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By Dominici F., Zanobetti A., Zeger S.L.

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Corresponding results hold in the r vector-valued case. Suppose, for example, that X(t) = {Xl(t) . . . , dT(A)} may be shown to be asymptotically NC(0, ST) with the entry in row j and column k of ~a- being )~k(A)27r f dp~(t)qb[(t)dt. In the case that ~bj = th for all j, the covariance matrix of the large sample distribution is seen to be proportional to f(h), the spectral density matrix of the series. 7). A distinct advantage of working with the Fourier transform is that the large sample results are the same for the frequency-domain statistics, whereas time-domain statistics have drastically differing appearances and properties.

Brillinger function, in the case that X(t), t = 0 . . is a sequence of independent identically distributed random variables with finite variance. In related work, Chen and Hannan (1980) prove that the empirical distribution of the standardized values [dr(2¢rs/T)12/(2zrTf(21rs/T)), s = 1 , . . e. the exponential). There are situations in which one is interested in A s In probability and almost sure bounds are given in Whittle (1959) and Brillinger (1975a) for example. 1) is considered in Fisher (1929) and Whittle (1954).

X(n) is gl,n(v)= ~ ajx*(v+j). 1) is often attributed to W. P. Cleveland (1972) but it is actually a special case of a simple fact about geometry. If $1 and $2 are two subspaces with $1 C $2 and P1 and P2 are projection operators onto $1 and $2 then PlY = P1P2Y. That is, to project o n Sl we can project onto $2 first and then project the result onto Sl. To see this note that Y = P2Y + 02 where 02 is in the orthogonal complement of S2 and therefore in the orthogonal complement of Sl. Then P1P2Y = PI(Y - 02) = PlY - Plo2 = Ply.

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