An Introduction to Heavy-Tailed and Subexponential Distributions by Sergey Foss Dmitry Korshunov & Stan Zachary

Author:Sergey Foss, Dmitry Korshunov & Stan Zachary
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Similarly, it may be checked that the lognormal distribution satisfies conditions of Theorem 3.30 and, therefore, belongs to the class .

3.6 Conditions for Subexponentiality in Terms of Truncated Exponential Moments

Note that some heavy-tailed distributions, for example those with tail functions of the form do not satisfy the conditions of Theorem 3.30 (in this case γ = 1) and we need a more advanced technique for proving the subexponentiality of such distributions. The next two theorems, due to Pitman [40], relate the classes and to the asymptotic behaviour of truncated exponential moments with special indices.

Theorem 3.31.

Let F be a distribution on . Suppose that the hazard rate function r exists, is eventually non-increasing and that r(x) → 0 as x →∞. Then F is subexponential ( ) if and only if

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