Multivariate iid central limit theorem

For each fixed $u\in {\mathbb{R}}^d$ set $Y_k=u\cdot (X_k-m)$. These are iid: inverse images under the continuous projection preserve the finite independence identities and the common law. By [F6], $\mathrm{\mid }{Y}_k{\mathrm{\mid }}^2\le \mathrm{\parallel }u{\mathrm{\parallel }}^2\mathrm{\parallel }{X}_k-m{\mathrm{\parallel }}^2$; the latter is integrable since $\mathrm{\parallel }{X}_k-m{\mathrm{\parallel }}^2\le 2\mathrm{\parallel }{X}_k{\mathrm{\parallel }}^2+2\mathrm{\parallel }m{\mathrm{\parallel }}^2$. Each centered coordinate product is integrable by $2\mathrm{\mid }ab\mathrm{\mid }\le a^2+b^2$, and symmetry of these products gives ${\mathrm{\Sigma }}_{ij}={\mathrm{\Sigma }}_{ji}$. Finite linearity gives $\mathbb{E}{Y}_k=0$ and $\mathbb{E}{Y}_k^2=u^T\mathrm{\Sigma }u=:v\ge 0$. Because this holds for every $u$, the symmetric matrix $\mathrm{\Sigma }$ is positive semidefinite. Consequently [F2] supplies the target law $G=N_d(0,\mathrm{\Sigma })$.

If v>0, [F1] gives $(nv{)}^{-1\mathrm{/}2}{\sum }_{k=1}^n{Y}_k\Rightarrow N(0,1)$. Scaling by $\sqrt{v}$ is continuous; [F7] gives $n^{-1\mathrm{/}2}{\sum }_k{Y}_k\Rightarrow N(0,v)$, which is the u-projection of G by [F2]. If v=0, [F4] applied to $Y_k^2$ gives $Y_k=0$ almost surely for each k. For each fixed n the union of the finitely many exceptional null sets is null, so the projected row sum is zero almost surely and has exactly the law N(0,0). Thus the same projection convergence holds without dividing by v.

The preceding convergence holds for every fixed u to the projections of the same specified probability G. [F3] therefore proves the vector conclusion. AC is inherited in the scalar CLT, Gaussian construction and Cramer–Wold theorem. A common null set across all u is neither claimed nor needed. When Sigma=0 all projections are in the zero-variance case. For d=1 this agrees with the scalar result after scaling.