Law of Large Numbers for geometrically decaying sequences

Question

Let $(X_n)_n$ be a sequence of i.i.d. random variables, and let $\rho \in (0,1)$. Is there any asymptotic Theorem for the following random variable:

$$ Y = \lim_{n\rightarrow \infty}\sum_{i=0}^n \rho^iX_i $$ in the fashion of a Law of large numbers? My conjecture is that $Y \rightarrow^{a.s.} \frac{\mathbb{E}[X]}{1-\rho}$, but I am not able to find a proof nor a reference about it.

Answer 1

To see why this is not possible, let $\psi_Y$ be the cumulant generating function for $Y$. It is defined for any real number $t$ by

$$\psi_Y(t) = \log E\left[e^{itY}\right].$$

If the sum in the question converges to a random variable $Y,$ then taking $X$ to be distributed like all the $X_i$ and independent of them we see

$$Y \sim \lim_{n\to\infty} X + \rho X_0 + \rho^2 X_1 + \cdots = X + \lim_{n\to\infty} \rho\left(X_0 + \rho X_1 + \cdots\right) = X + \rho Y .$$

This entails

$$\psi_Y(t)=\psi_X(t) + \psi_Y( \rho t ) $$

for all $t.$ We may recover $\psi_X$ from the distribution $Y$ via

$$\psi_X(t) = \psi_Y(t) - \psi_Y(\rho t)$$

and, since the c.g.f $\psi_X$ determines the distribution of $X,$ we can reverse the process: $Y$ determines the common distribution of the $X_i.$

Consequently there can be no analog of the laws of large numbers, which posit convergence to universal limiting distributions (under mild assumptions about the $X_i$), effectively wiping out almost all information about the underlying distribution used to arrive at the limit.