Abstract
A strong law is proved for weighted sumsS n=Σa in X i whereX i are i.i.d. and {a in} is an array of constants. When sup(n −1Σ|a in |q)1/q<∞, 1<q≤∞ andX i are mean zero, we showE|X|p<∞,p l+q −1=1 impliesS n /n \(\xrightarrow{{a.s.}}\)0. Whenq=∞ this reduces to a result of Choi and Sung who showed that when the {a in} are uniformly bounded,EX=0 andE|X|<∞ impliesS n /n \(\xrightarrow{{a.s.}}\)0. The result is also true whenq=1 under the additional assumption that lim sup |a in |n −1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a in} are uniformly bounded,E|X|1/α<∞ impliesS n /n \(\xrightarrow{{a.s.}}\)0 for α>1, but this is not true in general for 1/2<α<1, even when theX i are symmetric. In that case the additional assumption that (x 1/α log1/α−1 x)P(|X|⩾x)→0 asx↑∞ provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a in}.
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Cuzick, J. A strong law for weighted sums of i.i.d. random variables. J Theor Probab 8, 625–641 (1995). https://doi.org/10.1007/BF02218047
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DOI: https://doi.org/10.1007/BF02218047