Independent-copy symmetrization of random series

Independent random elements: Let $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$ be random elements on a common probability space. For each $i$, write $\sigma (X_i):=\{X_i^{-1}(B):B\in {\mathrm{\Sigma }}_i\}\subseteq \mathcal{F}\mathrm{.}$ The family $(X_i{)}_{i\in I}$ is independent when the sigma-algebras $(\sigma (X_i){)}_{i\in I}$ are independent in the sense of def-independent-sigma-algebras-and-events. When ${\mathrm{\Sigma }}_i$ is a Borel sigma-algebra, this agrees with the notation $\sigma (X_i)$ from def-sigma-algebra-generated-by-a-function.

Symmetric real random variables: A real random variable $X$ is symmetric if its law as defined in def-law-or-distribution-of-a-random-element equals the law of $-X$. Equivalently, $\mathbb{P}(X\in B)=\mathbb{P}(X\in -B)$ for every Borel $B\subseteq \mathbb{R}$, where $-B=\{-b:b\in B\}$. No existence of an expectation is assumed in this definition. In particular atoms, including an atom at zero, are allowed.

Almost-sure convergence of a random series: For real random variables $(X_n{)}_{n\ge 1}$, the series ${\sum }_{n\ge 1}{X}_n$ converges almost surely if its partial sums $S_n$ converge to a finite real limit on an event of probability one, as in def-almost-sure-convergence-of-random-variables. With $S_0=0$ from def-partial-sums-and-sample-means, its convergence event is $C={\bigcap }_{r\ge 1}{\bigcup }_{N\ge 1}{\bigcap }_{j\ge i\ge N}\{\mathrm{\mid }{S}_j-S_i\mathrm{\mid }<1\mathrm{/}r\}\mathrm{.}$ This is exactly the real Cauchy condition, with the indexing of thm-series-cauchy-criterion shifted by one. Measurable arithmetic makes every event in this countable expression measurable. For any fixed $m$, the union over $N$ may be restricted to $N\ge m$; then each difference uses only $X_{m+1},X_{m+2},\dots$. Thus $C$ is in the tail sigma-algebra, without assuming independence. Under independence, cor-almost-sure-convergence-of-an-independent-series-is-a-zero-one-event gives $\mathbb{P}(C)\in \{0,1\}$. Set $S={\lim }_n{S}_n$ on $C$ and $S=0$ off $C$. The functions ${\mathbf{1}}_C{S}_n$ converge everywhere to $S$, so thm-sequential-suprema-infima-limsup-liminf-and-pointwise-limits-are-measurable and thm-arithmetic-and-lattice-operations-preserve-measurability make $S$ measurable. For Borel sets $B_n$, the event $\{X_n\in B_n\text{ infinitely often}\}={\bigcap }_m{\bigcup }_{n\ge m}\{X_n\in B_n\}$ is likewise tail measurable. Changing finitely many summands adds an eventually constant finite difference to $S_n$; divided by deterministic $c_n>0$ tending to infinity that difference tends to zero, so the normalized limsup is unchanged. The sign of the unnormalized limsup need not be unchanged: the all-zero sequence has limsup zero, while changing its first term to $1$ makes the limsup of partial sums equal to $1$.

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique: Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be sigma-finite measure spaces. Then: 1. the set function $E\mapsto (\mu \times \nu )(E)$ of def-product-measure-on-sigma-finite-spaces is a measure on $\mathcal{A}\otimes \mathcal{B}$; 2. for measurable rectangles, $(\mu \times \nu )(A\times B)=\mu (A)\nu (B);$ 3. the measure $\mu \times \nu$ is sigma-finite; and 4. it is the unique measure on $\mathcal{A}\otimes \mathcal{B}$ with the rectangle formula.

Disjoint groups of an independent sigma-algebra family remain independent: Let $({\mathcal{F}}_i{)}_{i\in I}$ be an independent family of sigma-algebras on a probability space, and let $J_0,\dots ,J_{m-1}\subseteq I$ be pairwise disjoint index sets. For each $r<m$, define ${\mathcal{G}}_r:=\sigma \left({\bigcup }_{i\in J_r}{\mathcal{F}}_i\right)\mathrm{.}$ Then the sigma-algebras ${\mathcal{G}}_0,\dots ,{\mathcal{G}}_{m-1}$ are independent.

Measurable coordinatewise functions preserve independence: Let $(X_i{)}_{i\in I}$ be an independent family of random elements $X_i:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S_i,{\mathrm{\Sigma }}_i)$. For each $i$, let $g_i:(S_i,{\mathrm{\Sigma }}_i)\to (T_i,{\mathcal{T}}_i)$ be measurable. Then the family $(g_i\circ X_i{)}_{i\in I}$ is independent.

For $n\ge 1$, if $X_0,\dots ,X_{n-1}$ are independent real random variables and $g_i:\mathbb{R}\to \mathbb{R}$ are Borel measurable with each $g_i(X_i)$ integrable, then ${\prod }_{i<n}{g}_i(X_i)$ is integrable and $\mathbb{E}[{\prod }_{i<n}{g}_i(X_i)]={\prod }_{i<n}\mathbb{E}[g_i(X_i)]$ in $\mathbb{R}$ (Expectations factor over finite products of independent random variables).

Variance and covariance identities for random variables: Let $X,Y$ be square-integrable real random variables on one probability space. Then $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2,$ $\mathrm{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\mathrm{.}$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.