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.^[1]

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 ${\cal {D}}$ may be called subexponential if its tails are heavy enough so that^[2]^{: Definition 1.1}

\lim _{x\to +\infty }{\frac {{\mathbb {P}}(X_{1}+X_{2}>x)}{{\mathbb {P}}(X>x)}}=2,\qquad X_{1},X_{2},X\sim {\cal {D}},\qquad X_{1},X_{2}{\hbox{ independent.}}

See Heavy-tailed distribution § Subexponential distributions. 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

^ Fourier hyperfunction in the Encyclopedia of Mathematics
^ "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.

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

[1] Fourier hyperfunction in the Encyclopedia of Mathematics

[GK-2] "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.

[1]

[2]