The definition of lacunary strongly convergence is extended to the definition of lacunary strong $(A_{\sigma },p)$-convergence with respect to invariant mean when $A$ is an infinite matrix and $p=\left(p_i\right)$ is a strictly positive sequence. We study some properties and inclusion relations.

The definition of lacunary strong convergence is extended to a definition of lacunary strong convergence with respect to a sequence of modulus functions in a Banach space. We study some connections between lacunary statistical convergence and lacunary strong convergence with respect to a sequence of modulus functions in a Banach space.