# Interchanging limits for doubly indexed random sequences

I've encountered the following problem which seems to be quite standard but for which I can't find any proper references (asking on mathematics SE didn't bring up any answers so I'm reposting my question here):

Suppose that for $n \in \mathbb{N}$ we have a sum of random variables $S(n)= \sum_{r=1}^{\infty} X_r(n)$, such that $X_r(n)$ and $X_s(n)$ are uncorrelated for $r \neq s$ and $S(n)$ is uniformly bounded in $L^2$. Denote by $S_k(n)$ the partial sums $\sum_{r=1}^k X_r(n)$. If we know that

1. for each $k$, $$S_k(n) \xrightarrow{d} N(\mu_k,\sigma^2_k), \qquad (n \to \infty),$$ where $\xrightarrow{d}$ is convergence is distribution and $N(\mu_k,\sigma^2_k)$ denotes a Gaussian with mean $\mu_k$ and variance $\sigma^2_k$

and

1. the sequences $(\mu_k)$ and $(\sigma^2_k)$ both converge to some limits $\mu$ and $\sigma^2$ (so that in particular $N(\mu_k,\sigma^2_k) \xrightarrow{d} N(\mu,\sigma^2)$,

what are sufficient conditions to conclude that $S(n) \xrightarrow{d} N(\mu,\sigma^2)$ as $n\to \infty$? In other words, with some abuse of notation, i.e. writing $\lim$ for "limit in distribution", I'm looking for conditions which imply that $$\lim_{n \to \infty} \lim_{k \to \infty} S_k(n) = \lim_{k \to \infty} \lim_{n \to \infty} S_k(n).$$ What about the more general situation where the partial sums are replaced by some doubly indexed random variables and the limiting Gaussians with some generic (continuous) random variable?

• I must be missing something. $S_k(n)$ converges with probability 1 to $S(n)$ as $k\to\infty$ (since an infinite sum is by definition the limit of the partial sums). So if $S_k(n)$ converges in distribution to $N(\mu_n,\sigma^2_n)$, then $S(n)$ has distribution $N(\mu_n, \sigma^2_n)$ (since convergence with probability 1 implies convergence in distribution). But then assumption 2 gives you that the distribution of $S_n$ converges to $N(\mu, \sigma^2)$. So what is the issue? – James Martin Apr 29 '15 at 18:06
• @JamesMartin I'm very sorry, actually I interchanged the roles of $n$ and $k$ in my first writeup of the question. I have corrected the mistake and also added some clarification. – r_faszanatas Apr 30 '15 at 13:27

Theorem. For each integers $$m$$ and $$n$$, let $$X_n$$, $$X_n^{(m)}$$ and $$X^{(m)}$$ be real-valued random variables such that
• for each integer $$m$$, $$X_n^{(m)}\to X^{(m)}$$ in distribution, and
• for each positive $$\varepsilon$$, $$\lim_{m\to\infty}\limsup_{n\to\infty}\mathbb P\{|X_n-X_n^{(m)}| >\varepsilon \}=0$$.
Then there exists a real valued random variable $$X$$ such that $$X_n\to X$$ in distribution.
Applying this in your setting, you have to check that the convergence $$\lim_{k \to \infty}\limsup_{n\to \infty}\mathbb P\left\{\left|\sum_{i=k +1}^{ +\infty}X_i(n) \right|\gt \varepsilon\right\} =0$$ takes place for each positive $$\varepsilon$$. If $$X_i(n)$$ and $$X_j(n)$$ are uncorrelated for each $$n$$, this would hold if $$\lim_{k \to \infty}\limsup_{n\to \infty} \sum_{i\geqslant k}\mathrm{Var}(X_i(n))=0.$$