A necessary and sufficient condition for convergence in law of random sums of random variables under nonrandom centering

Abstract:

Let $\{X_n\}$ be a sequence of independent, identically distributed (i.i.d.) random variables with common mean $\mu \ne 0$ and variance $\sigma ^2 > 0$. Let $\{S_n\}$ be a sequence of nonnegative integer-valued random variables such that for each $n$ the random variables $S_n$, $X_1$, $X_2$, … are independent. Then $(X_1 + \dots + X_{S_n} - n \mu ) / \sqrt {n\sigma ^2} \stackrel {\mathcal {L}}{\rightarrow } {}$ (some) $Z$ if and only if $(S_n - n)/\sqrt {n} \stackrel {\mathcal {L}}{\rightarrow } {}$ (some) $U$, in which case the distribution of $Z$ is that of $X+Y$, where $X$ and $Y$ are independent random variables, $X$ being $\mathcal {N}(0, 1)$ and $Y$ having the same distribution as $\mu U/\sigma$.