Let

$\sigma $ be a mapping of the set of positive integers into itself, let

${V}_{\sigma}$ be the space of bounded sequences all of whose

$\sigma $-means are equal, and let

$\sigma $-lim x be the common value of all

$\sigma $-means on x. In this paper the author generalizes the idea of strong almost convergence of

*I. J. Maddox* [Math. Proc. Camb. Philos. Soc. 83, 61-64 (1978;

Zbl 0392.40001)]: a bounded sequence

$x=\left({x}_{k}\right)$ is said to be strongly

$\sigma $-convergent to a number L if and only if

$\left(\right|{x}_{k}-L\left|\right)\in {V}_{\sigma}$ with

$\sigma $-limit zero. He characterizes those real infinite matrices which map all convergent sequences (all sequences of

$\sigma $-bounded variation) into sequences strongly

$\sigma $-convergent to zero (strongly

$\sigma $- convergent). The concept of sequences of

$\sigma $-bounded variation was introduced by the author in an earlier paper [Q. J. Math., Oxf. II. Ser. 34, 77-86 (1983)].