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.


Gaussian processes

Moments of stochastic processes

Autocovariance and autocorrelation


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


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.