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.