# Sample mean and sample covariance canonical forms

Let $$(x_1, x_2,\dotsc, x_n)$$ be a sequence of vectors: ($$\forall i=1,\dots,n$$) $$\begin{pmatrix} x_i^1 \\ x_i^2\\ \vdots\\ x_i^m \end{pmatrix}$$

In statistics, one often has to compute the sample mean vector: $$X = \frac{1}{n}\sum_{i=1}^n x_i$$

and the sample covariance (or variance-covariance) matrix: $$\mathbb{V} = \frac{1}{n-1}\sum_{i=1}^n (x_i - X)(x_i - X)^T$$ where $$x^T$$ is the transpose of $$x$$.

My questions are:

1. what canonical form of information would I suggest to represent the sequence $$(x_1, x_2, \dotsc, x_n)$$ in order to compute the sample mean vector and the sample covariance matrix?

2. How can I verify that all the desirable properties of canonical information are satisfied: Existence and Uniqueness, Completeness, Elementary, Empty, Combination, Update, and Compactness and Efficiency.

3. What are the minimum number of observations $$n$$ for which $$X$$ and $$\mathbb{V}$$ are defined?