Etemadi strong law for pairwise independent iid variables

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.

Independence forces covariance to vanish: If $X$ and $Y$ are independent square-integrable real random variables, then $\mathrm{Cov}(X,Y)=0.$

Thus independence implies zero covariance. The converse is false in general.

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.

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$: Identify $\mathbb{Z}$ with its canonical copy inside $\mathbb{R}$, along the embeddings $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$ (lem-nat-embeds-int, lem-int-embeds-rat, lem-rat-embeds-dense, def-integers). Then for every real $x$ there is exactly one integer $m$ with

$m\le x<m+1.$

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$.

Two independent ingredients are needed and neither may be dropped. Existence is the Archimedean property (thm-of-archimedean) together with the well-ordering of $\mathbb{N}$ (thm-well-ordering-principle): the first says that $x$ is caught between two integers at all, the second picks the least integer above $x$. Uniqueness is the discreteness of $\mathbb{Z}$: no integer lies strictly between $m$ and $m+1$.

This lemma is stated once here and reused. It is what turns "the nearest integer to $x$" from a picture into an object, and the companion page's oscillator $\psi (x)={\inf }_{n\in \mathbb{Z}}\mathrm{\mid }x-n\mathrm{\mid }$ is computed from it in one line.

Integer powers $a^m$: Let $a\in \mathbb{R}$, where $\mathbb{R}$ is the ambient ordered field (def-ordered-field, def-field).

Natural exponents. By the recursion theorem (thm-recursion) applied to the set $\mathbb{R}$, the starting element $1$ and the function $f(x)=x\cdot a$, there is a unique function $\mathbb{N}\to \mathbb{R}$, written $n\mapsto a^n$, with

$a^0=1,a^{n+1}=a^n\cdot a(n\in \mathbb{N})\mathrm{.}$

Thus $a^1=a$, $a^2=a\cdot a$, and so on. Note that this is defined for every $a$, including $a=0$.

Negative exponents. If $a\mathrm{\ne }0$ and $n\in \mathbb{N}$ with $n\ge 1$, set

$a^{-n}:=(a^n{)}^{-1}\mathrm{.}$

Why that is legitimate. The right-hand side presupposes that $a^n$ is invertible, that is, that $a^n\mathrm{\ne }0$. This is a proof obligation and not an observation, and it is discharged by claim 2 of lem-power-laws: for $a\mathrm{\ne }0$ in a field, $a^n\mathrm{\ne }0$ for every $n\in \mathbb{N}$, proved there by induction on $n$ from the fact that a field has no zero divisors (lem-of-no-zero-divisors). That lemma is a statement about the operation introduced here, so it depends on this definition and is recorded in this item's justified_by rather than in its deps (SCHEMA §3). Given $a^n\mathrm{\ne }0$, the value $(a^n{)}^{-1}$ is a single well-determined element, because multiplicative inverses in a field are unique (lem-of-inverse-unique).

Integer exponents. Every integer $m$ (def-integers) is either $\iota (n)$ or $-\iota (n)$ for a unique natural $n$, where $\iota$ is the embedding $\mathbb{N}\to \mathbb{Z}$ (lem-nat-embeds-int, def-int-operations). This too is a citation and not a slogan: the order on $\mathbb{Z}$ is total (thm-int-ordered-ring), so $m\ge 0$ or $m<0$; the image of $\iota$ is exactly the set of nonnegative integers, and each of them is $\iota (n)$ for a unique natural $n$ (lem-nat-embeds-int); and if $m<0$ then $-m>0$, by compatibility of the order with addition (thm-int-ordered-ring), so $-m=\iota (n)$ and $m=-\iota (n)$, with $n$ unique because $\iota$ is injective. The two clauses above therefore define $a^m$ for every $m\in \mathbb{Z}$ whenever $a\mathrm{\ne }0$, and for every $m\in \mathbb{N}$ for arbitrary $a$. The exponent domains are disjoint: the natural clause handles $m\ge 0$, including $m=0$, while the negative clause requires $n\ge 1$ and handles only $m<0$.

For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges: Let $r\in \mathbb{R}$ and let $r^k$ be the integer power (def-integer-power), so that $r^0=1$ for every $r$, including $r=0$.

1. If $\mathrm{\mid }r\mathrm{\mid }<1$ then the series $\sum r^k$ converges (def-series) and ${\sum }_{k=0}^{\mathrm{\infty }}{r}^k=\frac{1}{1-r}\mathrm{.}$
2. If $\mathrm{\mid }r\mathrm{\mid }\ge 1$ then $\sum r^k$ diverges.

The series starts at $k=0$ and its first term is $r^0=1$; in particular ${\sum }_{k=0}^{\mathrm{\infty }}{2}^{-k}=2$, while the series starting at $k=1$ sums to $1$. Which starting index is meant has to be said, and it is said here.

Summability of truncated normalized variances: For identically distributed integrable real $(X_n)$ and $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$, ${\sum }_{n\ge 1}\mathrm{Var}(Y_n)\mathrm{/}{n}^2\le 2\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$. No independence is required.

Chebyshev's inequality for random variables: If $X$ is a square-integrable real random variable and $a>0$, then $\mathbb{P}(\mathrm{\mid }X-\mathbb{E}[X]\mathrm{\mid }\ge a)\le \frac{\mathrm{Var}(X)}{a^2}\mathrm{.}$

First Borel-Cantelli lemma for events: Let $(A_n{)}_{n\in \mathbb{N}}$ be events in a probability space. If ${\sum }_{n=0}^{\mathrm{\infty }}\mathbb{P}(A_n)<+\mathrm{\infty },$ then $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=0.$

No independence hypothesis is needed.

Cesaro limit of truncated means: For identically distributed integrable real $(X_n)$, with $Y_k=X_k{\mathbf{1}}_{\{\mathrm{\mid }{X}_k\mathrm{\mid }\le k\}}$, one has $n^{-1}{\sum }_{k=1}^n\mathbb{E}{Y}_k\to \mathbb{E}{X}_1$.

Iid linear truncation occurs only finitely often: For identically distributed integrable real $(X_n)$, put $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le n\}}$. Almost surely $Y_n=X_n$ for all sufficiently large $n$. Consequently $n^{-1}{\sum }_{k=1}^n(X_k-Y_k)\to 0$. Independence is unnecessary.