The Space of Square-Integrable Random Vectors

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$. Random vectors are random vectors in $\mathbb{R}^{d}$ on $(\Omega,\mathcal{F},P)$; their pointwise operations $X+Y$, $aX$ (for $a\in\mathbb{R}$) and $X-Y$, the random variables $\lVert X\rVert^{2}$ and $X\cdot Y$, the constant random vector $\mathbf{0}$, and the relation $X\sim_{P}Y$ meaning $P(X=Y)=1$ are as in Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form.

1. (Square-integrable random vectors)(i) A random vector $X$ is square-integrable if $\mathbb{E}[\lVert X\rVert^{2}]<\infty$. The set of all square-integrable random vectors is denoted $\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$; it contains $\mathbf{0}$ and is closed under $X+Y$, $aX$ and $X-Y$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (ii).

2. (Classes)(ii) For $X\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$, the class of $X$ is the set $[X]$ of all random vectors $X'$ with $X'\sim_{P}X$, as in Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (iii); every member of $[X]$ is square-integrable, $X\in[X]$, two classes are equal or disjoint, and $[X]=[Y]$ exactly when $X\sim_{P}Y$, by that clause. $L^{2}(\Omega;\mathbb{R}^{d})$ denotes the set of all classes $[X]$ with $X\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$, with the operations

$$[X]+[Y]=[X+Y],\qquad a[X]=[aX]\qquad(a\in\mathbb{R}),$$

which are well defined and make $L^{2}(\Omega;\mathbb{R}^{d})$ a real vector space with zero vector $[\mathbf{0}]$, by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (v).

3. (Inner product and norm)(iii) For $X,Y\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ set

$$\bigl\langle[X],[Y]\bigr\rangle_{L^{2}}=\mathbb{E}[X\cdot Y],\qquad\bigl\lVert[X]\bigr\rVert_{L^{2}}=\sqrt{\mathbb{E}[\lVert X\rVert^{2}]}\ ,$$

the real number $\mathbb{E}[X\cdot Y]$ depending only on the classes by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (ii) and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (iii). By Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (iv), together with Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (iii) for the identification of $U\sim_{P}\mathbf{0}$ with $[U]=[\mathbf{0}]$, this pairing is bilinear, symmetric and positive definite on $L^{2}(\Omega;\mathbb{R}^{d})$, so that $L^{2}(\Omega;\mathbb{R}^{d})$ is a real inner product space, and $\lVert[X]\rVert_{L^{2}}$ is its norm, since $\langle[X],[X]\rangle_{L^{2}}=\mathbb{E}[\lVert X\rVert^{2}]$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form (ii).

4. (Law of a class)(iv) The law $\mathcal{L}(X)$ of $X\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ depends only on $[X]$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (iv), and is called the law of the class $[X]$, written $\mathcal{L}([X])$.

5. (Notational convention)(v) An element of $L^{2}(\Omega;\mathbb{R}^{d})$ is denoted by the same symbol as a representative of it: one writes $X$ for $[X]$, $\langle X,Y\rangle_{L^{2}}$ for $\langle[X],[Y]\rangle_{L^{2}}$, $\lVert X\rVert_{L^{2}}$ for $\lVert[X]\rVert_{L^{2}}$, and $\mathcal{L}(X)$ for $\mathcal{L}([X])$.