New PDF release: An Introduction to Heavy-Tailed and Subexponential

By Sergey Foss, Dmitry Korshunov, Stan Zachary

ISBN-10: 1441994726

ISBN-13: 9781441994721

This monograph presents an entire and entire advent to the idea of long-tailed and subexponential distributions in a single size. New effects are offered in an easy, coherent and systematic manner. all of the regular houses of such convolutions are then bought as effortless results of those effects. The publication makes a speciality of extra theoretical elements. 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 ebook worthwhile.

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Additional info for An Introduction to Heavy-Tailed and Subexponential Distributions

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28) holds whenever each of the distributions Fi is long-tailed. 11). Let ξ1 , . . , ξn be independent random variables with respective distributions F1 , . . , Fn . For any fixed a > 0, we have the following lower bound: F1 ∗ . . ∗ Fn (x) ≥ = n ∑ P{ξk > x + (n − 1)a, ξ j ∈ (−a, x] for all j = k} k=1 n ∑ F k (x + (n − 1)a) ∏ Fj (−a, x]. 29) j=k For every ε > 0 there exists a such that Fj (−a, a] ≥ 1 − ε for all j. Thus, for all x > a, n F1 ∗ . . ∗ Fn (x) ≥ (1 − ε )n−1 ∑ F k (x + (n − 1)a). k=1 Since the function F 1 + .

12 that, for any heavy-tailed distribution F on R+ , lim inf x→∞ F ∗ F(x) = 2. F(x) In particular, if F is heavy-tailed on R+ and if F ∗ F(x) →c F(x) as x → ∞, where c ∈ (0, ∞], then necessarily c = 2. This observation leads naturally to the following definition. 1. Let F be a distribution on R+ with unbounded support. We say that F is subexponential, and write F ∈ S, if F ∗ F(x) ∼ 2F(x) as x → ∞. 1) Now let ξ1 and ξ2 be independent random variables on R+ with common distribution F. Then the above definition is equivalent to stating that F is subexponential if P{ξ1 + ξ2 > x} ∼ 2P{ξ1 > x} as x → ∞.

Proof. For (i), note that for each i we may choose a function hi , increasing to infinity, such that fi is hi -insensitive, and then define h by h(x) = mini hi (x). ˆ ¯ For (ii), note that we may take h(x) = min(h(x), h(x)) where h¯ is such that f is ¯h-insensitive. Finally we note that a further important use of h-insensitivity is the following. 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.

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


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