By Sergey Foss, Dmitry Korshunov, Stan Zachary

ISBN-10: 1441994726

ISBN-13: 9781441994721

This monograph presents an entire and finished advent to the idea of long-tailed and subexponential distributions in a single size. New effects are awarded in an easy, coherent and systematic method. all of the average houses of such convolutions are then got as effortless effects of those effects. The publication specializes in extra theoretical features. A dialogue of the place the parts of purposes at present stand in incorporated as is a few initial mathematical fabric. Mathematical modelers (for e.g. in finance and environmental technological know-how) and statisticians will locate this publication valuable.

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

**Sample text**

F 1 (x) + . . 28) now follows by letting ε → 0. 29 yields the following corollary. 30. Let the distribution F on R be long-tailed (F ∈ L). Then, for any n ≥ 2, lim inf x→∞ F ∗n (x) ≥ n. 29. 31. Let the distributions F and G on R be such that F is long-tailed (F ∈ L). Then, lim inf x→∞ F ∗ G(x) ≥ 1. 30) 26 2 Heavy-Tailed and Long-Tailed Distributions Proof. Let ξ and η be independent random variables with respective distributions F and G. For any fixed a, F ∗ G(x) ≥ P{ξ > x − a, η > a} = F(x − a)G(a).

Thus distributions which are regularly varying at infinity are o(x)-insensitive. Intermediate Regularly Varying Distributions It turns out that the property of o(x)-insensitivity characterises a slightly wider class of distributions than that of distributions whose tails are regularly varying, and we now discuss this. 46. A distribution F on R is called intermediate regularly varying if lim lim inf ε ↓0 x→∞ F(x(1 + ε )) = 1. 46) Any regularly varying distribution is intermediate regularly varying.

Thus since F is long-tailed we have lim sup x→∞ F ∗ G(x) ≤ 1. 31, we obtain the desired equivalence. In order to further study convolutions of long-tailed distributions, we make repeated use of two fundamental decompositions. Let h > 0 and let ξ and η be independent random variables with distributions F and G respectively. Then the tail function of the convolution of F and G possesses the following decomposition: for x > 0, F ∗ G(x) = P{ξ + η > x, ξ ≤ h} + P{ξ + η > x, ξ > h}. 33) since if ξ ≤ h and η ≤ h then ξ + η ≤ 2h ≤ x.

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

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