Stationary stochastic processes (SSP)

A stochastic process $v(t,s)$ is stationary if:

1. $m_v(t)=m_v$ $\forall{t}$: the mean function is a constant
2. $\lambda_v(t,\tau ) = \lambda_v(\tau )$ $\forall{t}$: the covariance is function only of the time lag $\tau$, not of the time $t$. In summary the partial indicators of SSP are time invariant.

Importance of SSPs

• SSPs are typical of many practical situations
• SSPs are easier to study
• Non stationarity typically arises as an additive contribution: $v(t,s) = v_{SSP}(t,s) + v_{NSSP}(t,s)$ so the study of SSPs is useful also when modeling non stationary processes.

Questions from past exams

Specify which conditions the mean function and the covariance function must satisfy to say that the process is (weak-sense) stationary. Explain intuitively what stationarity means.

See above.

Give the definition of (weak-sense) stochastic process and then the definition of (weak sense) stationary stochastic process. Let $y_1(t)$ and $y_2(t)$ be two stationary stochastic processes that are uncorrelated each other (that is, $\mathbb{E}[(y_1(t) − m_{y_1}(t))(y_2(r) − m_{y_2}(t))] = 0$, $\forall t, r$). Prove that the process $𝑧(𝑡) = 𝑦_1(𝑡) + 𝑦_2(𝑡)$ is stationary too.

See above for the definition of stationary stochastic process.

In order for $𝑧(𝑡) = 𝑦_1(𝑡) + 𝑦_2(𝑡)$ to be stationary then:

1. $m_z(t)=m_z$ $\forall{t}$
2. $\lambda_z(t,\tau ) = \lambda_z(\tau )$ $\forall{t}$

If we compute the mean function of $z$: $$m_z(t)=\mathbb{E}[z(t)]=\mathbb{E}[𝑦_1(𝑡) + 𝑦_2(𝑡)]=\mathbb{E}[𝑦_1(𝑡)]+\mathbb{E}[𝑦_2(𝑡)]=m_{y_1}+m_{y_2}=m_z$$

While the covariance function: $$\lambda_z(t,\tau)=\mathbb{E}[(z(t)-m_z)(z(t-\tau)-m_z)]=\mathbb{E}[(𝑦_1(𝑡)-m_{y_1}+𝑦_2(𝑡)-m_{y_2})(𝑦_1(𝑡-\tau)-m_{y_1}+𝑦_2(𝑡-\tau)-m_{y_2})]=\ \mathbb{E}[(𝑦_1(𝑡)-m_{y_1}+𝑦_2(𝑡)-m_{y_2})(𝑦_1(𝑡-\tau)-m_{y_1}+𝑦_2(𝑡-\tau)-m_{y_2})]$$