Finite weighted ensemble and covariance

Let $N\ge 1$ be an integer and let $a_1,\dots ,a_N\in [0,\mathrm{\infty })$ satisfy ${\sum }_{j=1}^N{a}_j=1$. The fixed tuple $(a_1,\dots ,a_N)$ is a finite ensemble of weights. For integers $m,n\ge 1$, let $X=(X_1,\dots ,X_N)\in ({\mathbb{R}}^m{)}^N$ and $Y=(Y_1,\dots ,Y_N)\in ({\mathbb{R}}^n{)}^N$. Define the weighted mean of $X$ and its $j$th fluctuation by

$\bar{X}:={\sum }_{j=1}^N{a}_j{X}_j,X_j^{\mathrm{\prime }}:=X_j-\bar{X}\mathrm{.}$

Using the same weights, define the cross-covariance tensor

$C(X,Y):={\sum }_{j=1}^N{a}_j{X}_j^{\mathrm{\prime }}\otimes Y_j^{\mathrm{\prime }}\in {\mathbb{R}}^{m\times n},$

where $x\otimes y$ is the matrix with entries $(x\otimes y{)}_{rs}=x_r{y}_s$. The covariance tensor of $X$ is $C(X):=C(X,X)$. If $D$ is a set and $X_j:D\to {\mathbb{R}}^m$, $Y_j:D\to {\mathbb{R}}^n$ are functions, apply these formulas pointwise at each $q\in D$ using the same fixed weights. No probability space, limiting operation, or interpretation as an empirical sample is part of this definition.

Flierl and Ferrari, MIT OCW 12.820, Chapter 3, §3.2 (joint moments and covariance), full lecture notes

Eyink, Johns Hopkins University Turbulence Notes, Chapter VIb, §(B) Reynolds Averaging and kinetic-energy balances, pp. 17–19