A paradox on the spectral representation of stationary random processes
aa r X i v : . [ s t a t . O T ] D ec A paradox on the spectral representation of stationaryrandom processes
Mohammad Mohammadi , Adel Mohammadpour ∗ ˙and Afshin Parvardeh Faculty of Mathematics & Computer Science, Amirkabir University of Technology Department of Statistics, University of Isfahan
Abstract
In this note our aim is to show a paradox in the spectral representation ofstationary random processes.
A random process X = { X t , t ∈ Z } with mean zero and finite variance is called stationaryif E ( X t X s ) = E ( X t + h X s + h ), for every t, s, h ∈ Z . We define the function γ ( . ) as, γ ( h ) = E ( X h X ).The spectral representation theorem says that for every stationary random process X = { X t , t ∈ Z } there exist a complex right continuous orthogonal-increment process Z such that, X t = Z ( − π,π ] e itx d Z ( x ) , with probability one. Therefore, γ ( h ) = Z ( − π,π ] e ihx d F ( x ) , (1)where F ( x ) = E | Z ( x ) − Z ( − π ) | , for every − π < x ≤ π . For more details see Brockwelland Davis (1994) Chapter 4.Let X t be a real stationary random process. From the spectral representation theoremwe have, X t = Z ( − π,π ] cos( tx )d Z ( x ) − Z ( − π,π ] sin( tx )d Z ( x ) , (2) ∗ Corresponding author, email: [email protected] Z is real part and Z is imaginary part of Z . Let F ( x ) = E | Z ( x ) − Z ( − π ) | , F ( x ) = E ( Z ( x ) − Z ( − π )) and F ( x ) = E ( Z ( x ) − Z ( − π )) ,therefore, F ( x ) = F ( x ) + F ( x ) , (3)for every − π < x ≤ π . From (2), we have X = R ( − π,π ] d Z ( x ) , with probability one.Hence V ar ( X ) = Z ( − π,π ] d F ( x ) . (4)On the other hand, from (1) for real stationary random process X , we have γ ( h ) = Z ( − π,π ] cos( hx )d F ( x ) . Now, from (3) we have