Draft:Sub-exponential distribution

  • Comment: This article would need more references to establish notability - mention in text or reference books for example. It also needs to establish why this is more notable than
    the sub-exponential distribution in heavy-tailed distribution mentioned at the top. It needs an indication of which statements in the article are supported by the list of references in the References section. Newystats (talk) 02:48, 21 March 2023 (UTC)

This article generalize the space of random variables generated by exponential Orlicz function, which in turn can be regarded as generalization of space for random variables. The sub-exponential distribution discussed here is heavily related to sub-Gaussian distribution. Do not be confused with the sub-exponential distribution in heavy-tailed distribution.

In probability theory, a sub-exponential distribution is a probability distribution with exponential tail decay. Informally, the tails of a sub-exponential distribution decay at a rate similar to those of the tails of a exponential random variable. This property gives sub-exponential distributions their name.

Formally, the probability distribution of a random variable is called sub-exponential if there are positive constant C such that for every ,

.

The sub-exponential distribution is heavily related to sub-Gaussian distribution. In fact, the square of a sub-exponential is sub-Gaussian [1], which has an even stronger tail decay.

Sub-Exponential properties edit

Let   be a random variable. The following conditions are equivalent:

  1.   for all  , where   is a positive constant;
  2.  , where   is a positive constant;
  3.   for all  , where   is a positive constant.

Proof.   By the layer cake representation,

 
After a change of variables  , we find that
 

  Using the Taylor series for  :

 
and monotone convergence theorem, we obtain that
 
which is less than or equal to   for  . Take  , then
 

  By Markov's inequality,

 

Definitions edit

A random variable   is called a sub-exponential random variable if either one of the equivalent conditions above holds.

The sub-exponential norm of  , denoted as  , is defined by

 
which is the Orlicz norm of   generated by the Orlicz function   By condition   above, sub-exponential random variables can be characterized as those random variables with finite sub-exponential norm.

Relation with sub-Gaussian distributions edit

A random variable   is sub-Gaussian if and only if  is sub-exponential. Moreover,  .

Proof. This follows easily from the characterization of the random variables by the sub-exponential norm and sub-Gaussian norm. Indeed, by definition,

 
Hence, we find that  . Therefore, one of the norm is finite if and only if another one is. This shows that   is sub-Gaussian if and only if  is sub-exponential.

More equivalent definitions edit

The following properties are equivalent:

  • The distribution of   is sub-exponential.
  • Laplace transform condition: for some  ,   holds for all  .
  • Moment condition: for some  ,   for all  .
  • Moment generating function condition: for some  ,   for all   such that  .[2]

Example edit

If   has exponential distribution with rate  , i.e.  , then

 
Then   is sub-exponential since it satisfy condition   with  .

See also edit

Notes edit

  1. ^ Vershynin, R. (2018). High-dimensional probability: An introduction with applications in data science. Cambridge: Cambridge University Press. pp. 35–36.
  2. ^ Vershynin, R. (2018). High-dimensional probability: An introduction with applications in data science. Cambridge: Cambridge University Press. pp. 33–34.

References edit