ABSTRACT
Let {X1(t)}, {X2(t)},... be a sequence of independent samples from {X(t), t Î G}. Then for each Î v and , Î Hv (v < ),
N
áXi( t ) ,Xi( t)ñH á ( ) , ñHv dm × m(t, ) (43) is a consistent estimator for áK( ) , ñHv, in the sense that
E|Kn,N( ; , ) –áK( ) , ñH v|2 0 as n, N . (44)
REMARK A Moore group ([62]; [28], p. 15) is a locally compact group all of whose irreducible unitary representations are finite dimensional. The structure of such groups was described by C.C. Moore in [42]. Let G be a Moore group and {X(t), t Î G} be an asymptotically stationary process on G satisfying the conditions of Theorem 6.8. Then we may view that theorem as providing a consistent estimator for the discrete part of the asymptotically stationary covariance K(t). This notion of the discrete part can easily be made precise (for a group that is not necessarily a Moore group).
