Stationary stochastic process

Summary

• A stochastic process \(\{ Y_t \}\) is said to be stationary in the wide sense or weakly stationary or covariance stationary if
• \(E\left( Y_t \right)\) does not depend on \(t\)
• \(Var\left( Y_t \right)\) does not depend on \(t\)
• \(Cov\left( Y_t,Y_{t-k} \right)\) does not depend on \(t\) (it may depend on \(k\) ).
• A stochastic process is said to be stationary in the strict sense if the joint distribution of any sequence \({Y_t,Y_{t+1}, \ldots ,Y_{t+k}}\) does not depend on \(t\) for every \(k>0\) .
• If \(\{ Y_t \}\) is a stationary process, then we define a new demeaned process \(\{ y_t \}\) by subtracting the constant \(E\left( Y_t \right)\) from \(Y_t\) ,

\[y_t=Y_t-E\left( Y_t \right)\]

• \(\{ y_t \}\) is a stationary process as well with \(E\left( y_t \right)=0\) while the remaining moments are the same.