= Solution
The <strong law of large numbers> states that for independent identically distributed real variables with $\mathbb E|X_1|<\infty$, the sample average converges almost surely to $\mathbb EX_1$. The <central limit theorem> states that if additionally $0<\operatorname{Var}X_1=\sigma^2<\infty$, then
$$
\sqrt n(\overline X_n-\mathbb EX_1)\ \xrightarrow{d}\ N(0,\sigma^2).
$$
The vector form uses finite second moments and gives a centred multivariate Gaussian with the <covariance matrix> of one observation.
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