Some asymptotic results for the periodogram of a stationary time series

Let x 1 , x 2 ,…, x n , be n consecutive observations generated by a stationary time series {x}, t = 0, ±1, ±2,…, with E( x t 2 ) < ∈. The periodogram of the set of observations, which may be defined as a function I n of angular frequency with range [0, π] that is proportional to , plays an important part in methods of making inferences about the structure of {x t }, particularly its spectral distribution function or spectral density.