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2.4 Estimates for jF.z/j and jF.rI z/jSince our approach is based on Cauchy’s integral formula, we also need some estimates for the probabilitygenerating functions F.z/ and F.rI z/.Let .r/ WD p 1 .r/ 2 .r/.Lemma 2.6. .i/ For jxj 1=3 with 2,.ii/ For jtj ,F.r/r D e x2 =21 C 3 2 C 2 3x 3 C Om 6 3 1 C x6 2: (2.13)jF.rI e it /j e c 2.r/ 2 t 2 ; (2.14)where c 2 WD 2= 2 ..iii/ For jtj 2=3 and jxj 1=3 with 2,F.rI e it /e mit D e .r/2 t 2 =21 C . 1 .r/ m/i tC 1.r/ 3 2 .r/ C 2 3 .r/.i t/ 3 C O.x 4 t 2 C 4 t 6 C 2 t 4 / : (2.15)6Proof. Since r D 1 C x=, 2 and jxj 1=3 , we see that jr1j 2 1=6 < 1, and thuslog F.r/r mD X j1D. 1/ j 1 . j m/.r 1/ jjx22 C 3 2 C 2 3x 3 C O6 3 x4 2 ;because, by (2.10), j m D j x D O j 2 for j 1. This proves (2.13).For (ii), we start from the relationjq C pe it j 2 D q 2 C p 2 C 2pq cos t D 1 2pq.1 cos t/ .t 2 R/;which yields the inequality jq C pe it j 2 e 2pq.1 cos t/ . This and the inequality 1 cos t c 2 t 2 forjtj giveˇYˇF.rI e it /ˇˇ D jq j .r/ C p j .r/e it j e .r/2 .1 cos t/ e c 2.r/ 2 t 2 ;1jnfrom which (2.14) follows.Finally, for (iii), by Taylor expansion and (2.10), we havelog F.rI e it /e mit D . 1 .r/m/i t C X . 1/ k 1. k .r/ 1 .r//.e it 1/ kkk2D . 1 .r/m/i t.r/ 2t 2 C 1.r/23 2 .r/ C 2 3 .r/6.i t/ 3 C O 2 t 4 :This proves (2.15) and the lemma.10