Abstract
That is, the variables (1/\/n)Sn again satisfy E[exp(it(1/Vn)Sn)] =exp(-| t 2), and to achieve a finite lim sup they must be down additionally (and multiplicatively) by the (2 log log n)-112. For some reason the obvious corresponding statement for the case 7y < 2 does not seem to have been recorded, and it is the purpose of this note to do so. For 0<7y<2, the variables n-<'YSn again satisfy E [exp(itn-ir_Sn)] = exp(t | Y). Since the corresponding distribution function F(x) has tail behavior F(-x) + 1-F(x)(const) I x I -Y as I xI -* oo, instead of exponential decrease as in the 7y = 2 case, we can expect the cut down factors to appear otherwise than as multipliers.
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