By Robert J. Serfling

This paperback reprint of 1 of the easiest within the box covers a vast diversity of restrict theorems invaluable in mathematical information, in addition to tools of facts and methods of software. The manipulation of "probability" theorems to acquire "statistical" theorems is emphasised.

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I lim F,,(t) = F(t), each continuity point t of F, n+ OD and we introduced the notation X, 5 X and alternate terminology “weak iconvergence of distributions” and notation F,, =F. 40 PRELIMINARY TOOLS AND FOUNDATIONS We next consider a condition equivalent to (*) but expressed in terms of PI,Pz,... and P. First we need further terminology. For any set A, the boundary is defined to be the closure minus the interior and is denoted by 8A. For any measure P, a set A for which P(8A) = 0 is called a P-continuity set.

2. D. ) the stochastic process which is a random element of the space C[O, 1). When convenient,we suppress the o notation. ) and y = y ( . ) in C[O, 13. Denote by A? the class of Bore1 sets in C[O, 11 relative to p. Denote by Q,,the probability distribution of Y , ( . ) induced by the measure P through the relation BEA?. ). This will be given by Donsker’s Theorem below. ) is the Wiener measure, that is, the probability distribution of one coordinate of the random path traced by a particlein “Brownianmotion,”or formally the probability measuredefined by the properties: &(a, (a) W ( { x ( .

Th),all r. lim #&Al,. n+ w . With t = 1, and since At,. 1A that X,, % X. The converse is proved by a similar argument. H Some extensions due to Wald and Wolfowitz (1944) and to Varadarajan (1958) are given in Rao (1973), p. 128. Also, see Billingsley (1968), p. 3 Uniformity of Convergence in Distribution An important question regarding the weak convergence of F,,to F is whether the pointwise convergences hold uniformly. The following result is quite useful. Theorem (Pblya), f'f F, * F and F is continuous, then lim supIF,(t) ,-+a - F(t)I = 0.

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