... formula to (6. 1 5), we obtain the following local limit theorem if xm =o(n) as n-+oo, then b (m, n , p) - (2rcnpq) ' exp { - ixm - ( xm) } , (6 6) where ~(x) = nPq v=3 ~ xv Pv- 1-( -q)v-1 v(v-1) (npq) ... DEVIATIONS Chap Theorem 6. 1 If x>,0 and x = o (n ) as n + oo, then P(Z„>x) - exp x3 - 7x n2 \n ) 1-G(x) 1+O C (6. 1.9) x+l~ n2 and P (Zn < - x) G(-x) x3 = e...
... >O and f (2r - 1) ( Tk ) = v(2r- 1) (Zk) Then ~ 00 x 2(r -1 ) sin (2T'kx)dF(x )- f(2r-2) (0)-f(2r-2)(,tk)=2 ( -1 )r- - 00 = V (2r-2)( ) - v(2r-2)('Lk) - Arguing as before, 00 x rdF(x) - Co < cc and ... we use the fact that 1- k (t) = for I t1 < 2T, so that 11 h ( 1- k)ll =11 (f-g)u ( 1- k )11 , for any function u c V with the property that u(t) =11 -it for jtj...
... that Ih1(t)I SBN , so that (4. 4.15) implies that 00 lI < I BN exp{-c(2ma-2)} - ~Ih (t)I dt=o(1) l (4. 4.16) Combining (4. 4.13), (4 4. 14) , (4 4.16) we obtain (4. 4.11) Since S ... ()a m b n- m = m-na
... a, and therefore, as x-+ oo, F„(-x) = o(x -s) , G(-x) = o(x -a) , 1- F„ (x) = o (x - a) , 1- G (x) = o (x - a) Taking so that p > S -1 > a -1 , we have IF„(x)-G(x)IP= o(Ixl -Pa) so that (Fn - ... 1
... (- cn2 z) = exp (- cn2 Re z) < , (7. 2.15) I and therefore = O(exp(-nr7 (E ))) + (7. 2.16) JL on L2 and L4 Moreover, lexp (- un- z) I = exp (- cn - Re z) < , 1 (7 2.15) and therefore = O(exp(-ni ... (27r/nK"(z o))2 (1 +Bn -1 log8 n) Thus the first term on the right-hand side of (7. 2.25) is equal to a( 2itK"(zo)) -I- exp In (K(zo) - aTzo)}(1+Bn -1 log8 n) ,...
... R-1 f V(x-z)e-hwdWn(z) _ §~ n +1 x z J - 00 dWn(z)e-hz J - 00 Making the substitution ~ =1 1- z, R "+1 x f- Go Jf dW" (z) e -hz J 00 x =Rn+1 (8 2 _8) e-h(''-z)dn e -hsdV(~) (( 8) becomes V (h-Z) ... exp(-hdn-Iy)dFP(y)=(2n )- exp(-hin#y-iy )dyJ0 -Q n (0)+h6n* Jo e xp(-hdnly)Qn (y)dy (8 11) The last integral in (8. 3 11) is o Bn - hQnz exp (-hin y) dy = Bn - , (8. 3 12)...
... have (cf (9 5.2)) pn (x) n-u n2 _ - 7r exp (-4 nt +nKm (t)-n+itx)dt+B exp(-E2 n2 a) f-n-P (10 10) „ Application of the method of steepest descents The integrand of (10 10) is an entire function, ... ( 10. 4.11) = Bn a -2 Moreover, since z= ±in Re [n(?z 2- •rz)] =2n(~2-n-2 -2 •t ~) < in' u= - n2« , (10. 4.12) since ~ < n - '`/ p (n) and i < n - °/ p (n) (10...