Tauberian Theorems for Summability Transforms

we then write sn → s(A), where A is the A method of summability. Appropriate choices of A= [an,k] for n,k ≥ 0 give the classical methods [2]. In this paper, we present various summability analogs of the strong law of large numbers (SLLN) and their rates of convergence in an unified setting, beyond the class of random-walk methods. A convolution summability method introduced in the next section as an extension of the random-walk method generalizes the classical Euler, Borel, Taylor, and Meyer-König type matrix methods [16]. This corresponds to the distribution of sums of independent and identically distributed (i.i.d.) integer-valued random variables. The identically distributed condition is relaxed to some extent in the convolution summability method to include other summability methods. Chow [5] gave the following analog of almost sure convergence of the Borel B and the Euler Ep methods of summability to μ requiring the finiteness of the second moment for sequence of i.i.d. random variables.