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I am told that the MV CLT can be proved using the Cramér–Wold device.

The theorem is as follows (from Flury's "A First Course in Multivariate Statistics") Suppose $\bf{X}_1, \bf{X}_2, \ldots$, $\bf{X}_n$ are independent, identically, distributed, p-variate random vector, with mean vectors $\bf{µ}=E[\bf{X}_i]$ and covariance matrices $\sigma = Cov[\bf{X}_i]$. Let $\bf{\overline{X}_N} = \frac{1}{N}\sum_{i=1}^n\bf{X}_i$. Then $\sqrt{N}(\bf{\overline{X}_N}-\bf{\mu}) \overset{d}{\to} N(0, \sigma))$.

Cramér–Wold says I can show the above result by showing

$$a^t\sqrt{N}(\bf{\overline{X}_N}-\bf{\mu}) \overset{d}{\to} a^t N(0,\sigma) = N(0, a\sigma a^t) \forall a \in \mathbb{R}^p$$

Through properties of expectation and variance I have shown

$$E\left(a^t\sqrt{N}(\bf{\overline{X}_N}-\bf{\mu})\right) = 0$$ and $$Var\left(a^t\sqrt{N}(\bf{\overline{X}_N}-\bf{\mu})\right) = a\sigma a^t$$

So whatever $a^t\sqrt{N}(\bf{\overline{X}_N}-\bf{\mu})$ converges to, it must have the properties I want. How can I guarantee it is a univariate normal?

It occurs to me that it might be useful to write

$$a^t\sqrt{N}(\bf{\overline{X}_N}-\bf{\mu}) = \sum_{i=1}^p\left(a_i \sqrt{n} (\overline{x}_{ni} - \mu_i)\right)$$

Then by the univariate central limit theorem, each term converges in distribution to some normal. Can we then say that their sum converges in distribution to some normal (we already know it will have the mean and variance we are looking for)? If not, what else shouldI be looking at?

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