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- W2014735916 abstract "In this paper we extend well-known results by Baum and Katz (1965) and others on the rate of convergence in the law of large numbers for sums of i.i.d. random variables to general zero-mean martingales S N . For 1 2 < α ⩽ 1 , p> 1 α and f ( x ) = | x | (two-sided case) or = x + or x − (one-sided case), it is e.g. shown that if, for some γ ϵ ( 1 α, 2 ] and q> (pα − 1) (γα − 1) , sup n⩾1 n −1 ∑ j=1 n E(|X j | γ |S 0 ,…S n ) q <∞ and an additional mixing condition holds in the one-sided case, then ∑ n⩾1 n pα−2 P(f(S n )>εn α <∞ for all ε> 0 holds iff ∑ n⩾1 ∑ j⩾n j pα−2 P(f(X n )>εj α )<∞ for all ε>0 , X 1 , X 2 , … being the increments of S N . The latter condition reduces to the well-known moment condition Ef ( X 1 ) p <∞, if X 1 , X 2 , … are i.i.d. Our results also extend recent ones by Irle (1985, 1987)." @default.
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- W2014735916 date "1990-12-01" @default.
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- W2014735916 title "Convergence rates in the law of large numbers for martingales" @default.
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- W2014735916 doi "https://doi.org/10.1016/0304-4149(90)90090-f" @default.
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