Sum of linear combination of product of exponentials is exponential

Question

This problem has arisen in my research: suppose that $V_i \sim \text{ED}$ are iid exponential distributions (ED) with mean $1$ and let $\lambda$ be a nonnegative number. Is it true that $$ \sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}V_{0} \cdots V_k}{k!} \sim \text{ED}? $$ This passes the sanity check, as the expected value of both sides is equal to $1$, and if we let $\lambda = 0$, then the left hand side equals $V_0$, which is exponential. Other than that, I am not sure how to approach this problem, as I do not know how to deal with the product of ED's.

Answer 1

Not a complete answer, sorry, but a few ideas (to long for a comment). Note that what you have is a product of $K+1$ iid random variables, where $K$ is a random variable (rv) with a poisson distribution with parameter $\lambda$. That can be used for another "sanity check", a simulation (using exponentials of rate 1):

set.seed(7*11*13)
N <- 1000000

prods <- rep(0, N)
ks <- rpois(N, 1)+1

for (i in 1:N) {
    k  <-  ks[i]
    prods[i]  <-  prod( rexp(k, 1))
}

qqplot( qexp(ppoints(N)), prods)

The resulting qqplot (not shown here) is far from a straight line, so this do not look to be an exponential of rate 1. The mean is right, the variance to large, there is a right tail much longer than for an exponential. What can be done theoretically? The Mellin transform https://en.wikipedia.org/wiki/Mellin_transform is adapted to products of independent random variables. I will compute only for the exponential with rate 1. The Mellin transform of $V_0$ then is $$ \DeclareMathOperator{\E}{\mathbb{E}} M_1(s) = \E V_0^s = \int_0^\infty x^s e^{-x}\; dx = \Gamma(s+1) $$ so the Mellin transform of a product of $k+1$ iid exponentials is $$ M_{k+1}(s) = \Gamma(s+1)^{k+1} $$ Since $K$ has a poisson distribution with parameter $\lambda$, the Mellin transform of the random product of a random number $K+1$ factors, is $$ M(s) = \E M_{K+1}(s) = \E \Gamma(s+1)^{K+1}= \Gamma(s+1)e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k}{k!}\Gamma(s+1)^k=e^{-\lambda}\Gamma(s+1) e^{\lambda \Gamma(s+1)} $$ but I cannot find an inverse of this transform. But note that if $X$ is a nonnegative random variable with Mellin transform $M_X(t)$, then defining $Y=\log X$ we find that $$ K_Y(t)=\E e^{tY} = \E e^{t\log X}=\E e^{\log (X^t)}=\E X^t =M_X(y) $$ so the Mellin transform of $X$ is the moment generating function of its logarithm $Y$. So, using that we can approximate the distribution of $X$ with saddlepoint approximation methods, How does saddlepoint approximation work? and search this site.