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Summability methods and almost sure convergence. (English) Zbl 0551.60037
The authors investigate almost sure convergence for sequences of i.i.d. random variables under different methods of summability. We say \(s_ n\to s\) (P), if \(\sum^{\infty}_{j=0}s_ jP(S_ n=j)\to s\) as \(n\to \infty\), where \(S_ n:=\xi_ 1+...+\xi_ n\), and \(\xi_ 1,\xi_ 2,..\). are integer-valued independent random variables. The summability method (P) is called a random-walk method. This method is related to the family of circle-methods C, defined as follows: \(s_ n\to s\) (C), if \(\sum^{\infty}_{j=0}s_ jc_ j(n)\to s\) for given weights \(c_ j(n)\). For instance: \(c_ j(n):=\sqrt{(2\pi n)^{-1}a}\) \(\exp \{- \frac{1}{2}a(j-n)^ 2/n\}\) gives the Valiron methods \(V_ a\), and \(c_ j(n):=e^{-n}n^ j/j!\) gives the Borel method B. The paper contains among others, the following
Theorem. For \(X,X_ 0,X_ 1,..\). i.i.d. the following are equivalent:
(1) Var X\(<\infty\), \(EX=m.\)
(2) \(X_ n\to m\) a.s. (P), for P some (any) random walk.
(3) \(X_ n\to m\) a.s. \((V_ a)\), for some (all) \(a>0.\)
(4) \(X_ n\to m\) a.s. (C), for some (any) circle-method.
Reviewer: Z.Jurek

60F15 Strong limit theorems
Full Text: DOI
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