Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$. Throughout, random vectors are random vectors in $\mathbb{R}^{d}$ on $(\Omega,\mathcal{F},P)$, with coordinates $X_{i}$; the random vectors $X+Y$, $aX$ and $X-Y$ (for real $a$) and the random variables $\lVert X\rVert^{2}$, $X\cdot Y$ and $\lVert X-Y\rVert^{2}$ are those of Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii); $\mathbf{0}$ denotes the constant map $\Omega\to\mathbb{R}^{d}$ with value the origin, a random vector since each of its preimages is $\varnothing$ or $\Omega$; and $X\sim_{P}Y$ means $P(X=Y)=1$, as in Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iv). Finite sums of real numbers are those of Finite Sum Notation in a Field. Then the following hold.

1. (Coordinates)(i) A random vector $X$ satisfies $\mathbb{E}[\lVert X\rVert^{2}]<\infty$ if and only if every coordinate $X_{1},\dots,X_{d}$ is a square-integrable random variable, and in that case $\mathbb{E}[\lVert X\rVert^{2}]=\sum_{i=1}^{d}\mathbb{E}[X_{i}^{2}]$.

2. (Operations)(ii) Let $X,Y$ be random vectors with $\mathbb{E}[\lVert X\rVert^{2}]<\infty$ and $\mathbb{E}[\lVert Y\rVert^{2}]<\infty$, and let $a\in\mathbb{R}$. Then $\mathbb{E}[\lVert X+Y\rVert^{2}]<\infty$, $\mathbb{E}[\lVert aX\rVert^{2}]<\infty$ and $\mathbb{E}[\lVert X-Y\rVert^{2}]<\infty$; the random variable $X\cdot Y$ is integrable with respect to $P$, with

$$\mathbb{E}[X\cdot Y]=\sum_{i=1}^{d}\langle X_{i},Y_{i}\rangle_{2}\qquad\text{and}\qquad|\mathbb{E}[X\cdot Y]|\le\sqrt{\mathbb{E}[\lVert X\rVert^{2}]}\,\sqrt{\mathbb{E}[\lVert Y\rVert^{2}]}\ ;$$

and $\mathbb{E}[X\cdot X]=\mathbb{E}[\lVert X\rVert^{2}]$.

3. (Almost sure equality)(iii) The relation $\sim_{P}$ on the set of random vectors is reflexive, symmetric and transitive; consequently, writing $[X]=\{X':X'\text{ a random vector},\ X'\sim_{P}X\}$, two such sets are either equal or disjoint, and $[X]=[Y]$ exactly when $X\sim_{P}Y$. If $X\sim_{P}X'$ and $Y\sim_{P}Y'$, then $X+Y\sim_{P}X'+Y'$, $aX\sim_{P}aX'$ for every $a\in\mathbb{R}$, $\mathbb{E}[\lVert X\rVert^{2}]=\mathbb{E}[\lVert X'\rVert^{2}]$, and, when $\mathbb{E}[\lVert X\rVert^{2}]<\infty$ and $\mathbb{E}[\lVert Y\rVert^{2}]<\infty$, $\mathbb{E}[X\cdot Y]=\mathbb{E}[X'\cdot Y']$.

4. (The mean-square form)(iv) Let $U,V,W$ be random vectors with $\mathbb{E}[\lVert U\rVert^{2}]$, $\mathbb{E}[\lVert V\rVert^{2}]$ and $\mathbb{E}[\lVert W\rVert^{2}]$ finite, and let $a\in\mathbb{R}$. Then

$$\mathbb{E}[(U+V)\cdot W]=\mathbb{E}[U\cdot W]+\mathbb{E}[V\cdot W],\qquad \mathbb{E}[(aU)\cdot W]=a\,\mathbb{E}[U\cdot W],\qquad \mathbb{E}[U\cdot V]=\mathbb{E}[V\cdot U],$$

$0\le\mathbb{E}[U\cdot U]$, and $\mathbb{E}[U\cdot U]=0$ if and only if $U\sim_{P}\mathbf{0}$.

5. (The vector space of classes)(v) Let $\mathbf{L}^{2}$ denote the set of random vectors $X$ with $\mathbb{E}[\lVert X\rVert^{2}]<\infty$, and let $L^{2}$ denote the set of all sets $[X]$ of claim 3 with $X\in\mathbf{L}^{2}$; every member of such a set $[X]$ lies in $\mathbf{L}^{2}$ by claim 3. Then $[X]+[Y]=[X+Y]$ for $[X],[Y]\in L^{2}$ and $a[X]=[aX]$ for $a\in\mathbb{R}$ and $[X]\in L^{2}$ are well defined operations on $L^{2}$ by claims 2 and 3, and with them $L^{2}$ is a real vector space whose zero vector is $[\mathbf{0}]$ and in which the additive inverse of $[X]$ is $[(-1)X]$.