Preprint

Application of Proposition 2.8 and simplification. From (2.30), we see that the probabilities p j .r/satisfy the condition of the Proposition 2.8, which, together with the two estimates,e 1.r/ 1.r/ mm!1 2 m m.m 1/C 1 .r/ 1 .r/ 2D O. 1 .r/ 1=2 /; (2.31)D 1 jm 1 .r/ C O 1 .r/j 2 C jm 1 .r/j; 1 .r/ 2(see (2.19)) for m 0, implies that P.S n .r/ D m/ D Q 1 C O.E 2 /, whereQ 1 WD e 1.r/ 1.r/ m1 C 2.r/ 11C p pm! 2 1 .r/ 21 .r/ 1 2 .r/= 1 .r/2 .r/E 2 WD 1 .r/1! 2 .r/;2 1 .r/ 2jm 1 .r/j 2 C 1 1 .r/ 3=2 C 2.r/.jm 1 .r/j C jm 1 .r/j 2 / C 3 .r/ 1 .r/ 5=2 : (2.32)Now from (2.11) and (2.29) it follows thatjm 1 .r/j D Ox1 2 C x2m.m 0/: (2.33)By substituting this, (2.27), and (2.30) into (2.32), we deduce that E 2 D O.E 3 /, whereE 3 WD 3=2 2 C 431 C x2.1 C x 4 /: mOn the other hand, by the estimatewe havee 1.r/ 1.r/ mm! 1jmp21 .r/ D O 1 .r/j 2 C 1 1 .r/ 3=2.m 0/;Q 1 D e 1.r/ 1.r/ m 1jm pm! 1 2 .r/= 1 .r/ C O 1 .r/j 2 C 1; 1 .r/ 3=2where the O-term is also bounded above by E 3 . Thus we obtainFurther simplification.haveP.S n .r/ D m/ D e 1.r/ 1.r/ m 1 pm! 1 2 .r/= 1 .r/ C O.E 3/: (2.34)Consider first the dominant term on the right-hand side of (2.34). By (2.33), we.m 1 .r// 2mD O 22! x4 1 C x2D O. 1=3 / D o.1/: (2.35)mThuse 1.r/ 1.r/ mm!D 1 mm! eD 1 mm! e mexp m log 1m m 21 C O x4 1 C x2m 1 .r/C mm 2!! 1.r/m: (2.36)16