Casella and Lerner's Theory of Point Estimation (2nd edition) provides a definition of the multivariate Central Limit Theorem, for which no proof is given.

Theorem 8.21 (Multivariate CLT) Let $\mathbf{X}_\nu = (X_{1\nu}, \dots, X_{r \nu}$) be iid with mean vector $\zeta = (\zeta_1, \dots, \zeta_r)$ and covariance matrix $\Sigma = \vert \vert \sigma_{ij} \vert \vert$, and let $\overline{X}_{in} = (X_{i1} + \dots + X_{in})/n$. Then, $$[ \sqrt{n} (\overline{X}_{1n} - \zeta_1), \dots, \sqrt{n} (\overline{X}_{rn} - \zeta_r)]$$ tends in law to the multivariate normal distribution with mean vector $\mathbf{0}$ and covariance matrix $\Sigma$.

What would be its derivation?

Casella and Lerner's Theory of Point Estimation (2nd edition) provides a definition of the multivariate Central Limit Theorem, for which no proof is given.

Theorem 8.21 (Multivariate CLT) Let $\mathbf{X}_\nu = (X_{1\nu}, \dots, X_{r \nu}$) be iid with mean vector $\zeta = (\zeta_1, \dots, \zeta_r)$ and covariance matrix $\Sigma = \vert \vert \sigma_{ij} \vert \vert$, and let $\overline{X}_{in} = (X_{i1} + \dots + X_{in})/n$. Then, $$[ \sqrt{n} (\overline{X}_{1n} - \zeta_1), \dots, \sqrt{n} (\overline{X}_{rn} - \zeta_r)]$$ tends in law to the multivariate normal distribution with mean vector $\mathbf{0}$ and covariance matrix $\Sigma$.

What would be its derivation?

Casella and Lerner's "Theory of Point Estimation" (2nd edition) provides a definition of the multivariate Central Limit Theorem, for which no proof is given.



What would be its derivation?

Casella and Lerner's Theory of Point Estimation (2nd edition) provides a definition of the multivariate Central Limit Theorem, for which no proof is given.

Theorem 8.21 (Multivariate CLT) Let $\mathbf{X}_\nu = (X_{1\nu}, \dots, X_{r \nu}$) be iid with mean vector $\zeta = (\zeta_1, \dots, \zeta_r)$ and covariance matrix $\Sigma = \vert \vert \sigma_{ij} \vert \vert$, and let $\overline{X}_{in} = (X_{i1} + \dots + X_{in})/n$. Then, $$[ \sqrt{n} (\overline{X}_{1n} - \zeta_1), \dots, \sqrt{n} (\overline{X}_{rn} - \zeta_r)]$$ tends in law to the multivariate normal distribution with mean vector $\mathbf{0}$ and covariance matrix $\Sigma$.

What would be its derivation?