By Sergey Foss, Dmitry Korshunov, Stan Zachary
Heavy-tailed likelihood distributions are a tremendous part within the modeling of many stochastic structures. they're often used to adequately version inputs and outputs of desktop and information networks and repair amenities resembling name facilities. they're a necessary for describing possibility strategies in finance and in addition for coverage premia pricing, and such distributions take place clearly in versions of epidemiological unfold. the category contains distributions with strength legislation tails equivalent to the Pareto, in addition to the lognormal and sure Weibull distributions.
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Graduate scholars in addition to modelers within the fields of finance, assurance, community technology and environmental reports will locate this ebook to be a vital reference.
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Extra resources for An Introduction to Heavy-Tailed and Subexponential Distributions
Can the minimum min(ξ , η ) have a light-tailed distribution? 10. Suppose that ξ1 , . . , ξn are independent random variables with a common distribution F and that ξ(1) ≤ ξ(2) ≤ . . ≤ ξ(n) are the order statistics. (i) For k ≤ n, prove that the distribution of ξ(k) is heavy-tailed if and only if F is heavy-tailed. (ii) For k ≤ n − 1, prove that the distribution of ξ(k+1) − ξ(k) is heavy-tailed if and only if F is heavy-tailed. (iii) Based on (ii) and on Problem 8, prove that ξ(k) − ξ(l) has a heavy-tailed distribution if and only if F is heavy-tailed.
Fn (x)), then F is long-tailed. (iv) If F(x) = max(F1 (x), . . , Fn (x)), then F is long-tailed. (v) The distribution of min(ξ1 , . . , ξn ) is long-tailed. (vi) The distribution of max(ξ1 , . . , ξn ) is long-tailed. Proof. 16 to the corresponding tail functions. 16. 6 Long-Tailed Distributions and Integrated Tails In the study of random walks in particular, a key role is played by the integrated tail distribution, the fundamental properties of which we introduce in this section. 24. 21) (and hence x∞ F(y) dy < ∞ for any finite x) we define the integrated tail distribution FI via its tail function by F I (x) = min 1, ∞ x F(y)dy .
C1 − 1 FI (x) Letting c1 ↓ 1, we get lim inf x→∞ xF(x) ≥ α − 1. 47) we get, for any c2 < 1, lim sup x→∞ α −1 xF(x) c1− , ≤ 2 1 − c2 FI (x) Letting c2 ↑ 1, we get lim sup x→∞ xF(x) ≤ α − 1. 49) lead to xF(x) ∼ (α − 1)FI (x) as x → ∞, which implies regular variation of F with index −α . Assume now that F is a regularly varying distribution with index −α . Then, for every c > 0, 34 2 Heavy-Tailed and Long-Tailed Distributions FI (cx) = ∞ F(y)dy cx = c−1 ∼ cα −1 ∞ F(z/c)dz x ∞ x F(z)dz = cα −1 FI (x) as x → ∞, which means the regular variation of FI with index 1 − α .