Almost-sure conditional convergence

Kolmogorov three-series theorem: Let $(X_n{)}_{n\ge 1}$ be independent real random variables and fix $A>0$. Put $Y_n=X_n{\mathbf{1}}_{\{\mathrm{\mid }{X}_n\mathrm{\mid }\le A\}}$. Then ${\sum }_n{X}_n$ converges almost surely if and only if all three conditions hold: ${\sum }_n\mathbb{P}(\mathrm{\mid }{X}_n\mathrm{\mid }>A)<\mathrm{\infty },{\sum }_n\mathbb{E}{Y}_n\text{ converges in }\mathbb{R},{\sum }_n\mathrm{Var}(Y_n)<\mathrm{\infty }\mathrm{.}$ The conditions hold for some $A>0$ if and only if they hold for every $A>0$. No moment assumption is imposed on the untruncated variables.

Countably many independent copies of a prescribed law exist: Assume countable choice and dependent choice. Every probability measure $\nu$ on $(S,\mathrm{\Sigma })$ is the common law of a countable independent family of $S$-valued random elements.

The p-series for a real exponent p converges exactly when p is greater than one: For every real $p$, ${\sum }_{k\ge 1}\frac{1}{k^p}\text{ converges}⟺p>1.$

The alternating series test: if $(b_k)$ is nonincreasing with $b_k\to 0$ then ${\sum }_k(-1{)}^k{b}_k$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_n$: Let $({\epsilon }_k)$ be the alternating sequence of lem-alternating-sequence, that is the unique sequence of reals with ${\epsilon }_0=1$ and ${\epsilon }_{k+1}=-{\epsilon }_k$, which is what is usually written ${\epsilon }_k=(-1{)}^k$; let $e$ and $o$ be its even and odd index maps, so that ${\epsilon }_{e_j}=1$, ${\epsilon }_{o_j}=-1$, and every natural number is $e_j$ for exactly one $j$ or $o_j$ for exactly one $j$. Let $(b_k)$ be a sequence of reals that is nonincreasing (def-monotone-sequence) and converges to $0$ (def-real-limit); then $b_k\ge 0$ for every $k$. Write $t_n:={\sum }_{k<n}{\epsilon }_k{b}_k$ for the partial sums (def-series). Then: 1. the series $\sum {\epsilon }_k{b}_k$ converges; write $L$ for its sum; 2. $t_{e_j}\le L\le t_{o_j}$ for every $j\in \mathbb{N}$, and for every $n\in \mathbb{N}$ the sum $L$ lies between the two consecutive partial sums $t_n$ and $t_{n+1}$; 3. $\mathrm{\mid }L-t_n\mathrm{\mid }\le b_n$ for every $n\in \mathbb{N}$. Claim 3 is the error bound: the partial sum $t_n$, which uses the $n$ terms ${\epsilon }_0{b}_0,\dots ,{\epsilon }_{n-1}{b}_{n-1}$, differs from the sum by at most the first term omitted. Only claim 1 is a corollary of thm-dirichlet-test. Claims 2 and 3 are not: they come from the interlacing of the even-index and odd-index partial sums, and that argument is carried out below rather than smuggled into the Dirichlet estimate, which produces no bracketing at all.