Asymptotic properties of certain functionals of the periodogram

The asymptotic properties of the periodogram of a weakly stationary time series for the triangular array of fundamental frequencies is studied. For linear Gaussian processes, results are obtained relating the asymptotic distribution of certain Riemann sums of the periodogram of the process to those of the periodogram of the innovation process. PERIODOGRAM; WEAKLY STATIONARY TIME SERIES; GAUSSIAN; ASYMPTOTIC DISTRIBUTION 1. Background A weakly stationary non-deterministic time series {X,, t = 0, ? 1, 2, ... can be shown to have a linear representation X, = ,= o cEt-, where co = 1 and {e,} the so called innovation process consists of uncorrelated variables with Set = 0, varEt, = a'. The spectral density of the process is then given by S(f) = a2A(f) A(f) where A(f) = c,exp (27rifu). A time series {Xt} is said to be Gaussian if for any set of time points {t,, t2, "", tkQ, then {Xt,,1X2, *X, ,Xtk} are jointly normally distributed. Further, if a Gaussian time series is non-deterministic, so that it has a moving average representation, and if the innovations {et) are independent, then the innovation process is also Gaussian. The usefulness of this comes from Fisher (1929) where it is shown that for any sequence of independent, normally distributed random variables with mean zero and finite variance a2, then the periodogram intensities, I,(v/n), v = 1, **., [J(n 1)], are independent and have exponential distributions with a2 as their common mean. The objective of this section then is to relate, asymptotically at least, the properties of the periodogram of the Gaussian time series {Xt} to the known properties of the periodogram of the innovation process, It is useful then to define It is useful then to define Received in revised form 8 October 1973. 578 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 06:02:46 UTC All use subject to http://about.jstor.org/terms Asymptotic properties of certain functionals of the periodogram 579 T(f) = Inx(f)S(f)l,(f), 0, 5'-= I then p then im sup = 0. In the present paper, we need a result which is slightly stronger than this. Lemma 1. If {X,} is a weakly stationary, non-deterministic time series with the linear representation X, = '=o cus, such that (i) {e,} is Gaussian and (ii) ~u, I c I 0, then sup T,(A) = o,(n-' ) for every 6' < 6. Proof. The proof is an adaptation of the proof of Walker's (1965).