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.
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