D.S.G. Pollock 255The assumptions regardingdA(ω)anddB(ω)are analogous to those regarding the
random variablesαjandβj, which are their prototypes. It is assumed thatA(ω)and
B(ω)represent a pair of stochastic processes of zero mean, which are indexed on
the continuous parameterω. Thus:
E{
dA(ω)}
=E{
dB(ω)}
=0. (6.33)It is also assumed that the two processes are mutually uncorrelated and that non
overlapping increments within each process are uncorrelated. Thus:
E{
dA(ω)dB(λ)}
=0 for all ω,λ,E{
dA(ω)dA(λ)}
=0ifω =λ,E{
dB(ω)dB(λ)}
=0ifω =λ.(6.34)The variance of the increments is given by:V{
dA(ω)}
=V{
dB(ω)}
= 2 dF(ω). (6.35)The functionF(ω), which is defined provisionally over the interval[0,π],is
described as the spectral distribution function. The properties of variances imply
that it is a non decreasing function ofω. In the case where the processy(t)is purely
random,F(ω)is a continuousdifferentiablefunction. Its derivativef(ω), which is
nonnegative, is described as the spectral density function.
The domain of the functionsA(ω),B(ω)may be extended from[0,π]to[−π,π]
by regardingA(ω)as an even function such thatA(−ω)=A(ω)and by regarding
B(ω)as an odd function such thatB(−ω)=−B(ω). Then,dZ∗(ω)=dZ(−ω),in
accordance with (6.32). From the conditions of (6.34), it follows that:
E{
dZ(ω)dZ∗(λ)}
=E{
dZ(ω)dZ(−λ)}
=0ifω =λ,E{
dZ(ω)dZ∗(ω)}
=E{
dZ(ω)dZ(−ω)}
=dF(ω),(6.36)where the domain ofF(ω)is now the interval[−π,π].
The sequence of the autocovariances of the processy(t)may be expressed in terms
of the spectrum of the process. From (6.36), it follows that the autocovariance of
y(t)at lagτ=t−sis given by:
γτ=C(yt,ys)=E{∫ωeiωtdZ(ω)∫λeiλsdZ(λ)}=∫ω∫λeiωteiλsE{
dZ(ω)dZ(λ)}=∫ωeiωτE{
dZ(ω)dZ∗(ω)}=∫π−πeiωτdF(ω).(6.37)