Infra-exponential
A growth rate is said to be infra-exponential or subexponential if it is dominated by all exponential growth rates, however great the doubling time. A continuous function with infra-exponential growth rate will have a Fourier transform that is a Fourier hyperfunction.
Examples of subexponential growth rates arise in the analysis of algorithms, where they give rise to sub-exponential time complexity, and in the growth rate of groups, where a subexponential growth rate implies that a group is amenable.
A positive-valued, unbounded probability distribution [math]\displaystyle{ \cal D }[/math] may be called subexponential if its tails are heavy enough so that[1]
- [math]\displaystyle{ \lim_{x\to+\infty} \frac{{\Bbb P}(X_1+X_2\gt x)}{{\Bbb P}(X\gt x)}=2,\qquad X_1, X_2, X\sim {\cal D},\qquad X_1, X_2 \hbox{ independent.} }[/math]
See Heavy-tailed distribution. Contrariwise, a random variable may also be called subexponential if its tails are sufficiently light to fall off at an exponential or faster rate.
References
- ↑ "Subexponential distributions", Charles M. Goldie and Claudia Klüppelberg, pp. 435-459 in A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, eds. R. Adler, R. Feldman and M. S. Taggu, Boston: Birkhäuser, 1998, ISBN 978-0817639518.
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