Sum of squared Poisson probability masses

Question

Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Poisson distribution with parameter $\lambda$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2,$$ as a function of $\lambda$. In other words I am interested in (the exponential of) the second-order Renyi entropy of a Poisson distribution.

Background:

• I'd like to use this to evaluate Brier and spherical scores.

• Czado et al. (2009) write that this expression can be evaluated analytically, but don't give further information, and I'm kind of stuck.

• No, this is not homework, although I imagine it could be ;-) Any hints or pointers to literature would be almost as much appreciated as a full solution.

• Here is the analogous question for the Negative Binomial distribution.

Answer 1

Don't hesitate to use WolframAlpha to get the sum of a series. Or do you need a mathematical proof ?

This gives $\exp(-2\lambda)I_0(2\lambda)$. The link to the documentation of the Bessel function $I_0$ is this one.

Actually the proof here would just mean the series representation of $I_0$.

If you want to use R to evaluate this Bessel function, you can do it with the help of the gsl package:

> library(gsl)
> lambda <- 1
> exp(-2*lambda)*bessel_I0(2*lambda)
[1] 0.3085083
> sum(dpois(0:100, lambda)^2)
[1] 0.3085083