Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 47 / 2 / b / Solution

The even time-series spectral density makes the sine part vanish. At lag zero its integral is the area of the triangle, so ${\gamma }_0={\pi }^2$. For any nonzero integer $k$, integration by parts givesTherefore the triangular spectral density of a stationary time series has autocovariance functionNo additional factor $1/(2\pi )$ belongs in this inverse integral under the convention stated above.