Khinchin weak law for integrable IID variables

Zero truncation at a positive level: For a real random variable $X$ and a deterministic level $A>0$, its zero truncation is $X^{(A)}=X{\mathbf{1}}_{\{\mathrm{\mid }X\mathrm{\mid }\le A\}}\mathrm{.}$ The threshold event is measurable because $X$ is measurable and $[-A,A]$ is Borel; its indicator and the product are measurable by thm-arithmetic-and-lattice-operations-preserve-measurability. Thus $X^{(A)}$ is a real random variable as in def-random-element-and-real-random-variable. It equals $X$ at both cutoff endpoints and is zero outside the interval. Since $\mathrm{\mid }{X}^{(A)}\mathrm{\mid }\le A$, for every $0<p<\mathrm{\infty }$ its absolute $p$th moment is at most $A^p\mathbb{P}(\mathrm{\Omega })=A^p$. This is not clipping to the endpoints.

IID finite-variance weak law: Let $(X_k{)}_{k\ge 1}$ be IID square-integrable real random variables, with $\mu =\mathbb{E}{X}_1$ and ${\sigma }^2=\mathrm{Var}(X_1)$. For $S_n={\sum }_{k=1}^n{X}_k$, $\mathbb{E}\mathrm{\mid }{S}_n\mathrm{/}n-\mu {\mathrm{\mid }}^2={\sigma }^2\mathrm{/}n,$ and $S_n\mathrm{/}n\to \mu$ in $L^2$ and in probability. Also $\mathbb{P}(\mathrm{\mid }{S}_n\mathrm{/}n-\mu \mathrm{\mid }\ge \epsilon )\le {\sigma }^2\mathrm{/}(n{\epsilon }^2)$ for $\epsilon >0$.

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.

Dominated convergence: Let $f$ and $(f_n)$ be measurable complex-valued functions such that $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single nonnegative measurable function $g$ with $\int gd\mu <+\mathrm{\infty }$. Then $f\in L^1(\mu )$, $\int \mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu ⟶0,$ and hence $\int f_{n}d\mu ⟶\int fd\mu \mathrm{.}$

Markov's inequality for random variables: If $X:\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ is a nonnegative random variable on a probability space and $a>0$, then $\mathbb{P}(X\ge a)\le \frac{\mathbb{E}[X]}{a}\mathrm{.}$

Finite-measure $L^r$ includes into $L^p$ for $p<r$: Let $(X,\mathcal{A},\mu )$ be a measure space with $\mu (X)<\mathrm{\infty }$. 1. If $1\le p<r<\mathrm{\infty }$ and $f\in {\mathcal{L}}^r(\mu )$, then $f\in {\mathcal{L}}^p(\mu )$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_p\le \mu (X{)}^{1\mathrm{/}p-1\mathrm{/}r}\mathrm{\parallel }f{\mathrm{\parallel }}_r\mathrm{.}$ 2. If $1\le p<\mathrm{\infty }$ and $f\in L^{\mathrm{\infty }}(\mu )$, then $f\in {\mathcal{L}}^p(\mu )$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_p\le \mu (X{)}^{1\mathrm{/}p}\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$

Linearity, monotonicity, and the modulus bound for expectation: Let $X,Y$ be integrable real or complex random variables on one probability space. 1. For scalars $a,b$, $\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]\mathrm{.}$ 2. If $X$ and $Y$ are real-valued and $X\le Y$ almost surely, then $\mathbb{E}[X]\le \mathbb{E}[Y]\mathrm{.}$ 3. $\mathrm{\mid }\mathbb{E}[X]\mathrm{\mid }\le \mathbb{E}[\mathrm{\mid }X\mathrm{\mid }]\mathrm{.}$