The structure of the class of subexponential distributions
Tóm tắt
LetX
1,X
2, ...,X
n
be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.)F and denote byX
1:n
≦X
2:n
≦...≦X
n:n
the order statistics of the sample. We characterize the class of d.f.F for which
$$P(X_{1:n} + X_{2:n} + \ldots + X_{n - i:n} > x) \sim P(X_{n - i:n} > x) as x \to \infty $$
for fixedn andi (i≦n-1), and we show that it is independent ofn. This leads to the genesis of a new class of d.f.L
i
; we show that the sequence (L
i
)
∞
=0 is strictly decreasing and we illustrate how the classesL
i
determine the probabilistic structure of the classL of subexponential distributions.
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