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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 $k \to \infty$?

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    $\begingroup$ Yes: the distribution of $(S_k - \mu N_k) / (\sigma \sqrt{N_k})$ is a mixture of the distributions of standardised sums $Y_k = (\sum_{j=1}^k X_j - k\mu) / (\sigma \sqrt{k})$. Since the distribution of $Y_k$ converges to standard normal, the same is true for the mixture. $\endgroup$ Commented Jan 21, 2020 at 10:46
  • $\begingroup$ Otherwise you can redo the CLT proof with the characteristic function $\mathbb{E}(e^{i\alpha (S_k-\mu N_k)/\sigma\sqrt{N_k}})=\mathbb{E}(\mathbb{E}(e^{i\alpha (X-\mu )/\sigma\sqrt{N_k}})^{N_k})\approx \mathbb{E}((1-\frac{\alpha^2}{2N_k})^{N_k})\approx e^{-\alpha^2/2}$. $\endgroup$
    – RaphaelB4
    Commented Jan 21, 2020 at 13:54

1 Answer 1

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$\newcommand{\ep}{\varepsilon} \newcommand{\Si}{\Sigma}$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.

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  • $\begingroup$ The answers are correct! $\endgroup$ Commented Jan 21, 2020 at 19:14

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