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~ ~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~ ~~ ~~~~~~~~~~~ ~~~~~~~~~ ~ ~~~ ~~~ ~~~~~~~ ~~~~~~ ~~~~ ~ ~ ~~~ ~~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~ ~~~~~ ~~~~~~~~~~~~~ ~ ~~~~~~~ ~ ~ ~ ~~ ~~ ~~~ ~~~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~ ~~~~~ ~~~~~~~~~ ~ ~~~ ~~ ~~~ ~ ~~~ ~~~~~~~~~~~ ~~~~~~ ~~ ~~~ ~~~ ~~ ~~ ~ ~~~~ ~ ~~~ ~ ~~~ ~~ ~~~ ~~~~~~~ ~ ~ ~~~ ~~~ ~ ~ ~~~~~~~~ ~~ ~~~ ~~~~~~~ ~~~ ~~~~~~~~~~~~~ ~ ~ ~~~ ~ ~~~ ~ ~ ~~ ~~~~ ~ ~~~~~~~~~ ~ ~ ~~~ ~ ~ ~~~ ~ ~~~ ~~~~ ~ ~~~~~ ~~ ~~ ~~~ ~~~~ ~ ~~~ ~~~ ~~~~~~ ~~~ ~~~ ~~~~~ ~~~ ~~~~~~~~~~~~~~~ ~~~ ~~ ~~~ ~~~ ~~ ~~~ ~~~ ~~ ~~~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~~~~ ~~~~ ~~~~~~~~~ ~ ~~~~~~ ~ ~~ ~ ~~~ ~ ~ ~ ~ ~~~ ~ ~ ~ ~ ~~~ ~~~~~~~ ~~~ ~ ~ ~~~ ~ ~~~ ~ ~~~~ ~ ~~~~ ~ ~~~~~~~ ~ ~ ~ ~ ~ ~~~~ ~~~~~ ~~~~~~~~ ~ ~~~ ~~ ~~~~ ~~~~~~~~~~ ~~ ~~~~~~~~~~ ~~~ ~ ~ ~~~~~~~ ~~ ~~~ ~ ~~~~ ~ ~ ~~~ ~~ ~~~~ ~~~~ ~~~ ~ ~~~~ ~~~~~~ ~~~ ~ ~~~~~~~ ~ ~~~ ~ ~~ ~ ~ ~ ~ ~ ~ ~~ ~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~~~~~~~~ ~~~ ~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~ ~ ~~~~~~ ~ ~~~~~~~~~~ ~~ ~~~~~ ~ ~~~~~~ ~~~ ~~~ ~~~~~ ~ ~ ~ ~~~ ~~~~ ~~ ~ ~~~ ~ ~~~ ~~~~~~~ ~ ~~~ ~ ~~~ ~~ ~ ~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~ ~ ~~~ ~ ~~~ ~~~ ~~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~~~~ ~ ~~~~~~~ ~ ~~~~ ~~~ ~~ ~~~~ ~~~~~~~~~ ~~~~~~~ ~ ~~~~~~~~ ~ ~~ ~~~ ~~~ ~ ~ ~~~~~~~ ~~~ ~~ ~~~~ ~~ ~~~~~ ~~ ~~~~ ~~ ~~~~ ~~~ ~~~~~~~~~~~~~~ ~~ ~~~~~ ~~~ ~~~ ~ ~~~~ ~~ ~~~~ ~ ~ ~~~~ ~~~ ~ ~~~ ~~~ ~ ~~~~~~~~~~ ~ ~~~~~~~ ~ ~~~~ ~~~ ~ ~~~~~~~~ ~~~ ~~ ~~~~ ~ ~ ~~~ ~~ ~~~~ ~~ ~ ~~~ ~ ~~~ ~~~ ~ ~~~ ~~~~~ ~ ~~~~~~~ ~ ~~~~ ~~~~~~ ~ ~~ ~~~~~~~ ~~~~~~~~~~~ ~ ~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~ ~~~ ~~~~~ ~ ~~~ ~~~ ~ ~~~~~ ~ ~~~ ~~ ~~~~ ~~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~ ~~~~ ~~~ ~~~ ~ ~ ~~~~~ ~ ~ ~ ~~~ ~~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~ ~~~~~ ~ ~ ~~~~~ ~ ~ ~ ~ ~~ ~~~~~~~ ~ ~ ~ ~~~~~~~~~~~~ ~~~~~~~~ ~~ ~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~~~ ~~~~~~~ ~ ~~~~~~ ~ ~~~~~~~ ~ ~ ~~ ~~ ~~~ ~~~ ~~ ~~~ ~ ~ ~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~~~ ~~~~~ ~ ~~~ ~~~ ~ ~ ~~~ ~~~ ~~~ ~ ~ ~ ~~~~~ ~~~~ ~~~ ~ ~ ~ ~~~~~~~ ~~~ ~~~~~~~~~~~~~ ~ ~~~~~~~~~~ ~ ~~~~~ ~ ~ ~ ~~ ~~ ~~~ ~~~~~ ~~~ ~~~ ~~ ~~~~~~ ~~~ ~~~~ ~ ~~~~~ ~~~~~~ ~~~~~~ ~~~~~~~ ~~~~~~~ ~ ~~~~~~~~ ~~ ~ ~ ~ ~ ~~~~~ ~~~~ ~~ ~ ~~ ~~~~~ ~ ~ ~~~~ ~~~~~~~ ~ ~~~ ~~~~ ~ ~~~ ~~ ~ ~~~~~~~~ ~~~ ~~~~ ~ ~~~ ~~ ~ ~~~~ ~ ~~~~~ ~~~~~~~~~~~ ~ ~~~~~ ~ ~~~ ~~~~ ~ ~~ ~~~~ ~ ~ ~ ~~ ~~ ~~~ ~~~~~ ~ ~ ~~~~~~~~~ ~ ~ ~ ~~ ~~ ~ ~ ~ ~~ ~~ ~~~ ~~~ ~~~~ ~ ~ ~~ ~ ~ ~~~ ~~~~ ~~~ ~ ~~~ ~~ ~~~ ~ ~ ~~~~~~ ~ ~~~~ ~~~ ~~~ ~~~~~~~~~~~~~ ~~~ ~~~~~~~~ ~ ~ ~~~ ~~~ ~~~ ~~ ~~~ [...]... o(1) e -2 "Z du (11 11 5) x as n-> oo, if x satisfies (11 11 2) Thus P(n-2S©+n-ZY©>xjn-ZIY©I < n 2 " - 2)=P(n - S©+n - Y©>x)(1+o(1))+0 (1) f x e - Z" 2 du (11 11 6) assuming (11 11 2) Moreover, © P(n S©>x) i P(n S©+n-+Y©>x+n2«-Z In -ZI 1, I< n 2a-2) and 2a-2< 3-2 x) e - -j "z du 2m -l ( )z fx (11 11 8) Then, from (11 11 6) and (11 11 7), P(n -2 S©>x)... e -2 "2 du+o(1) f~e -2 " 2 du x +nz a '/z - x (11 11 9) Writing 2 - 2 , we have y=n " x+y e -2 u2 du = By exp (-2 x 2 ) exp (xy)(1 +o(1)) x Since a < 6, xy < n3 ,-1 2 = n - Ea , n-ES y= n 2a2< n-'< x -1 (11 11 10} 214 NARROW ZONES OF NORMAL ATTRACTION Chap 11 and so x+y e '"2du=Bx-1 eXp ( - 2 x 2) 11 - ES (11 11 11) x In view of this, (11 11 9) gives P(n - ZS n >x) >, (I +o(1))(2rc )-2 e -2 " 2 du (11. .. THEOREM 11 11 213 we show that, if this is a zone of normal attraction for Z©, then it is a zone of normal attraction for Zn We have Y©EN(0, n"), Y©n - zeN(0, n' - (11 11 3) and P(n - 2 S©+n - Y©>x) _ =P(n - DIY©I