Cauchy averages admit no deterministic weak centering

Exact tail criterion for a truncated-centered IID weak law: For IID real random variables $(X_n{)}_{n\ge 1}$ and $S_n={\sum }_{k=1}^n{X}_k$, there exist deterministic real constants $({\mu }_n)$ with $S_n\mathrm{/}n-{\mu }_n\to 0$ in probability if and only if $n\mathbb{P}(\mathrm{\mid }{X}_1\mathrm{\mid }>n)⟶0.$ When this condition holds, ${\mu }_n=\mathbb{E}[X_1{\mathbf{1}}_{\{\mathrm{\mid }{X}_1\mathrm{\mid }\le n\}}]$ works. Neither existence of an untruncated mean nor convergence of $({\mu }_n)$ is asserted.

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.

Law or distribution of a random element: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element. Its law or distribution is the set function ${\mathbb{P}}_X:\mathrm{\Sigma }\to [0,+\mathrm{\infty }],{\mathbb{P}}_X(B):=\mathbb{P}(X^{-1}(B))\mathrm{.}$ Thus the law of $X$ records the probability of each measurable target set by pulling it back to an event in the original probability space.

Probability laws correspond to distribution functions: Assume the Axiom of Countable Choice. 1. Let $X$ be a real random variable, let ${\mathbb{P}}_X$ be its law, and let $F_X(x)=\mathbb{P}(X\le x)$. Then $F_X$ is nondecreasing and right-continuous, satisfies ${\lim }_{x\to -\mathrm{\infty }}{F}_X(x)=0,{\lim }_{x\to +\mathrm{\infty }}{F}_X(x)=1,$ and obeys ${\mathbb{P}}_X((a,b])=F_X(b)-F_X(a)(a<b)\mathrm{.}$ 2. Conversely, if $F:\mathbb{R}\to \mathbb{R}$ is nondecreasing and right-continuous with ${\lim }_{x\to -\mathrm{\infty }}F(x)=0,{\lim }_{x\to +\mathrm{\infty }}F(x)=1,$ then there is a unique Borel probability measure $\mu$ on $\mathbb{R}$ such that $\mu ((a,b])=F(b)-F(a)(a<b),$ equivalently $F(x)=\mu ((-\mathrm{\infty },x])(x\in \mathbb{R})\mathrm{.}$