By Sergey Foss, Dmitry Korshunov, Stan Zachary

ISBN-10: 1441994726

ISBN-13: 9781441994721

This monograph presents a whole and finished creation to the speculation of long-tailed and subexponential distributions in a single measurement. New effects are provided in an easy, coherent and systematic means. the entire normal homes of such convolutions are then acquired as effortless effects of those effects. The booklet specializes in extra theoretical features. A dialogue of the place the parts of functions at present stand in incorporated as is a few initial mathematical fabric. Mathematical modelers (for e.g. in finance and environmental technology) and statisticians will locate this e-book necessary.

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

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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). 31) For every ε > 0 there exists a such that G(a) ≥ 1 − ε . Thus, for all x, F ∗ G(x) ≥ (1 − ε )F(x − a).

If F ∈ S∗ then G ∈ S∗ . The following theorem asserts in particular that S∗ is a subclass of SR . 27. If F ∈ S∗ , then F ∈ SR and FI ∈ S. We do not provide a proof for this result now. Instead of that we recall the notion of an integrated weighted tail distribution and state sufficient conditions for its tail to be subexponential. 28. Let F be a distribution on R and let μ be a non-negative measure on R+ such that ∞ 0 F(t) μ (dt) is finite. 25), we can define the distribution Fμ on R+ by its tail: F μ (x) := min 1, ∞ 0 F(x + t) μ (dt) , x ≥ 0.

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). 31) For every ε > 0 there exists a such that G(a) ≥ 1 − ε . Thus, for all x, F ∗ G(x) ≥ (1 − ε )F(x − a). 30) once more follows by letting ε → 0. We now have the following corollary. 32. Let the distribution F on R be such that F is long-tailed (F ∈ L) and let the distribution G be such that G(a) = 0 for some a.

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


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