The multivariate method of moments for a determinate limit

Gaussian projections: let $Y\sim N_d(m,\mathrm{\Sigma })$, $u\in {\mathbb{R}}^d$, $c:=u\cdot m$ and $s:=u^{\mathsf{T}}\mathrm{\Sigma }u\ge 0$. By [F6] the projection $u\cdot Y$ is normal with mean $c$ and variance $s$, and by [F7] the standard normal has moments of every order; hence $\mathbb{E}\mathrm{\mid }u\cdot Y{\mathrm{\mid }}^k<\mathrm{\infty }$ for every $k$, and by [F9] and linearity of expectation, $\mathbb{E}[(u\cdot Y{)}^k]={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=k}\left(\genfrac{}{}{0px}{}{k}{\alpha }\right)u^{\alpha }\mathbb{E}[Y^{\alpha }]$, a finite sum of finite mixed moments. Indeed each coordinate has all absolute moments by [F7]; for a multi-index of total degree $q>0$, ${\prod }_i\mathrm{\mid }{Y}_i{\mathrm{\mid }}^{{\alpha }_i}\le {\max }_i\mathrm{\mid }{Y}_i{\mathrm{\mid }}^q\le {\sum }_i\mathrm{\mid }{Y}_i{\mathrm{\mid }}^q$, proving mixed absolute integrability. The degree-zero product is $1$.

Tightness of the sequence: for each coordinate $i$, hypothesis (i) applied to the multi-index with a single $2$ in place of $i$ gives $\mathbb{E}[X_{n,i}^2]\to \int x_i^{2}d\mu$, so $C:={\max }_i{\sup }_n\mathbb{E}[X_{n,i}^2]<\mathrm{\infty }$. Fix $\epsilon >0$ and choose $R>0$ with $dC\mathrm{/}{R}^2<\epsilon$; the cube $K:=[-R,R{]}^d$ is compact by [F10] and ${\mathbb{R}}^d\setminus K$ is contained in the union of the $d$ coordinate slabs $\{\mathrm{\mid }{x}_i\mathrm{\mid }>R\}$, so by [F2] and the union bound of [F10], ${\mu }_n({\mathbb{R}}^d\setminus K)\le {\sum }_{i=1}^d{\mu }_n(\{\mathrm{\mid }{x}_i\mathrm{\mid }>R\})\le {\sum }_{i=1}^d\mathbb{E}[X_{n,i}^2]\mathrm{/}{R}^2\le dC\mathrm{/}{R}^2<\epsilon$ for every $n$. Hence $\{{\mu }_n\}$ is tight.

Matching of projection moments: let $\nu$ be a Borel probability on ${\mathbb{R}}^d$ with the same mixed moments as $Y$, i.e. $\int w^{\alpha }d\nu =\mathbb{E}[Y^{\alpha }]$ for every multi-index $\alpha$, and let $W\sim \nu$. Then for every $u\in {\mathbb{R}}^d$ and every $k\ge 0$ the multinomial expansion [F9] gives $\mathbb{E}[(u\cdot W{)}^k]={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=k}\left(\genfrac{}{}{0px}{}{k}{\alpha }\right)u^{\alpha }\int w^{\alpha }d\nu ={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=k}\left(\genfrac{}{}{0px}{}{k}{\alpha }\right)u^{\alpha }\mathbb{E}[Y^{\alpha }]=\mathbb{E}[(u\cdot Y{)}^k]$ by step 1.1; moreover $\mathbb{E}[(u\cdot W{)}^{2k}]=\mathbb{E}[(u\cdot Y{)}^{2k}]<\mathrm{\infty }$, so [F5] gives $\mathbb{E}\mathrm{\mid }u\cdot W{\mathrm{\mid }}^k\le (\mathbb{E}[(u\cdot W{)}^{2k}]{)}^{1\mathrm{/}2}<\mathrm{\infty }$. Thus all moments of the projection $u\cdot W$ are finite and equal those of $u\cdot Y$.

Extraction along any subsequence: let $({\mu }_{n_k})$ be an arbitrary subsequence of $({\mu }_n)$. By step 1.2 the subfamily $\{{\mu }_{n_k}\}$ is tight as well, so by [F1], applicable to the Polish space verified in [F10], it has a further subsequence ${\mu }_{n_{k_j}}$ converging weakly to some Borel probability ${\mu }^{\mathrm{\prime }\mathrm{\prime }}$ on ${\mathbb{R}}^d$; by [F3] there are random elements $Y_j,Y$ on a common probability space with laws ${\mu }_{n_{k_j}},{\mu }^{\mathrm{\prime }\mathrm{\prime }}$ and $Y_j\to Y$ almost surely.

