By Sergey Foss, Dmitry Korshunov, Stan Zachary
This monograph offers a whole and accomplished advent to the idea of long-tailed and subexponential distributions in a single measurement. New effects are awarded in an easy, coherent and systematic manner. all of the usual homes of such convolutions are then bought as effortless results of those effects. The booklet specializes in extra theoretical points. A dialogue of the place the components of functions presently 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 booklet priceless.
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Additional resources for An Introduction to Heavy-Tailed and Subexponential Distributions
Example text
The function l(ε ) := lim inf x→∞ F(x(1 + ε )) F(x) decreases in ε > 0, due to the monotonicity of F. 46) implies that there exists a positive δ such that l(ε ) ≤ 1 − 2δ for any ε > 0. Hence, for any positive integer n, we can find xn such that F(xn (1 + 1/n)) ≤ (1 − δ )F(xn ) Without loss of generality we may assume the sequence {xn } to be increasing. Now put h(x) = x/n for x ∈ [xn , xn+1 ). Then h(x) = o(x) as x → ∞. However, lim inf x→∞ F(x + h(x)) F(xn + h(xn )) ≤ lim inf n→∞ F(x) F(xn ) = lim inf n→∞ F(xn (1 + 1/n)) F(xn ) ≤ 1−δ, which contradicts the o(x)-insensitivity of F.
For the lognormal distribution, one can take h(x) = o(x/ ln x) in order to have h-insensitivity. For the 1−α ). Weibull distribution with parameter α ∈ (0, 1), one can √take h(x) = o(x In many practical situations, the class of so-called x-insensitive distributions – those which are h-insensitive for the function h(x) = x1/2 – is of special interest. Among these are intermediate regularly-varying distributions (in particular regularly-varying distributions), lognormal distributions and Weibull distributions with shape parameter α < 1/2.
In the case where we do not have h(x) < x/2 for all x, small variations are required to the above proof. If F is subexponential, then we may consider instead ˆ ˆ the function hˆ given by h(x) = min(h(x), x/2). 3) holds with h replaced by h, and so also in its original form. 4), which does not affect the argument. 6 implies that, as in the case of non-negative subexponential summands, the most probable way for large deviations of the sum ξ1 + ξ2 to occur is that one summand is small and the other is large; for (very) large x, the main contribution to the probability P{ξ1 + ξ2 > x} is made by the probabilities of the events {ξ1 + ξ2 > x, ξi ≤ h(x)} for i = 1, 2.