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Let us estimate the second one. ;; -exp(x2/2)/x. ;; C3As/2. 24). 8. S{3. ;; 2 [B =: If we put (3 =: c11ln I1n cl lIn cl cI 1-{3 + {3 + c( 8 0) c2 1ln cI 2 . S{3 + c3· 2/7 then the Theorem is proved with I =: 5/7. Notice that the simulation algorithm of the transition function of a process with shifted centres may be developed with only a little more complication than a traditional one of the uniform distribution on the sphere. It should be remembered that the problem of checking if a shifted ball is a part of D is in 30 CHAPTER 1 general more complicated than in the traditional method.

1- 1 ::::; y) = X2 -1 2k-l 1 -Jrr1In [ 2(k-l) J . 2(k-l) . -x2 = d [ -In(a(r-l)) 1'-1 -1'- - 2 In 2 dx 1'+1 J + -1'- In(r+l) , where k ::::; km' a ::::; am ::::; 1/(2(k -1»; d ::::; dm ::::; am(2k -1); l' = rm = (1 + 2(k -1) X x exp(-A)/h. As Bl = -(1' -1) In(a(r -1»/1' ~ (k -1) A In(2k -1) exp( -A) then for all A such that A ~ 2 In(2(k-l) In(2k-l)/8) the inequality B ~ 8/2 holds. On the other hand, if we put A = M In kn then 1'+ 1 B2 = - - In(r+l) - 2 In 2 [ + 7k -IJ [ + k-lJ k l' ~ 2 M In 2 2 In 2.

C· w(e:). i=O Thereby Tl~ is a c· w(e:)-biased estimator of u(x). If Exv < 00 then the estimator is realizable. 5 is an outline of the method by which the estimators of solutions of a large number of problems may be constructed. In a concrete situaEx 19 MARKOV PROCESSES AND INTEGRAL EQUATIONS tion some conditions of the Theorem may be weakened or changed. In Chapter 2 we shall construct a realizable estimator of a solution of the Dirichlet problem for the sufficiently arbitrary elliptic operator.

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