Independent And Stationary Sequences Of Random Variables - Chapter 12 ppsx

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(12. 12.1) = jFn (x_z) g i (z)dz , and we shall consider the expression ( 12. 12.2) F§(x2)-F.(xl) ° Suppose that 82 > 0 ; then jIzI>n~' -e 2 Fn (x- z)g l (z) = B exp(-Znl-2E2) If x l [(1+ 712" '-l)]-z J exp [-2 u 2 /(1+n 2 "' -l )]du(1+208 4) (12 12 7) If 84 is sufficiently small compared with the number ~l in (12. 11 11), then (12. 12 7) contradicts (12 11 9) Thus (12. .. +OPn4 (12 8 1) 12 8 EXPANSION OF R AS A TAYLOR SERIES 237 We take K = [10/(2 -a)] + 1 (12. 8.2) and note that, for p < K, JIyI>yn y"p(y)dy < C 6 exp for sufficiently large (12 7.5), n [-c9n27p5(n)2] < C6n-10 (12. 8.3) Thus (12 8 1) becomes, using (12. 8 2), (12 8 3) and K hp R=1+ E ap -+ 6C7 n 10 , p=2 P! ( 12. 8 4) where the ap are the moments of the Xj (i < n) Moreover, 00 m=E(Xj)=R-1 f- (12 8 5)... (12 8 13) implies that h = m+OC8n-1P2(n)-s-3 (12 8 17) „ 9 Further transformations Turning to (12. 7.1) we now choose h so that (12. 9 1) x=mn2 , where 1 < x< n"/P7 (n) , P7 (n) % P2 (n) 20 • (12. 9 2) 12 9 FURTHER TRANSFORMATIONS From (12 8 17) this implies that h = n-Zx+9Csn-1 p2 (n)-s-3 , 239 ( 12. 9.3) which in view of (12 9.2) is consistent with (12 3 3) Moreover, Rn e -h m" = exp [n (log R -. .. 27r n ©, -n -pi Chap 12 exp [-2 (n+n2a')t2] exp[nKs 0 +3(t) - n+ itx]dt+ +B exp (-E 1 n2a') (12. 11 4) Following the computations of „„ 9 6, 6 that, for 1