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

ISBN-10: 3540108238

ISBN-13: 9783540108238

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Read or Download Analytical Methods in Probability Theory: Proceedings of the Conference Held at Oberwolfach, Germany, June 9–14, 1980 PDF

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Extra resources for Analytical Methods in Probability Theory: Proceedings of the Conference Held at Oberwolfach, Germany, June 9–14, 1980

Example text

LEMMA i. +Une; such that uniformly in t e (0,i) [nt] a=l,2 ..... n~ = { [ Uni; 0 ~ t ~ i} i=l {n_½(/t Bn(Y) 1 Bn(Y) 0 I-C~ dy - 0/ l-y t dy 0/ 1 log ~ dy +Op(1)) 30 t + f 0 PROOF. 2) it suffices to show that the sequence of stochastic processes { [ Yi:n/Sn; ~ = 1,2 ..... 10). 11) ; 0 Sn {1 [nt] i=l < t < 1I = -_ In_½[n ~ ] n ½ (i - n-T~) i (-~---n Yi:n_ log = 1 Y. 9). s. 10). Lemma 1 implies Theorem B almost immediately. PROOF OF THEOREM B. 15) Gn(t ) = f 0 t B (y) 1 B (y) n d y - ((l-t)log(l-t)+t)f ~ dy .

Is ergodic, we must have P(1)(A(1)) = 0 or I. 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.

T h e o r e m 2 . The l i m i t i n g =O,j=l ..... - ,lp by f o l l o w i n g the explicit As a c o r o l l a r y , Z . liD. ,ip X2 . T = > ~;11'''"ip il,. ~ip independence) that of a ~ 2 ( r ) , with a degree of freedom P P r = ~ Mj - I M. + p - 1 j=! j=l 3 Proof : It can be noted that the preceding result is very similar to the derivation of the ~ 2 statistic in a multivariate contingency table. Indeed the same methods are then valid (see Everitt, [8]). We will, in the following paragraph, give some examples of the use of the preceding statistics for tests of independence.

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