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 ${𝒟}$ may be called subexponential if its tails are heavy enough so thatTemplate:R/superscript

$\underset{x\to +\mathrm{\infty }}{\lim }\frac{\mathbb{\mathbb{P}}(X_1+X_2>x)}{\mathbb{\mathbb{P}}(X>x)}=2,X_1,X_2,X\sim {𝒟},X_1,X_2\text{ independent.}$

See Template:Slink. 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

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