Repeated convolution of probability distributions

Question

Question

Let $$S_k=\sum_{i=1}^k X_i$$ be the sum of $k$ independent random variables. I am interested in closed-form expressions of the pdf of $S_k$.

In general, the pdf is given by the $k$-fold convolution of the individual probability distributions such that $$f_S=(f_1*\ldots*f_k)(s),$$ where $f_i$ is the probability distribution of the $i^{\rm th}$ random variable.

In particular, $S_k$ is gamma-distributed if the individual random variables are gamma-distributed with the same scale parameter.

What other distributions exist that satisfy the same property but have at least two parameters?

Motivation

I am performing simulations involving a sum of random variables. Performing the convolutions numerically is too expensive computationally, which is why I would like to find out more about distributions that are "self-repeating" under convolution.

Some thoughts

Formally, we can define the characteristic function $\hat{f}(k;{\bf a})$ of a probability distribution $f(x;{\bf a})$, where $\bf a$ is a set of parameters characterising the distribution.

By the convolution theorem, the product of the characteristic functions corresponds to the convolution of the probability distributions.

The family of distributions I am interested in thus satisfies $$\hat{f}(k;{\bf a})\times \hat{f}(k;{\bf b}) = \hat{f}(k;g({\bf a},{\bf b})),$$ where $g$ is an arbitrary function that is symmetric with respect to exchange of $\bf a$ and $\bf b$.

Unfortunately, this formalism has not helped me come up with an appropriate family of distributions.

Answer 1

You might want to look into infinite divisibility. Many probability distributions are infinitely divisible, which is easy to recognize from their characteristic function. Among them, the stable distributions have two parameters, scale and location.

Examples are normal distribution, Cauchy distribution, Lévy distribution, and $\alpha$-stable distributions.

Answer 2

There are many distributions that fit the bill. These are families of infinitely divisible distributions, closed under convolution. Examples:

• normal distribution
• gamma distribution
• Poisson distribution
• inverse Gaussian distribution
• normal inverse Gaussian
• Levy alpha-stable distribution
• Levy distribution

to name a few.

Answer 3

Two other answers mention infinite divisibility, but that's not needed. The list given in Sasha's and Memming's answers are good as far as they go, but we can add some distributions that are not infinitely divisible.

The family of binomial distributions is closed under convolution of probability mass functions, and is not infinitely divisible. If $X\sim\operatorname{Bin}(n,p)$ and $Y\sim\operatorname{Bin}(m,p)$ and these are independent, then $X+Y\sim\operatorname{Bin}(n+m,p)$.

The negative binomial distributions are also closed under convolution. Here we need to attend to an issue of conventions. To say $X$ is negative-binomially distributed with parameters $n$, $p$ could mean

• by one convention, that $X$ is the number of independent Bernoulli trials needed to get $n$ successes, with probability $p$ of success on each trial; or
• by another convention, that $X$ is the number of failures before the $n$th success in independent Bernoulli trials with probability $p$ of success on each trial.

By the first convention, the distribution is supported on the set $\{n,n+1,n+2,\ldots\}$.

By the second convention, the distribution is supported on the set $\{0,1,2,\ldots\}$.

By either convention, it is closed under convolution.

By the second convention, it is infinitely divisible; by the first, it is not.

The second convention seems to be currently favored in Wikipedia's article about this: http://en.wikipedia.org/wiki/Negative_binomial_distribution