# Stochastic processes and their moments

## Introduction to processes

### Stochastic processes

In a stochastic process we have a mapping from a variable (time) to a random variable.

#### Discrete and continuous time

Time could be discrete, or continuous.

Temperature over time is a stochastic process, as is the number of cars sold each day.

#### Discrete and continous state space

The state space for temperature is continous, the number of people on the moon is discrete.

### Stochastic evolution

We can describe processes by their evolution.

\(p(x_t|x_{t-1}...)\)

### Gaussian processes

### Moments of stochastic processes

### Autocovariance and autocorrelation

#### Autocovariance

\(AC(a,b)=cov(X_a, X_b)\)

#### Autocorrelation

The autocorrelation between two time periods is their covariance, normlised by their variances

\(AC(a,b)=\dfrac{E[(X_a-\mu_a)(X_b-\mu_b)]}{\sigma_a \sigma_b}\)

This is also called serial correlation.