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where W˛./ is defined in (2.1).The sum of jı n;m j˛ over the remaining range of m, namely, jxj .= 1 / 1=18 can be estimated bymeans of Lemma 2.2 and we obtainX ˛˛1 3˛=2 :jxj>.= 1 / 1=18 jı n;m j˛ D OLower and upper bounds for W˛./. To finish the proof, we need to estimate the order of W˛./. First,since C 1 D O./ and C 2 D O./, we have the upper bound01W˛./ D O @ X ˛ e ˛ m.1 C jxj 3 /˛ A D O ˛.1 ˛/=2 :m!m0Next, we see that the same estimate holds from below sinceW˛./ 1 X˛ eˇˇˇˇ m '.x/2m! 2 ˇjx x 0 j1=2 ˇˇ 3 ˛ 18ˇ2 C o.1/ Xjx x 0 j1=20 OX˛=2 @ ˛jxx 0 j1=2e ˛x2 .1˛˛ e m/=2as ! 0 and ! 1. This completes the proof of Theorem 2.1.2.10 Kolmogorov distance and the point metricm!1A O ˛ .1 ˛/=2 ;The methods of proof we used above can be readily amended for the consideration of other distances. Webriefly discuss the Kolmogorov distance and the point metric and start with the following lemma.Lemma 2.11. For m D C x 0P.S n m/P.S n D m/XjmProof. First of all, sincee jj ! D p22 1 … m ./ C ˆ 1C O p 1 ˆ.x/2 p 2 xe .1 /x2 =2 3 3=2 C 2 1=2 ; (2.49)e mm! D 22 2 … m ./ C e .1 /x2 =2p2 x2 .1 /x2 1 !1 C e x2 =2p1 C O 3 2 C 2 1 : (2.50)P.S n m/Xjme jj ! D 1 Z e mit F.eit / e .ei t 1/dt;2 1 e it20