In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, one definition of a subexponential distribution is as a
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
whose tails decay at an exponential rate, or faster: a real-valued distribution
is called subexponential if, for a random variable
,
:
, for large
and some constant
.
The subexponential norm,
, of a random variable is defined by
:
where the infimum is taken to be
if no such
exists.
This is an example of a
Orlicz norm. An equivalent condition for a distribution
to be subexponential is then that
Subexponentiality can also be expressed in the following equivalent ways:
#
for all
and some constant
.
#
for all
and some constant
.
# For some constant
,
for all
.
#
exists and for some constant
,
for all
.
#
is
sub-Gaussian.
References
* ''High-Dimensional Statistics: A Non-Asymptotic Viewpoint'', Martin J. Wainwright, Cambridge University Press, 2019, .
Continuous distributions
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