Joint Distribution, Expectations, and Block Independence for Independent Random Variables

Throughout, $r$ is a natural number with $r\ge 1$, $\mathbb{R}^r$ is Euclidean space, $\mathbb{R}$ denotes the real numbers, and $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra.

Define the $r$-fold product Borel $\sigma$-algebra $\mathcal{B}_r$ on $\mathbb{R}^r$ iteratively: $\mathcal{B}_1=\mathcal{B}(\mathbb{R})$ and, for $2\le k\le r$, $\mathcal{B}_k=\mathcal{B}_{k-1}\otimes\mathcal{B}(\mathbb{R})$, the product $\sigma$-algebra on $\mathbb{R}^{k}$, identifying the Cartesian product $\mathbb{R}^{k-1}\times\mathbb{R}$ with $\mathbb{R}^{k}$ via $((x_1,\dots,x_{k-1}),x_k)\mapsto(x_1,\dots,x_k)$. Given probability measures $\nu_1,\dots,\nu_r$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, define the probability measure $\nu_1\otimes\cdots\otimes\nu_r$ on $(\mathbb{R}^r,\mathcal{B}_r)$ iteratively by Existence and Uniqueness of the Product Measure (probability measures are $\sigma$-finite). Call a function $\varphi:\mathbb{R}^r\to\mathbb{R}$ jointly Borel if it is measurable from $(\mathbb{R}^r,\mathcal{B}_r)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $V_1,\dots,V_r$ be independent random variables on it with distributions $\nu_1,\dots,\nu_r$. Then the following hold.

Claim 1. The map $V:\Omega\to\mathbb{R}^r$ given by $V(\omega)=(V_1(\omega),\dots,V_r(\omega))$ is measurable from $(\Omega,\mathcal{F})$ to $(\mathbb{R}^r,\mathcal{B}_r)$, and its distribution $P_V:\mathcal{B}_r\to[0,1]$, $P_V(C)=P(V\in C)$, is a probability measure equal to $\nu_1\otimes\cdots\otimes\nu_r$.

Claim 2. If $\varphi:\mathbb{R}^r\to\mathbb{R}$ is jointly Borel and either nonnegative or bounded, then $\varphi(V_1,\dots,V_r)=\varphi\circ V$ is a random variable and its expectation is defined (as an element of $[0,\infty]$ in the nonnegative case, and as a real number in the bounded case, where $\varphi\circ V$ is integrable), with

$$\mathbb{E}[\varphi(V_1,\dots,V_r)]=\int_{\mathbb{R}^r}\varphi\,d(\nu_1\otimes\cdots\otimes\nu_r).$$

In particular this expectation depends only on $\varphi$ and the distributions $\nu_1,\dots,\nu_r$.

Claim 3 (block independence). Let $I=\{i_1<\dots<i_p\}$ and $J=\{j_1<\dots<j_q\}$ be disjoint nonempty subsets of $\{1,\dots,r\}$, and let $\varphi:\mathbb{R}^p\to\mathbb{R}$ and $\psi:\mathbb{R}^q\to\mathbb{R}$ be jointly Borel. Then $\varphi(V_{i_1},\dots,V_{i_p})$ and $\psi(V_{j_1},\dots,V_{j_q})$ are independent random variables.

Claim 4. Each coordinate projection $(x_1,\dots,x_r)\mapsto x_i$ and the addition map $(x_1,\dots,x_r)\mapsto x_1+\cdots+x_r$ are jointly Borel; moreover, if $\varphi:\mathbb{R}^r\to\mathbb{R}$ is jointly Borel and $t:\mathbb{R}\to\mathbb{R}$ is Borel measurable, then $t\circ\varphi$ is jointly Borel.