# On an infinite linear combination of chi-squared random variables

Question

Let $$Z_i\sim\chi_{(1)}^2$$ be i.i.d. chi-squared random variables with 1 degree of freedom. We define: $$W_{\infty} = \sum_{k = 1}^{\infty} \frac{Z_k}{2^{k}}$$ I have interest in computing the distribution of this random variable and more specifically the probability $$P(W_{\infty} > 1$$).

References

This problem has some references, in particular from this paper by Mathai:

https://www.ism.ac.jp/editsec/aism/pdf/034_3_0591.pdf

Reference [4]:

In my case, following his notations $$\alpha = \frac{1}{2}$$, $$X_i\sim\Gamma(\frac{1}{2},\frac{1}{2})$$ are i.i.d. and $$n\to\infty$$. The main problem is that I didn't manage to find Prabhu's work about this fact, and in general I didn't manage to find any other references to this problem.

My attempt

I was trying to approach this problem using the Levy-criteria for the convergence in distribution, since the variables are i.i.d. and so the characteristic function of $$W_\infty$$ is "simply" an infinite product: $$\phi_{W_{\infty}}(t) = \prod_{k = 0}^{\infty}\frac{1}{\sqrt{1-\frac{it}{2^k}}}$$ On the other hand I know that for every $$n\in\mathbb{N}$$, $$W_n = \sum_{k=1}^{n}\frac{Z_k}{2^k}$$ is distributed as a generalized chi-squared distribution of a very particular case, and maybe this could help in finding a closed-expression for the distribution of its limit.

https://en.wikipedia.org/wiki/Generalized_chi-squared_distribution

Code and simulation

I'm sure that $$W_n$$ converges in distribution by simulating it using R, in particular the following code:

it = 10000
n = 300
mat = matrix(rep(0,n*it),n,it)
w = vector()
for(i in 1:n) {
mat[i,] = rchisq(it,1)
}
for(i in 1:n) {
mat[i,] = mat[i,]/2^i
}
for(j in 1:it) {
w[j] = sum(mat[,j])
}
hist(w,freq = F,breaks = seq(0,10,0.1))


produces the following output: $W_{300}$" />

which is the distribution of $$W_{300}$$. I don't really know if this is a well known distribution, and I don't really know if its density function has a closed form. I don't know how to proceed. Thank you in advance for your help!

New approximation of $$P(W_{\infty} > 1)$$

Using the software R and the library "CompQuadForm":

I manage to approximate the value of $$P(W_{\infty} > 1)$$ with the code:
library(CompQuadForm)
approx = davies(q, lambda, h = rep(1, length(lambda)), delta = rep(0,length(lambda)), sigma = 0, lim = 500000, acc = acc1)$Qq sprintf("%.30f",approx)  which gives: $$P(W_{\infty} > 1) \approx 0.371741079532780016592141691945$$ which I think are the real first 30 digits. (I'm sure for the first 15 since I set an accuracy of $$10^{-15}$$). Of course I would prefer an "analytic" form of this number.. • It is easy enough to calculate many moments, then you can use stats.stackexchange.com/questions/141652/… Commented Nov 30, 2023 at 20:55 • Thank you for your reply! Although as I pointed out at the end of the question, I would prefer an exact solution, not an approximation. To obtain an exact solution using this strategy I should compute a general form for the$k$-th moment, which I've already tried and it's very difficult. Commented Dec 1, 2023 at 6:43 •$Z_i\sim{}Gamma(1/2,1/2)$. A non-closed form PDF of$n\$ weighted gamma random variables is given in Eq. (2) here. Commented Dec 2, 2023 at 14:25