Let $X_1, X_2,\ldots, X_n \in \mathcal{R}^d$ and be zero-mean, unit variance random variables. Here the dimension ($d$) is a function of the sample size($n$) i.e, $d=f(n)$. For example $d = \sqrt{n}$.

1. When can we apply vectorized central-limit-theorem to the following sum?

$$ S_n=\frac{X_1+X_2+\cdots+X_n}{\sqrt{n}} $$

2. Where can one find a comprehensive literature for above?

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    $\begingroup$ To have a guess, MVN means all linear combinations are normal. This could be a way to approach the "finite variance" needed for CLT to work. Basically you're trying to show that $\sqrt {n}t^T\overline {X}_n$ is quadratic in $t $ as $n\to\infty $. For some functions $d=f (n) $ this won't work - my guess is anything which grows faster than $\sqrt {n} $ will have infinite variance (unless you have strong negative correlation between component of $X_i $) $\endgroup$ – probabilityislogic Apr 25 '16 at 1:02
  • $\begingroup$ I don't see why $d>\sqrt{n}$ should give infinite variance. The normalization from $t$ should take care of it. $\endgroup$ – Vivek Bagaria Apr 25 '16 at 1:18
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    $\begingroup$ The asymptotic covariance matrix will be $d \times d$, which means it will be $\sqrt{n} \times \sqrt{n}$. But since $n$ is tending to $\infty$ and the CLT is an asymptotic result, I don't see how one can successfully define an infinite dimensional square matrix. It might be a matrix of finite values, but it will have countably infinite entries. $\endgroup$ – Greenparker Apr 25 '16 at 3:10
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    $\begingroup$ Greenparker: An infinite dimensional rv is gaussian iff every finite subset is a gaussian. This helps us deal with infinite matrices. $\endgroup$ – Vivek Bagaria Apr 25 '16 at 9:35
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    $\begingroup$ It's not apparent that the increase in $d$ is any problem at all provided you explain how you intend to add vectors of different dimensions. If you fix $k\ge 1$ and simply ignore all components beyond the $k^\text{th}$ one (and presuming $d(n)$ is a nondecreasing function so that a limit can even be defined), you may apply the usual CLT (assuming it holds for your variables) to demonstrate the first $k$ components of $S$ converge in distribution to an MVN. $\endgroup$ – whuber Apr 25 '16 at 14:31

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