Infra-exponential

{{See also|Time complexity#Sub-exponential time}}

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.[https://encyclopediaofmath.org/wiki/Fourier_hyperfunction Fourier hyperfunction] in the Encyclopedia of Mathematics

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 $\backslash cal\; D$ may be called subexponential if its tails are heavy enough so that{{r|GK|at=Definition 1.1}}

:$\backslash lim\_\{x\backslash to+\backslash infty\}\; \backslash frac\{\{\backslash Bbb\; P\}(X\_1+X\_2>x)\}\{\{\backslash Bbb\; P\}(X>x)\}=2,\backslash qquad\; X\_1,\; X\_2,\; X\backslash sim\; \{\backslash cal\; D\},\backslash qquad\; X\_1,\; X\_2\; \backslash hbox\{\; independent.\}$

See {{slink|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.