- Source: 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: Definition 1.1
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.