2.9 Second-order description of stochastic processes

Second-order description

Practical and incomplete statistics

Mean

The mean function of a random process $X_t$ is defined as the expected value of $X_t$ for all $t$ 's.

$${\mu }_{X_t}()=(X_t)=\begin{cases}\int_{()} \,d x & \text{if $$}\end{cases}$$∞ ∞ x f X t x continuous k ∞ ∞ x k p X t x k discrete

Autocorrelation

The autocorrelation function of the random process $X_t$ is defined as

$$R_X(t_2, t_1)=(X_{t_2}\overline{X_{t_1}})=\begin{cases}\int_{()} \,d x_2 & \text{if $$}\end{cases}$$∞ ∞ x 1 ∞ ∞ x 2 x 1 f X t 2 X t 1 x 2 x 1 continuous k ∞ ∞ l ∞ ∞ x l x k p X t 2 X t 1 x l x k discrete

If $X_t$ is second-order stationary, then $R_X(t_2, t_1)$ only depends on $t_2-t_1$ .

If $R_X(t_2, t_1)$ depends on $t_2-t_1$ only, then we will represent the autocorrelation with only one variable $\tau =t_2-t_1$

Properties

1. $R_X(0)\ge 0$
2. $R_X(\tau )=\overline{R_X(-\tau )}$
3. $\left|R_X(\tau )\right|\le R_X(0)$

Wide Sense Stationary

A process is said to be wide sense stationary if ${\mu }_X$ is constant and $R_X(t_2, t_1)$ is only a function of $t_2-t_1$ .

If $X_t$ is strictly stationary, then it is wide sense stationary. The converse is not necessarily true.

Autocovariance

Autocovariance of a random process is defined as

$$C_X(t_2, t_1)=((X_{t_2}-{\mu }_X(t_2))\overline{X_{t_1}-{\mu }_X(t_1)})=R_X(t_2, t_1)-{\mu }_X(t_2)\overline{{\mu }_X(t_1)}$$

The variance of $X_t$ is $\mathrm{Var}(X_t)=C_X(t, t)$

Two processes defined on one experiment ( [link] ).

Crosscorrelation

The crosscorrelation function of a pair of random processes is defined as

$$R_{XY}(t_2, t_1)=(X_{t_2}\overline{Y_{t_1}})=\int_{()} \,d y$$∞ ∞ x ∞ ∞ x y f X t 2 Y t 1 x y

$$C_{XY}(t_2, t_1)=R_{XY}(t_2, t_1)-{\mu }_X(t_2)\overline{{\mu }_Y(t_1)}$$

Jointly Wide Sense Stationary

The random processes $X_t$ and $Y_t$ are said to be jointly wide sense stationary if $R_{XY}(t_2, t_1)$ is a function of $t_2-t_1$ only and ${\mu }_X(t)$ and ${\mu }_Y(t)$ are constant.