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Extra info for Conditional logistic analysis of case-control studies with complex sampling

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The process {x,}, where (22) is called a finite {one-sided) moving average. Let b0, bx,... be an infinite sequence such that Xjt0^7 < o o . Then the infinite one-sided moving average { x j , where (23) is a well-defined stationary process with mean 0 and variance 7 Using the definition of {x,} a n d the condition o n Yj=ob], one can obtain an explicit expression for since it is assumed that Eet = 0 a n d This expression for yx may be used to show that the condition given in (20) is satisfied since which clearly tends to zero with η uniformly in p.

The characteristic function of a r a n d o m vector ζ is defined as "'*}, (2) where u is a vector of real numbers of the same dimension as z. ) The expectation on the right-hand side of (2) always exists no matter what the distribution of z. Let z(k) = ( x f l , . . , x i k) and zx(k) = ( x f l + T, . . , * i k + T) ; then our previous definition for a stationary time series reduces to

The proof, however, requires some further considerations. 2 2 Returning to the original line of argument assume 0 < σ (οο) < σ χ. Let k oo {k) (k) and let the processes x and u tend to the limiting processes ξΐ and r\x, respectively, in the sense discussed in the previous section of this chapter. Since k) k) = 0 E(x\ u\ ) for all /c, it must be true that Εξίηί = 0 as well. Since {k) {k E(x u lT) k k) = 0 = E(x\ lX ) for all k, any τ > 0, it must be true as well that Thus the whole process is orthogonal to {ί/J.

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