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\(0.\) (4) \(X_ n\to m\) a.s. (C), for some (any) circle-method.