Learning Notes

# Definition

### Stochastic Process

A stochastic process $X_T$ is defined to be a function that maps the set $T$ to random variable $R_v$.

$\displaystyle X_T : T \mapsto R_v$

### State Space

Let $S$ and $B$ be the values of the random variables and B be its Borel field. State space is denoted by $(S, B)$

### σ-Algebra/σ-Field

Let  $\sigma$ be the $\sigma$-field of some set $\mathcal{F}$, it then must follow

1. $\emptyset \in \sigma$
2. $\text{if } A \in \sigma , \text{then} A^c \in \sigma$
3. $\text{if } A_i \subset A, \text{then} \bigcup_i A_i = A$

E.g. $\mathcal{F} = {a, b, c, d, e}$ then an example of the sigma-algebra generated from $\mathcal{F}$ is $\sigma(\mathcal{F})=\{\emptyset, \{a, b\}, \{c, d, e\}, \{a\}, \{b, c, d, e\}, \{a, b, c, d, e\}\}$.

### Borel Field

Borel Field or Borel-sigma-algebra generated from $\mathcal{R}$ is defined as the sigma-algebra of all open subsets of $\mathcal{R}$.

### Probability Space

$( \Omega, \mathcal{F} , P)$ defines a probability space

$\Omega \text{ is the sample space} \\ \mathcal{F} \text{ is the } \sigma \text{-algebra subsets of event in sample space } \Omega \\ P \text{ is the probability measures}$

### Probability Measure

Probability measure P is a function which associates a number P(A) to each set $A\in \mathcal{F}$ with the following properties:

1. $P(A) \in [0, 1]$
2. $P(\Omega) = 1$
3. Associative properties on disjoint sets $A_i \in \mathcal{F}$, that is

$\displaystyle P\left(\bigcup_i A_i \right) = \sum_{i} P(A_i)$

Note: See $P(\mathcal{F})$ as the probability that the events $\mathcal{F}$ occurs.