Chapter 2: Events

2.1. Model for experiment: $(\Omega,\mathcal{F},\mathbb{P})$ . 2.2. The intuitive meaning. 2.3. Examples of $(\Omega,\mathcal{F})$ pairs. 2.4. Almost surely (a.s.) 2.5. Reminder: $\limsup$ , $\liminf$ , $\downarrow\lim$ , etc. 2.6. Definitions. $\limsup E_{n}$ , ( $E_{n}$ ,i.o.). 2.7. First Borel-Cantelli Lemma (BC1). 2.8. Definitions. $\liminf E_{n}$ , ( $E_{n}$ , ev). 2.9. Exercise.

Events

Given a probability triple $(\Omega,\mathcal{F},\mathbb{P})$ , $\Omega$ is called a sample space, $\omega\in\Omega$ is called a sample point, and $A\in\mathcal{F}$ is called an event, so that $\mathcal{F}$ is the family of events.

Almost surely (a.s.)

A statement $S$ about outcomes is said to be true almost surely (a.s.) or with probability $1$ (w.p.1) if $F=\{\omega:S\mbox{ is true }\}\in\mathcal{F}$ and $\mathbb{P}(F)=1$ .

If for $n\in\mathbb{N}$ , $F_{n}\in\mathcal{F}$ and $\mathbb{P}(F_{n})=1$ , then $\mathbb{P}(\cap_{n\in\mathbb{N}}F_{n})=1$ .

( $E_{n}$ , i.o.), $\liminf E_{n}$ ( $E_{n}$ , ev)

For a sequence of events $\{E_{n}\}$ , we define $\begin{eqnarray*} (E_{n},\mbox{ i.o.}) & = & \limsup E_{n}\\ & = & \cap_{m\in\mathbb{N}}\cup_{n\ge m}E_{n}, \end{eqnarray*}$ and $\begin{eqnarray*} (E_{n},\mbox{ ev}) & = & \liminf E_{n}\\ & = & \cup_{m\in\mathbb{N}}\cap_{n\ge m}E_{n}. \end{eqnarray*}$

$I_{\limsup E_{n}}(\omega)=\limsup I_{E_{n}}(\omega)$ .
$I_{\liminf E_{n}}(\omega)=\liminf I_{E_{n}}(\omega)$ .

Fatou Lemma

$\mathbb{P}(\liminf E_{n})\le\liminf\mathbb{P}(E_{n})$ . Also, since $\mathbb{P}$ is finite, $\mathbb{P}(\limsup E_{n})\ge\limsup\mathbb{P}(E_{n})$ .

First Borel-Cantelli Lemma (BC1)

If $\sum_{n\in\mathbb{N}}\mathbb{P}(E_{n})<\infty$ then $\mathbb{P}(\limsup E_{n})=\mathbb{P}(E_{n},\mbox{ i.o.})=0$ .

Chapter 2: Events

Chapter 2: Events

Events

Almost surely (a.s.)

\limsup E_{n} ( E_{n} , i.o.), \liminf E_{n} ( E_{n} , ev)

First Borel-Cantelli Lemma (BC1)

( $E_{n}$ , i.o.), $\liminf E_{n}$ ( $E_{n}$ , ev)