Stationary of cy

Problem

Suppose that \(y_1, \ldots ,y_T\) is a stationary process. Define a new process \(x_1, \ldots ,x_T\) from

\[x_t=cy_t\]

for \(t=1, \ldots ,n\) where \(c\) is a given constant. Show that \(x_t\) is a stationary process.

Hint: For two random variables \(X\) and \(Y\) and for constants \(c\) and \(d\) we have

\[cov\left( cX,dY \right)=cdCov\left( X,Y \right)\]

Solution

We must show that

• \(E\left( x_t \right)\) does not depend on \(t\)
• \(Var\left( x_t \right)\) does not depend on \(t\)
• \(Cov\left( x_t,x_{t-k} \right)\) does not depend on \(t\) (it may depend on \(k\) )
• Since

\[E\left( x_t \right)=E\left( cy_t \right)=cE\left( y_t \right)\]

and \(E\left( y_t \right)\) does not depend on \(t\) ( \(y\) is stationary) it follows that \(E\left( x_t \right)\) does not depend on \(t\) .

• Since

\[Var\left( x_t \right)=Var\left( cy_t \right)=c^2Var\left( y_t \right)\]

and \(Var\left( y_t \right)\) does not depend on \(t\) ( \(y\) is stationary) it follows that \(Var\left( x_t \right)\) does not depend on \(t\) .

• Since

\[Cov\left( x_t,x_{t-k} \right)=Cov\left( cy_t,cy_{t-k} \right)=c^2Cov\left( y_t,y_{t-k} \right)\]

and \(Cov\left( y_t,y_{t-k} \right)\) does not depend on \(t\) ( \(y\) is stationary) it follows that \(Cov\left( x_t,x_{t-k} \right)\) does not depend on \(t\) .