By Sergey Foss, Dmitry Korshunov, Stan Zachary
This monograph presents a whole and complete creation to the speculation of long-tailed and subexponential distributions in a single measurement. New effects are provided in an easy, coherent and systematic manner. the entire regular houses of such convolutions are then bought as effortless outcomes of those effects. The publication specializes in extra theoretical elements. A dialogue of the place the components of purposes at present stand in integrated 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 necessary.
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Additional info for An Introduction to Heavy-Tailed and Subexponential Distributions
We say that F is whole-line subexponential, and write F ∈ SR , if F is long-tailed and F ∗ F(x) ∼ 2F(x) as x → ∞. Equivalently, a random variable ξ has a whole-line subexponential distribution if ξ + has a subexponential distribution. e. S ⊆ SR ⊆ L. We now have the following theorem which provides the foundation for our results on convolutions of subexponential distributions. 6. Let the distribution F on R be long-tailed (F ∈ L) and let ξ1 , ξ2 be two independent random variables with distribution F.
0 50 3 Subexponential Distributions Therefore, always lim inf x→∞ 1 F(x) x 0 F(x − y)F(y)dy ≥ 2m, where m = Eξ + and ξ has distribution F. If F is heavy-tailed, then (see Lemma 4 in ) lim inf x→∞ 1 F(x) x 0 F(x − y)F(y)dy = 2m. 12) These observations provide a motivation for the following definition. 22. Let F be a distribution on R with right-unbounded support and finite mean on the positive half line. We say that F is strong subexponential, and write F ∈ S∗ , if x 0 F(x − y)F(y)dy ∼ 2mF(x) as x → ∞, where m = Eξ + and ξ has distribution F.
27. Let the distribution F be such that F(x) = 2−2n for x ∈ [2n , 2n+1 ). Then F is not long-tailed since F(2n − 1)/F(2n ) = 4 for any n, so that F(x − 1)/ F(x) → 1 as x → ∞. But we have x−2 ≤ F(x) ≤ 4x−2 for any x > 0. In particular, F I (x) ≥ x−1 and thus F(x) = o(F I (x)) as x → ∞. 25, FI is longtailed. We now formulate a more general result which will be needed in the theory of random walks with heavy-tailed increments, and is also of some interest in its own right. Let F be a distribution on R and μ a non-negative measure on R+ such that ∞ 0 F(t)μ (dt) < ∞.
An Introduction to Heavy-Tailed and Subexponential Distributions by Sergey Foss, Dmitry Korshunov, Stan Zachary