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. $\gamma_v(t,\tau ) = \gamma_v(\tau )$ $\forall{t}$: 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.

Intuitively this definition respond to the question: what does it mean for a random process to remain the “same” over time? Obviously, the exact values will be different, since the process is random. However a SSP has the partial indicators that are time invariant and so the characterization of the process does not change with time.

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.

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.

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

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

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: $$\gamma_z(t,\tau)=\mathbb{E}[(z(t)-m_z)(z(t-\tau)-m_z)]= \newline \mathbb{E}[((𝑦_1(𝑡)-m_{y_1})+(𝑦_2(𝑡)-m_{y_2}))((𝑦_1(𝑡-\tau)-m_{y_1})+(𝑦_2(𝑡-\tau)-m_{y_2}))]= \newline \mathbb{E}[(𝑦_1(𝑡)-m_{y_1})(𝑦_1(𝑡-\tau)-m_{y_1})+(𝑦_2(𝑡-\tau)-m_{y_2})(𝑦_2(𝑡-\tau)-m_{y_2})] = \newline \gamma_{y_1}(\tau)+\gamma_{y_2}(\tau)=\gamma_z(\tau) $$ Which is time invariant