The iid multivariate-normal sample and covariance conventions

Assume AC. Fix integers $n\ge 2$ and $p\ge 1$. Let ${\mathcal{S}}_{++}^p$ be the set of real symmetric $p\times p$ matrices $\mathrm{\Sigma }$ such that $v^T\mathrm{\Sigma }v>0$ for every nonzero $v\in {\mathbb{R}}^p$. The iid multivariate-normal sampling experiment has data space $\mathcal{X}=({\mathbb{R}}^p{)}^n$ with sigma-algebra $\mathcal{B}({\mathbb{R}}^p{)}^{\otimes n}$, parameter set $\mathrm{\Theta }={\mathbb{R}}^p\times {\mathcal{S}}_{++}^p$, and sampling law $P_{\mu ,\mathrm{\Sigma }}=N_p(\mu ,\mathrm{\Sigma }{)}^{\otimes n}$. The finite product is a probability measure by Probability measures and probability spaces and For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique. The coordinate projection $X_i$ is the complete vector observation for sampling unit $i$; its $p$ coordinates need not be independent.

For $x=(x_1,\dots ,x_n)\in \mathcal{X}$, define $\bar{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i,R(x)={\sum }_{i=1}^n(x_i-\bar{x})(x_i-\bar{x}{)}^T,S(x)=\frac{R(x)}{n-1},M(x)=\frac{R(x)}{n}\mathrm{.}$ The functions $x\mapsto \bar{x}$, $S(x)$, and $M(x)$ are continuous, hence Borel measurable under AC and AC supplies countable selections and prescribed serial paths; they are statistics in the experiment by Statistics and estimators. Here $S$ uses the $n-1$ covariance denominator, while $M$ is the $n$-denominator covariance candidate whose maximum-likelihood property is proved separately below.

This item instantiates the mathematical portion of the scientific model contract (Scientific statistical model contracts) as follows. One sampling and analysis unit is one complete observed vector $X_i$; index $i$ identifies the sampling unit and coordinate $a$ identifies a component within that vector. For fixed parameter label $\theta =(\mu ,\mathrm{\Sigma })$, $\mu$ and $\mathrm{\Sigma }$ are fixed, every $X_i$ is random, every component of each $X_i$ is observed, and there are no latent quantities, predictors, treatments, or randomization. The target population for this reference experiment is the abstract law $N_p(\mu ,\mathrm{\Sigma })$ of a vector-valued sampling unit, and the target parameter is $\theta$ itself; no separate scientific estimand or physical setting is asserted. The observation map records the complete vector tuple unchanged, and inference uses the joint product law, with no conditioning convention beyond that law. No topological support or empirical adequacy claim is needed; this is a mathematical reference model, not an assertion that observed scientific data follow it.