.4.11) is easily proved. Therefore
note that
P {max (X 1 , . . . , X n ) > x} = 1 − [B(x)] n
n−1
k
= [1 − B(x)] [B(x)] ∼ n[1 − B(x)]
k=0
as x → ∞
and so P {max(X 1 , . . . , X n ) > x} ∼ nP {X 1 > x} as x → ∞. From this result the
equivalence of (8.4.10) and (8.4.11) follows.
Denote by

1 x
B e (x) = {1 − B(y)} dy, x ≥ 0
µ 0
the equilibrium excess distribution function associated with B(x). Then the follow-
ing result can be proved:
ρ
Q(x) ∼ [1 − B e (x)] as x → ∞ (8.4.12)
1 − ρ
if and only if B(x) is subexponential. Here ρ = λµ/σ. This result is mainly of
theoretical importance. Unlike the asymptotic expansion (8.4.9) for the light-tailed