Author: Eiko

Tags: probability theory, time series, stochastic process, stationary, stationarity

Time: 2025-01-08 10:00:09 - 2025-01-29 01:08:40 (UTC)

Basic Concepts

Let $\{X_t,t\in T\}$ be a stochastic process / time series. We have some concepts of covariance matrices to understand the dependence between them.

• When $\mathrm{Var}(X_t)<\mathrm{\infty }$, we can define auto-covariance function

  $${\gamma }_X(r,s):=\mathrm{Cov}(X_r,X_s)=\mathbb{E}[(X_r-\mathbb{E}{X}_r)(X_s-\mathbb{E}{X}_s)],r,s\in T.$$

• $X_t$ is called stationary or weakly stationary / covariance stationary / second order stationary, if

  • The covariance is time-homogeneous, i.e. ${\gamma }_X(r,s)={\gamma }_X(r+h,s+h)$

  • The expectation is also time-homogeneous, i.e. $\mathbb{E}{X}_t=\mu$ for all $t\in T$.

  • Additionally, all $X_t$ are in $L^2$.

• When we are in the stationary setting, the covariance ${\gamma }_X(r,s)$ can be reduced to a single parameter function ${\gamma }_X(r-s)$, so

  $${\gamma }_X(h)=\mathrm{Cov}(X_{t+h},X_t).$$

  And the auto-correlation is then

  $$\begin{array}{rl}{\rho }_X(h)& :=\frac{\mathrm{Cov}(X_{t+h},X_t)}{\sqrt{\mathrm{Var}(X_{t+h})\mathrm{Var}(X_t)}}\\ & =\frac{{\gamma }_X(h)}{\sqrt{{\gamma }_X(0){\gamma }_X(0)}}\\ & =\frac{{\gamma }_X(h)}{{\gamma }_X(0)}.\end{array}$$

• $X_t$ is said to be strictly stationary if the joint distribution of $(X_{t_1},\dots ,X_{t_k})$ is the same as $(X_{t_1+h},\dots ,X_{t_k+h})$ for all $t_1,\dots ,t_k,h\in T$.