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

{\displaystyle {\cal {D}}}

may be called subexponential if its tails are heavy enough so that

lim

x

→

+

∞

P

(

X

1

+

X

2

>

x

)

P

(

X

>

x

)

=

2

,

X

1

,

X

2

,

X

∼

D

,

X

1

,

X

2

independent.

{\displaystyle \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.