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In this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47-63, 1988; Publ. Math. (Debr.) 76:77-88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f. Namely, for every modulus function f , we will prove that a f-strongly Cesàro convergent sequence is always f-statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f-statistically convergent, that is, we show that Connor-Khan-Orhan's result is sharp in this sense.

Some classical Hahn-Schur Theorem-like results on the uniform convergence of unconditionally convergent series can be generalized to weakly unconditionally Cauchy series. In this paper, we obtain this type of generalization via a summability method based upon the concept of almost convergence. We also obtain a generalization of the main result in Aizpuru et al. (2003) [3] using pointwise convergence of sums indexed in natural Boolean algebras with the Vitali-Hahn-Saks Property. In order to achieve that, we first study the notion of almost convergence through its original definition (which involves Banach limits), giving a description of the extremal structure of the set of all norm-1 Hahn-Banach extensions of the limit function on c to ∞ . We also show the existence of norm-1 HahnBanach extensions of the limit function on c to ∞ that are not extensions of the almost limit function and hence are not Banach limits.

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