Chapter 3: Random Variables

3.1. Definitions. $\Sigma$ -measurable function, $m\Sigma$ , $(m\Sigma)^{+}$ , $b\Sigma$ . 3.2. Elementary Propositions on measurability. 3.3. Lemma. Sums and products of measurable functions are measurable. 3.4. Composition Lemma. 3.5. Lemma on measurability of infs, lim infs of functions. 3.6. Definition. Random variable. 3.7. Example. Coin tossing. 3.8. Definition. $\sigma$ -algebra generated by a collection of functions on $\Omega$ . 3.9. Definitions. Law, Distribution Function. 3.10. Properties of distribution functions. 3.11. Existence of random variable with given distribution function. 3.12. Skorokod representation of a random variable with prescribed distribution function. 3.13. Generated $\sigma$ -algebras - a discussion. 3.14. The Monotone-Class Theorem.

Random variables

A function $h:(S,\Sigma)\to\mathbb{R}$ (or $[-\infty,\infty]$ ) is called $\Sigma$ -measurable if $h^{-1}(\mathcal{B})\subset\Sigma$ . $m\Sigma$ ,

and $b\Sigma$ are classes of general, non-negative and bounded measurable functions, respectively. $h$ is called Borel if it is $\mathcal{B}(S)$ -measurable where $S$ is a topological space.

It is enough to see that either collection of sets $\{h\le c\}_{c\in\mathbb{R}}$ , $\{h<c\}_{c\in\mathbb{R}}$ , $\{h\ge c\}_{c\in\mathbb{R}}$ , or $\{h>c\}_{c\in\mathbb{R}}$ belongs to $\Sigma$ .
If $h_{n}\in m\Sigma$ for $n\in\mathbb{N}$ , then $\inf h_{n},\sup h_{n},\liminf h_{n},\limsup h_{n}\in m\Sigma$ .
If $h_{n}\in m\Sigma$ for $n\in\mathbb{N}$ , then $\{s:\lim h_{n}(s)\mbox{ exists in }\mathbb{R}\}\in\Sigma$ , and for $r\in\mathbb{R}$ , $\{s:\lim h_{n}(s)=r\}\in\Sigma$ .

A random variable on $(\Omega,\mathcal{F})$ is an element of $m\mathcal{F}$ .

-algebras generated by functions

Given a collection of functions $\{Y_{j}:\Omega\to\mathbb{R}\}_{j\in J}$ , the $\sigma$ -algebra $\sigma(Y_{j}|j\in J)$ generated by $\{Y_{j}\}_{j\in J}$ is the smallest $\sigma$ -algebra $\mathcal{Y}$ on $\Omega$ such that for each $j\in J$ , $Y_{j}$ is $\mathcal{Y}$ -measurable.

Intuitively $\sigma(Y_{j}|j\in J)$ consists of exactly those events $F$ for which, for every $\omega$ , whether $\omega\in F$ is determined by the values of $\{Y_{j}(\omega)\}_{j\in J}$ .
$Z$ is $\sigma(Y)$ -measurable iff there is a Borel function $f:\mathbb{R}\to\mathbb{R}$ such that $Z=f(Y)$ .
- $Z$ is $\sigma(Y_{1},\ldots,Y_{n})$ -measurable iff there is a Borel function $f:\mathbb{R}^{n}\to\mathbb{R}$ such that $Z=f(Y_{1},\ldots,Y_{n})$ .
- $Z$ is $\sigma(Y_{j}|i\in J)$ -measurable iff there is a countable sequence $j_{n}\in J$ , $n\in\mathbb{N}$ , and a Borel function $f:\mathbb{R}^{\mathbb{N}}\to\mathbb{R}$ such that $Z=f(Y_{j_{1}},Y_{j_{2}},\ldots)$ .

Law and distribution functions

The law $\mathcal{L}_{X}$ of a random variable $X$ is $\mathcal{L}_{X}=\mathbb{P}\circ X^{-1}:\mathcal{B}\to[0,1]$ .

The distribution function $F_{X}$ of $X$ is $F_{X}(c)=$ $\mathcal{L}_{X}(-\infty,c]=$ $\mathbb{P}(X\le c)$ .

$F$ is a distribution function of some random variable iff $F:\mathbb{R}\to[0,1]$ is a non-decreasing right-continuous function such that $\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to+\infty}F(x)=1$ .
The measure $\mathcal{L}_{F}$ associated to $F$ such that $\mathcal{L}(-\infty,x]=F(x)$ for all $x\in\mathbb{R}$ is called the Lebesgue-Stieltjes measure.
(Skorokhod representation.) If $(\Omega,\mathcal{F},\mathbb{P})=([0,1],\mathcal{B}[0,1],\mbox{Leb})$ , then $\begin{eqnarray*} X^{+}(\omega) & = & \inf\{z|F(z)>\omega\}\\ & = & \sup\{z|F(z)\le\omega\} \end{eqnarray*}$ and $\begin{eqnarray*} X^{-}(\omega) & = & \inf\{z|F(z)\ge\omega\}\\ & = & \sup\{z|F(z)<\omega\} \end{eqnarray*}$ have distribution function $F$ and law $\mathcal{L}_{F}$ .

Monotone-Class Theorem

Let $\mathcal{H}$ be a vector space over $\mathbb{R}$ of bounded functions $f:S\to\mathbb{R}$ such that $1\in\mathcal{H}$ , and for $f_{n}\ge0$ in $\mathcal{H}$ , if $f_{n}\uparrow f$ , and $f$ is bounded, then $f\in\mathcal{H}$ .

Then, if for some $\pi$ -system $\mathcal{I}$ , $\mathcal{H}$ contains all indicator functions of sets of $\mathcal{I}$ , then $b\sigma(\mathcal{I})\subset\mathcal{H}$ .

Chapter 3: Random Variables

Chapter 3: Random Variables

Random variables

\sigma -algebras generated by functions

Law and distribution functions

Monotone-Class Theorem

-algebras generated by functions