Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1155053 | Statistics & Probability Letters | 2006 | 12 Pages |
Abstract
We show that if Xn are i.i.d. random variables with EX1=0,EX12=1 and (dk) is a positive numerical sequence obeying a condition similar to Kolmogorov's condition for the LIL, then letting Dn=âk=1ndk we have1DNâk=1NdkI{Sk/k⩽x}â12Ïâ«-âxe-t2/2dt.This shows that logarithmic means, used traditionally in a.s. central limit theory, can be replaced by other means lying much closer to ordinary (Cesà ro) averages, leading to considerably sharper results. Our results remain valid (under suitable regularity conditions) for independent random variables Xn satisfying the weak limit theorem an-1Sn-bnâ¶LH with an arbitrary distribution function H.
Keywords
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Physical Sciences and Engineering
Mathematics
Statistics and Probability
Authors
Siegfried Hörmann,