Asymptotic behavior of the empirical mean of weakly convergent processes.

by Alfred F.   Last Updated August 13, 2018 12:20 PM - source

Let $M,n \in \mathbb{N}$. For $i \in \{0,\dots,M\}$, let $(X_{n,i})_{n\geq 1}$ be $M$ independent sequences of real valued random variables, each weakly converging to a standard gaussian, i.e. $X_{n,i} \rightarrow \mathcal{N}(0,1)$ in distribution as $n \rightarrow \infty$ for all $1\leq i \leq M$.

For fixed $M$, we know that $\frac{1}{\sqrt{M}}\sum_{i=1}^M X_{n,i} \rightarrow \mathcal{N}(0,1)$ in distribution as $n \rightarrow \infty$. Under what condition on $M$ does this still hold when $M\sim a_n$ where $(a_n)_{n\geq 1}$ is a sequence of real numbers converging to infinity as $n\rightarrow \infty$ ? I am particularly interested in the case $M \sim \sqrt{n}$.

Thanks in advance !

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