Kolmogorov convergence criterion

Kolmogorov maximal inequality: Let $X_1,\dots ,X_n$ be independent centered square-integrable real random variables, $n\ge 1$, and $S_k={\sum }_{j=1}^k{X}_j$. For every $\lambda >0$, $\mathbb{P}({\max }_{1\le k\le n}\mathrm{\mid }{S}_k\mathrm{\mid }\ge \lambda )\le \frac{\mathrm{Var}(S_n)}{{\lambda }^2}=\frac{{\sum }_{j=1}^n\mathrm{Var}(X_j)}{{\lambda }^2}\mathrm{.}$ Thus controlling the whole finite maximum costs no larger bound than controlling the final sum by Chebyshev.

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 $1$.

A series converges iff for every $\epsilon >0$ there is $N$ with $\mathrm{\mid }{a}_{m+1}+\cdots +a_n\mathrm{\mid }<\epsilon$ for all $n>m\ge N$: Let $(a_k)$ be a sequence of reals, with partial sums $s_n={\sum }_{k<n}{a}_k$ (def-series). Then $\sum a_k$ converges if and only if $\text{for every real }\epsilon >0\text{ there is }N\in \mathbb{N}\text{ such that }\mid {\sum }_{k=m+1}^n{a}_k\mid <\epsilon \text{ for all }n>m\ge N\mathrm{.}$ The block ${\sum }_{k=m+1}^n{a}_k$ is the finite sum $a_{m+1}+\cdots +a_n$ of def-finite-sum, and it equals $s_{n+1}-s_{m+1}$. This is the Cauchy criterion transported from sequences to series. Its value is that it decides convergence without producing, or even naming, the sum.

Continuity from below for measures: Let $(E_n{)}_{n\in \mathbb{N}}$ be an increasing sequence of measurable sets for a measure $\mu$, so $E_n\subseteq E_{n+1}$. Then $\mu \left({\bigcup }_{n\in \mathbb{N}}{E}_n\right)={\sup }_{n\in \mathbb{N}}\mu (E_n)\mathrm{.}$ No finiteness hypothesis is required.

Continuity from above when one set has finite measure: Let $(E_n{)}_{n\in \mathbb{N}}$ be a decreasing sequence of measurable sets for a measure $\mu$. If $\mu (E_{n_0})<+\mathrm{\infty }$ for some $n_0$, then $\mu \left({\bigcap }_{n\in \mathbb{N}}{E}_n\right)={\inf }_{n\in \mathbb{N}}\mu (E_n)\mathrm{.}$

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.

Fatou's lemma holds for supplied nonnegative measurable functions (Fatou's lemma); the nonnegative integral is monotone (Monotonicity and nonnegative homogeneity of the nonnegative integral).

Integrable functions equal almost everywhere have equal integrals (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

A supplied countable union of measurable null sets is null (Finite and countable subadditivity of measures).

For square-integrable real random variables, $L^2$ convergence means $\mathbb{E}\mathrm{\mid }{S}_n-S{\mathrm{\mid }}^2\to 0$ ($L^p$ convergence for random variables).