Read e-book online An Introduction to Heavy-Tailed and Subexponential PDF

By Sergey Foss, Dmitry Korshunov, Stan Zachary

ISBN-10: 1441994726

ISBN-13: 9781441994721

This monograph offers a whole and finished advent to the speculation of long-tailed and subexponential distributions in a single measurement. New effects are offered in an easy, coherent and systematic means. all of the typical houses of such convolutions are then received as effortless outcomes of those effects. The e-book makes a speciality of extra theoretical points. A dialogue of the place the components of functions presently stand in integrated as is a few initial mathematical fabric. Mathematical modelers (for e.g. in finance and environmental technology) and statisticians will locate this ebook priceless.

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Extra resources for An Introduction to Heavy-Tailed and Subexponential Distributions

Example text

34. 19. Now note first that, as in the earlier proof, h(x) −∞ G(x − y)F(dy) + h(x) −∞ F(x − y)G(dy) ≤ G(x − h(x)) + F(x − h(x)), and second that h(x) −∞ G(x − y)F(dy) + ≥ h(x) −h(x) h(x) −∞ F(x − y)G(dy) G(x − y)F(dy) + h(x) −h(x) F(x − y)G(dy) ≥ F(−h(x), h(x)]G(x + h(x)) + G(−h(x), h(x)]F(x + h(x)) ∼ G(x + h(x)) + F(x + h(x)) as x → ∞, where the last equivalence follows since h(x) → ∞ as x → ∞. The required result now follows from the choice of the function h. 34, and (b) F is long-tailed and G(x) = o(F(x)) as x → ∞.

For any given positive function h, increasing to infinity, we may consider the class of those distributions whose (necessarily long-tailed) tail functions are h-insensitive. For varying h, this gives a powerful method for the classification of such distributions, which we explore in detail in Sect. 8. 5 Long-Tailed Distributions As discussed in the Introduction, all heavy-tailed distributions likely to be encountered in practical applications are sufficiently regular as to be long-tailed, and it is the latter property, as applied to distributions, which we study in this section.

Assume that F ∈ SR . Then for any n ≥ 2, F ∗n (x)/F(x) → n as x → ∞. In particular, F ∗n ∈ SR . The following converse result follows. 21. Let a distribution F on R+ with unbounded support be such that F ∗n (x) ∼ nF(x) for some n ≥ 2. Then F is subexponential. Proof. Take G := F ∗(n−1) . For any x we have the inequality G(x) ≥ F(x). On the other hand, G(x) ≤ F ∗n (x) ∼ nF(x). Hence the distributions F and G are weakly tail-equivalent. 11, as x → ∞, F ∗ G(x) ≥ (1 + o(1))(F(x) + G(x)) = F(x) + G(x) + o(F(x)).

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

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