A randomized central limit theorem

Question

Let $X_k$, $k = 1, 2, \dots$, be a sequence of i.i.d. random variables with finite second moments. Also, let $N_k \geq 1$, $k = 1, 2, \dots$, be a sequence of random variables taking integral values, such that $\lim_k N_k = \infty$ a.s.. Furthermore, assume that each $N_k$ is independent of the $X_k$'s.

If $S_k := \sum_1^{N_k} X_k$, does it follow that $(S_k - \mu N_k)/\sigma\sqrt{N_k}$ converges in distribution to the standard normal variable (where $\mu = \mathbb{E}[X_k]$ and $\sigma^2 = \mathbb{V}[X_k]$) as $k \to \infty$?

Answer 1

$\newcommand{\ep}{\varepsilon} \newcommand{\Si}{\Sigma}$The answer is yes, and it is enough that $N_k\to\infty$ just in probability or, equivalently, in distribution (rather than almost surely), which means that for each real $b$ $$P(N_k\le b)\to0$$ as $k\to\infty$.

Indeed, let $F_n$ be the cdf of $\sum_1^n(X_i-\mu)/(\sigma\sqrt n)$ and let $G_k$ be the cdf of $S_k$. Take any real $x$. By the central limit theorem, $F_n(x)\to\Phi(x)$ as $n\to\infty$, where $\Phi$ is the standard normal cdf. Take now any real $\ep>0$. Then there is some natural $A$ such that $|F_n(x)-\Phi(x)|<\ep/2$ for all $n>A$. Further, there is some natural $K$ such that $P(N_k\le A)<\ep/2$ for all $k>K$. Take now any such $A$ and $K$, and then take any natural $k>K$. Write
$$G_k(x)-\Phi(x)=\sum_{n=1}^\infty P(N_k=n)F_n(x)-\Phi(x)=\Si_1+\Si_2, $$ where $$|\Si_1|=\Big|\sum_{n=1}^A P(N_k=n)(F_n(x)-\Phi(x))\Big| \le\sum_{n=1}^A P(N_k=n)=P(N_k\le A)<\ep/2,$$ $$|\Si_2|=\Big|\sum_{n>A} P(N_k=n)(F_n(x)-\Phi(x))\Big| \le\sum_{n>A} P(N_k=n)\ep/2\le\ep/2,$$ whence $$|G_k(x)-\Phi(x)|<\ep$$ for all $k>K$. That is, $G_k(x)\to\Phi(x)$ as $k\to\infty$, as desired.