The main heroes in this subject: sequences of random variables (RVs) $X_{1}, X_{2}, \ldots$ (or $\left.\xi_{1}, \xi_{2}, \ldots\right)$. Can view as a single object: $\boldsymbol{X}=\left\{X_{j}\right\}_{j \in \mathbb{N}}$. Key general questions:
– How to specify the distribution of $\boldsymbol{X}$ ? On what space?
– For a sequence of functions $f_{n}: \mathbb{R}^{\mathbb{N}} \rightarrow \mathbb{R}$, what can one say about the behaviour of $f_{n}(\boldsymbol{X})$ as $n \rightarrow \infty$ ?
Ex. For $\boldsymbol{x}=\left\{x_{j}\right\}_{j \in \mathbb{N}}$, one can consider: (i) $f_{n}(\boldsymbol{x})=x_{n} ;$ (ii) $f_{n}(\boldsymbol{x})=\sum_{j=1}^{n} x_{j}$;
(iii) $f_{n}(\boldsymbol{x})=\frac{1}{n} \sum_{j=1}^{n} x_{j} ;$ (iv) $f_{n}(\boldsymbol{x})=\prod_{j=1}^{n} x_{j} ;(\mathrm{v})$ etc.
We need to recall what RVs and RVecs are. We will discuss measurable spaces, their products, product measures, the distribution of a sequence of RVs, fundamental probabilistic laws governing the behaviour of $f_{n}(\boldsymbol{X})$, and the magnificent martingale property (extending in a sense the independence assumptions) and its consequences.
This subject is about replacing hands-waving with bricklaying.

# Probability Fundamentals

Def. A family $\mathscr{A}$ of subsets of a set $\Omega$ is said to be an algebra on $\Omega$ if (A.1) $\Omega \in \mathscr{A}$,
(A.2) $A \in \mathscr{A} \Rightarrow A^{c} \in \mathscr{A}$,
(AF.3) $A_{1}, A_{2} \in \mathscr{A} \Rightarrow A_{1} \cup A_{2} \in \mathscr{A}$.
Remark. If $A_{1}, A_{2} \in \mathscr{A}$ then also $A_{1} \cap A_{2}, A_{1} \backslash A_{2} \equiv A_{1} \cap A_{2}^{c} \in \mathscr{A} .$ [Why? Use de Morgan’s laws.]
Def. A family $\mathscr{F}$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if (A.1) $\Omega \in \mathscr{F}$
(A.2) $A \in \mathscr{F} \Rightarrow A^{c} \in \mathscr{F}$,
(A.3) $A_{1}, A_{2}, \ldots \in \mathscr{F} \Rightarrow \bigcup_{n=1}^{\infty} A_{n} \in \mathscr{F}$
The pair $(\Omega, \mathscr{F})$ is called a measurable space.
Important: if $\left\{\mathscr{A}_{\gamma}\right\}_{\gamma \in \Gamma}\left(\left\{\mathscr{F}_{\gamma}\right\}_{\gamma \in \Gamma}\right)$ is a family of algebras $(\sigma$-algebras) on $\Omega$, then $\bigcap_{\gamma \in \Gamma} \mathscr{A}_{\gamma}\left(\bigcap_{\gamma \in \Gamma} \mathscr{F}_{\gamma}\right)$ is again an algebra (a $\sigma$-algebra) on $\Omega$.

# $\sigma$-algebras.

Recall a very important concept of generated $\sigma$-algebras.
Theorem $1\left(\boldsymbol{P f I}_{14}\right)$ For any family $\mathscr{G}$ of subsets of $\Omega$, there exists a unique $\sigma$-algebra, denoted by $\sigma(\mathscr{G})$ and called the $\sigma$-algebra generated by $\mathscr{G}$, s.t.
a) $\mathscr{G} \subseteq \sigma(\mathscr{G})$, and
b) if $\mathscr{H}$ is a $\sigma$-algebra on $\Omega$ and $\mathscr{G} \subseteq \mathscr{H}$, then $\sigma(\mathscr{G}) \subseteq \mathscr{H}$.

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