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By Geoffrey Grimmett (auth.)

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I. 12). ··, x m) and Y = (Yl' Y2,"" Ym) where each Xi and Yj equals either 0 or 1. Let A and B be increasing events in t§. We define the following events in the product space: we write A' for the set of all points x x Y E r , x r 2 for which x E A, and, for 0::; k ::; m, we write B~ for the set of points x x Y for which the composite vector (Yl, Y2, ... , Yk> Xk+l, ... , x m) belongs to B. We note that A' and B~ are increasing events in the product space. For each point x x Y E r , x r 2 , we say that the subset I of {1, 2, ...

LEI. 36). 34) to obtain d 1 dp Pp(A) :s;; p(1 _ p) Jvarp(N) varp«(t) and part (a) follows from the observation that N has the binomial distribution with parameters m and p, and JA has the Bernoulli distribution. Suppose now that A is increasing. 41) covp(N, JA) = covp(JA, JA) ~ + cOVp(N - JA' JA) varp(IA) by the FKG inequality, since N - JA is an increasing random variable. 37). 38). d. :.. log PI - log P as req uired. 42) by induction on the number of edges which are relevant to A. Suppose that A is increasing and depends only on the edges in the finite set E, and write m = lEI.

Log PI - log P as req uired. 42) by induction on the number of edges which are relevant to A. Suppose that A is increasing and depends only on the edges in the finite set E, and write m = lEI. 42) is valid. 42) is valid whenever m:s;; k, and consider the case when m = k + 1. Let 0 < p < 1, Y ~ 1, and let e be an edge in E. 43) + Pp,(Alm(e) = 0)(1 l)YpY + Pp(Alm(e) = 0)Y(1 - h(pY) = Pp,(Alm(e) = l)pY pY) :s;; Pp(Alm(e) = pY) by the induction hypothesis. 44) xYpY + yY(1 - pY) :s; {xp + y(1 - pW when x ~ y ~ 0; to see this, check that equality holds when x = y ~ 0 and that the derivative of the left-hand side with respect to x is at most the corresponding derivative of the right-hand side when x, y ~ O.

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