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The univariate version of the Lyapunov CLT (a version of the CLT for independent but not necessarily iid random variables) is as follows according to Wikipedia:

Lyapunov CLT from https://en.wikipedia.org/wiki/Central_limit_theorem#Multidimensional_CLT

Is there a multivariate version of the Lyapunov CLT (ideally one that doesn't require the means of the random variables to all be zero vectors)? I'm having a very hard time finding any information on this, even on Google Scholar.

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  • $\begingroup$ Your last paragraph is a little puzzling, because it suggests you are interested in a sequence of random matrices--that, after all, is what "means ... be[ing] zero matrices" implies. Indeed, even in the univariate case what version of the CLT requires all variables to have zero means?? $\endgroup$
    – whuber ♦
    Commented May 18, 2022 at 21:55
  • $\begingroup$ Sorry I meant zero vectors. I'll fix it. $\endgroup$
    – bob
    Commented May 18, 2022 at 22:23
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    $\begingroup$ I guess you are searching for this theorem: en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem See the Multidimensional version. $\endgroup$
    – Jiyuan Yang
    Commented Jan 8 at 8:53
  • $\begingroup$ I get a lot of results when I search for multivariate Lindeberg CLT. The univariate Lindberg condition is implied by the Lyapunov condition. $\endgroup$
    – Taylor
    Commented Jan 8 at 14:22

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It seems to me that the multivariate version may follow from the Cramér-Wold theorem. You have to check that the conditions stated in your question are satisfied on every possible linear combination of components of $\mathbf X_i$, so this seems to require suitable behaviour of the cumulative variances $s_n(\mathbf a)$ defined as $s_{n}(\mathbf a) = \sum_{i=1}^n \mathrm{Var} (\mathbf a^T \mathbf X_i)$.

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