Exact tail criterion for a truncated-centered IID weak law

Identical distribution and IID families: Let $(X_i{)}_{i\in I}$ be random elements with the same measurable target $(E,\mathcal{E})$. They are identically distributed if $\mathbb{P}(X_i\in B)=\mathbb{P}(X_j\in B)$ for all $i,j\in I$ and $B\in \mathcal{E}$, that is, their laws in def-law-or-distribution-of-a-random-element agree. They are independent and identically distributed (IID) if, in addition, the whole family is independent in def-independent-random-elements. Independence means mutual independence, not merely pairwise independence. No moment assumption is part of either definition. The empty family satisfies these universal conditions vacuously.

Independent-copy symmetrization of random series: Given an independent sequence $(X_n{)}_{n\ge 1}$ on $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, form the product probability space $({\mathrm{\Omega }}^2,\mathcal{F}\otimes \mathcal{F},\mathbb{P}\otimes \mathbb{P})$. Write $U_n(\omega ,{\omega }^{\mathrm{\prime }})=X_n(\omega )$, $V_n(\omega ,{\omega }^{\mathrm{\prime }})=X_n({\omega }^{\mathrm{\prime }})$, and $Z_n=U_n-V_n$. Then $(U_n)$ and $(V_n)$ are independent copies of the whole sequence, and the $Z_n$ are independent symmetric real random variables. Almost-sure convergence of ${\sum }_n{X}_n$ implies almost-sure convergence of ${\sum }_n{Z}_n$. If $\mathrm{\mid }{X}_n\mathrm{\mid }\le A$ almost surely for every $n$, with $0\le A<\mathrm{\infty }$, then $\mathrm{\mid }{Z}_n\mathrm{\mid }\le 2A$ almost surely, $\mathbb{E}{Z}_n=0$, and $\mathrm{Var}(Z_n)=2\mathrm{Var}(X_n)$.

Tail comparisons under independent-copy symmetrization: Let $X^{\mathrm{\prime }}$ be an independent copy of a real random variable $X$. For every $t>0$, $\mathbb{P}(\mathrm{\mid }X-X^{\mathrm{\prime }}\mathrm{\mid }>t)\le 2\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }>t\mathrm{/}2)\mathrm{.}$ There exists a finite $M\ge 0$ with $\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }\le M)\ge 1\mathrm{/}2$; for every such $M$, $\mathbb{P}(\mathrm{\mid }X-X^{\mathrm{\prime }}\mathrm{\mid }>t)\ge \frac{1}{2}\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }>t+M)\mathrm{.}$

One-sided maximal inequality for symmetric independent sums: For independent symmetric real random variables $X_1,\dots ,X_n$, $n\ge 1$, let $S_k={\sum }_{j=1}^k{X}_j$. For every real $a$, $\mathbb{P}({\max }_{1\le k\le n}{S}_k>a)\le 2\mathbb{P}(S_n>a)\mathrm{.}$ Consequently for every $t>0$, $\mathbb{P}({\max }_{1\le k\le n}\mathrm{\mid }{S}_k\mathrm{\mid }>t)\le 2\mathbb{P}(\mathrm{\mid }{S}_n\mathrm{\mid }>t)\mathrm{.}$ No moment assumptions are needed.

Truncation weak law for independent arrays: For each $n\ge 1$ let $X_{n,1},\dots ,X_{n,r_n}$ be independent real random variables on one probability space, with finite $r_n\ge 0$. Let deterministic $b_n>0$ tend to infinity and set $Y_{n,k}=X_{n,k}{\mathbf{1}}_{\{\mathrm{\mid }{X}_{n,k}\mathrm{\mid }\le b_n\}}$. If ${\sum }_{k=1}^{r_n}\mathbb{P}(\mathrm{\mid }{X}_{n,k}\mathrm{\mid }>b_n)\to 0,b_n^{-2}{\sum }_{k=1}^{r_n}\mathrm{Var}(Y_{n,k})\to 0,$ then $\frac{{\sum }_{k=1}^{r_n}{X}_{n,k}-{\sum }_{k=1}^{r_n}\mathbb{E}{Y}_{n,k}}{b_n}⟶0\text{in probability}\mathrm{.}$ No independence between rows is required.

Vanishing tail control bounds truncated second moments: Let $X$ be a real random variable with $n\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }>n)\to 0$ as positive integers $n\to \mathrm{\infty }$. Then $x\mathbb{P}(\mathrm{\mid }X\mathrm{\mid }>x)\to 0$ for real $x\to \mathrm{\infty }$, and $\frac{\mathbb{E}[X^2{\mathbf{1}}_{\{\mathrm{\mid }X\mathrm{\mid }\le n\}}]}{n}⟶0.$ Moreover $\mathbb{E}\mathrm{\mid }X{\mathrm{\mid }}^p<\mathrm{\infty }$ for every $0<p<1$.

Convergence in probability: For real random variables $(X_n)$ and $X$ on one probability space, write $X_n\to X$ in probability when, for every $\epsilon >0$, $\mathbb{P}(\mathrm{\mid }{X}_n-X\mathrm{\mid }>\epsilon )⟶0.$ This is precisely def-convergence-in-measure for the probability measure.

Finite and countable subadditivity of measures: Let $\mu$ be a measure and let $(E_k{)}_{k\in \mathbb{N}}$ be measurable. Then $\mu \left({\bigcup }_{k\in \mathbb{N}}{E}_k\right)\le {\sum }_{k=0}^{\mathrm{\infty }}\mu (E_k)\mathrm{.}$ For every $m\in \mathbb{N}$ one also has $\mu \left({\bigcup }_{k<m}{E}_k\right)\le {\sum }_{k<m}\mu (E_k),$ including $m=0$, where both sides are $0$.

Largest-summand bound for independent symmetric variables: Let $Y_1,\dots ,Y_n$ be independent symmetric real random variables, $n\ge 1$, and $S_n={\sum }_{k=1}^n{Y}_k$. For $t>0$, $\mathbb{P}(\mathrm{\mid }{S}_n\mathrm{\mid }\ge t)\ge \frac{1}{2}\mathbb{P}({\max }_{k\le n}\mathrm{\mid }{Y}_k\mathrm{\mid }\ge t)\mathrm{.}$ The same bound holds when both inequalities inside the probabilities are strict. If the $Y_k$ are IID and $p=\mathbb{P}(\mathrm{\mid }{Y}_1\mathrm{\mid }>t)$, then $\mathbb{P}(\mathrm{\mid }{S}_n\mathrm{\mid }>t)\ge \frac{1}{2}(1-(1-p{)}^n)\ge \frac{1}{2}(1-e^{-np})\mathrm{.}$