On a family of summability methods and one-sided Tauberian conditions
The summability methods used in this paper are defined in terms of a sequence \(p=(p_k)\) of non-negative reals for which \(p_k= e^{-g(k)}\), where \(g: C_3[0,\infty)\to \mathbb{R}\) satisfies \(g''(x)\searrow 0\), \(x^2g''(x) \nearrow \infty\), \(|g'''(x)/ g''(x)|= O(1/x)\), as \(x\to\infty\), and \(\phi(x)= [g''(x)]^{-1/2}\). The author uses Cesàro means with speed of convergence \((C_{1,\phi})\), generalized Nörlund means \((N,p,p)\), power series means \((P_\phi)\), running means \((M_\phi)\), and generalized (continuous) Valiron means \((V_\phi)\). Theorems A and B give three known inclusion theorems of the form \((M_\phi) \Leftrightarrow(C_{1,\phi})\), \((M_\phi) \Rightarrow (N,p,p)\), and \((N,p,p) \Rightarrow (P_\phi)\), together with the companion Tauberian results (subject to \(s_n=O(1)\)) for the two last-mentioned inclusions. The main result (Theorem 1) assumes the above hypotheses on \(p\) and \(g\), together with \(\phi(x)= o(x/\log x)\), and it is then proved that the methods \((M_\phi)\), \((C_{1,\phi})\), \((N,p,p)\), \((P_\phi)\), and \((V_\phi)\) are all equivalent for sequences \((s_k)\) satisfying \[ \lim_{h\searrow0} \liminf_{n\to\infty} \inf_{0<u\leq h} {1\over{h\phi(n)}} \sum_{n\leq k< n+u\phi(n)} s_k>-\infty. \] Theorem 2 concerns an inclusion theorem of the form \((M_\phi) \Rightarrow (V_\phi)\) (without any Tauberian condition). From his results the author derives strong laws and Erdös-Rényi laws for all the methods in question when applied to independent identically-distributed random variables.
scientific article; zbMATH DE number 861807