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By Harald Bergström (auth.), Daniel Dugué, Eugene Lukacs, Vijay K. Rohatgi (eds.)

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Hold spacings. ) sup 0

PROOF. a) We n e e d show subseauence (ii) x n ~ ~, gn(Xn) of the (iii) + ¢ '0 () xe integers when Xn ÷ x 0 and n ÷ ~ t h r o u g h for the cases: Xn + 0, a n X n ÷ ~' (iv) (i) x n ÷ x 0 E some (0,~), Xn ÷ x0 -< 0, a n X n --

This shows that J Since [~(t)] is ergodic, hence T is ergodic. To relate these results back to our original problem, define the real-valued function h on ~ by h(w(1),w (2)) = (w(1)(0),w~ 2)) = [ ~(0, w (i)) ,y0(w (2)) ]. Then it is straightforward to verify that h[T(w(1),w(2))] = [~(yl(w(2)),w(1)),yl(w(2))] or hT = [~(YI),YI] = (ZI,YI) , and in general hT n = [~(~n),Yn] = (Zn,Yn). Thus we have 22 THEOREM 2. d, positive random variables satisfying A2 and A3, then the [(Zn,Yn) ] process is erg0dic and we can estimate consistently all joint probabilities of the [~(t)] process b_~ means of observation of the [(Zn,Yn) ] process.

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