By Sergey Foss, Dmitry Korshunov, Stan Zachary
This monograph presents an entire and finished advent to the idea of long-tailed and subexponential distributions in a single size. New effects are provided in an easy, coherent and systematic approach. the entire usual houses of such convolutions are then acquired as effortless outcomes of those effects. The publication makes a speciality of extra theoretical features. A dialogue of the place the components of purposes presently 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 publication precious.
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Extra info for An Introduction to Heavy-Tailed and Subexponential Distributions
1 are the Pareto, Burr, and Cauchy distributions. If a distribution F on R+ is regularly varying at infinity with index −α < 0, then all moments of order γ < α are finite, while all moments of order γ > α are infinite. The moment of order γ = α may be finite or infinite depending on the particular behaviour of the corresponding slowly varying function (see below). If a function f is regularly varying at infinity with index α then we have f (x) = xα l(x) for some slowly varying function l. Hence it follows from the discussion of Sect.
52). We finish this section by observing that the exponential distribution, while itself light-tailed, is, in an obvious sense, on the boundary of the class of such distributions. We may construct examples of long-tailed (and hence heavy-tailed) distributions on R+ , say, whose tails are, in a sense, arbitrarily close to that of the exponential distribution. 19 to be any such that h(x) = o(lnα x) as x → ∞. Further, if we replace the logarithmic function by the mth iterated logarithm, we obtain again a long-tailed distribution.
On the other hand, lim inf x→∞ f (x + 1) f (2n + 1) 1 ≤ lim inf = , n→∞ f (x) f (2n ) 2 which shows that f is not long-tailed. 20 2 Heavy-Tailed and Long-Tailed Distributions h-Insensitivity We now introduce a very important concept of which we shall make frequent subsequent use. 18. Given a strictly positive non-decreasing function h, an ultimately positive function f is called h-insensitive (or h-flat) if sup | f (x + y) − f (x)| = o( f (x)) |y|≤h(x) as x → ∞, uniformly in |y| ≤ h(x). 19) implies that the function f is long-tailed, and conversely that any long-tailed function is h-insensitive for any constant function h.