Projections determine the Gaussian: keep the notation of steps 1.1 and 2.1 with $W\sim \nu$ and fix $u\in {\mathbb{R}}^d$. If $s=0$ then, by step 2.1 with $k=1,2$, $\mathbb{E}[u\cdot W]=c$ and $\mathbb{E}[(u\cdot W{)}^2]=\mathbb{E}[(u\cdot Y{)}^2]=s+c^2=c^2$, so $\mathrm{Var}(u\cdot W)=\mathbb{E}[(u\cdot W-c{)}^2]=0$ and $u\cdot W=c$ almost surely: by [F2], $\mathbb{P}(\mathrm{\mid }u\cdot W-c\mathrm{\mid }>1\mathrm{/}l)=0$ for every integer $l\ge 1$, and their countable union is the event $u\cdot W\mathrm{\ne }c$, of probability zero by [F10]. This is the law of $u\cdot Y$. If $s>0$, put $Z^{\mathrm{\prime }}:=(u\cdot W-c)\mathrm{/}\sqrt{s}$; by step 2.1 its moments satisfy $\mathbb{E}[(Z^{\mathrm{\prime }}{)}^k]=s^{-k\mathrm{/}2}{\sum }_{j=0}^k\left(\genfrac{}{}{0px}{}{k}{j}\right)(-c{)}^{k-j}\mathbb{E}[(u\cdot W{)}^j]=s^{-k\mathrm{/}2}{\sum }_{j=0}^k\left(\genfrac{}{}{0px}{}{k}{j}\right)(-c{)}^{k-j}\mathbb{E}[(u\cdot Y{)}^j]=\mathbb{E}[{\zeta }^k]$ for $\zeta \sim N(0,1)$, because $(u\cdot Y-c)\mathrm{/}\sqrt{s}\sim N(0,1)$ by [F6]; and $\mathbb{E}\mathrm{\mid }{Z}^{\mathrm{\prime }}{\mathrm{\mid }}^k<\mathrm{\infty }$ by step 2.1. [F8] therefore gives $Z^{\mathrm{\prime }}\sim N(0,1)$, so $u\cdot W\sim N(c,s)$, again the law of $u\cdot Y$.

Uniform integrability along the coupling: fix a multi-index $\alpha$ and let $f_j:={\prod }_i{Y}_{j,i}^{{\alpha }_i}$, so $f_j\to {\prod }_i{Y}_i^{{\alpha }_i}$ almost surely by step 2.2. By hypothesis (i) applied to the multi-index $2\alpha$, $\mathbb{E}[f_j^2]=\mathbb{E}[{\prod }_i{X}_{n_{k_j},i}^{2{\alpha }_i}]\to \int {\prod }_i{x}_i^{2{\alpha }_i}d\mu$, so ${\sup }_j\mathbb{E}{f}_j^2<\mathrm{\infty }$; hence ${\int }_{\{\mathrm{\mid }{f}_j\mathrm{\mid }>M\}}\mathrm{\mid }{f}_j\mathrm{\mid }\le M^{-1}{\sup }_j\mathbb{E}{f}_j^2$ tends to $0$ uniformly in $j$ as $M\to \mathrm{\infty }$, and the family $\{f_j\}$ is uniformly integrable by [F4].

Gaussian determinacy: if $\nu$ has the same mixed moments as $Y\sim N_d(m,\mathrm{\Sigma })$, then by step 3.1 every projection $u\cdot W$ of $W\sim \nu$ has the law of the projection $u\cdot Y$; by the Cramér-Wold clause of [F6] the laws of $W$ and $Y$ coincide, so $\nu =N_d(m,\mathrm{\Sigma })$.

Identification of the limit: with the notation of step 3.2, almost sure convergence implies convergence in measure on the finite measure space by [F4]; together with the uniform integrability of step 3.2, Vitali's theorem [F4] gives $\mathbb{E}[{\prod }_i{Y}_{j,i}^{{\alpha }_i}]\to \mathbb{E}[{\prod }_i{Y}_i^{{\alpha }_i}]$ and shows the limit is finite. The left-hand side equals $\mathbb{E}[{\prod }_i{X}_{n_{k_j},i}^{{\alpha }_i}]$, which tends to $\int {\prod }_i{x}_i^{{\alpha }_i}d\mu$ by hypothesis (i); hence $\int w^{\alpha }d{\mu }^{\mathrm{\prime }\mathrm{\prime }}=\int w^{\alpha }d\mu$ for every multi-index $\alpha$, ${\mu }^{\mathrm{\prime }\mathrm{\prime }}$ has finite mixed moments, and hypothesis (ii) gives ${\mu }^{\mathrm{\prime }\mathrm{\prime }}=\mu$.

Convergence of the full sequence: let $({\mu }_{n_k})$ be an arbitrary subsequence of $({\mu }_n)$. Steps 2.2, 3.2 and 4.2 applied to it produce a further subsequence converging weakly to $\mu$. Hence ${\mu }_n\Rightarrow \mu$: otherwise there are a bounded continuous real function $f$ on ${\mathbb{R}}^d$, a real $\epsilon >0$ and a subsequence with $\mid \int fd{\mu }_{n_k}-\int fd\mu \mid \ge \epsilon$ for all $k$ (Weak convergence of borel probability measures), yet that subsequence has a further subsequence converging weakly to $\mu$, along which $\int fd{\mu }_{n_{k_j}}\to \int fd\mu$, a contradiction.

Conclusion: step 5.1 proves the convergence assertion from hypotheses (i) and (ii), and steps 1.1, 2.1, 3.1 and 4.1 prove that every multivariate Gaussian law is moment-determinate. AC was used exactly through the Polish-space existence theorems of [F1], [F3] and the determinacy lemma [F8].