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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

##### MSC:
 60F15 Strong limit theorems
##### Keywords:
different methods of summability
Full Text:
##### References:
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