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For a sequence of dependent random variables ${\left(X_k\right)}_{k\in \mathbb{N}}$ we consider a large class of summability methods defined by R. Jajte in [] as follows: For a pair of real-valued nonnegative functions g,h: ℝ⁺ → ℝ⁺ we define a sequence of “weighted averages” $1/g\left(n\right){\sum }_{k=1}^n\left(X_k\right)/h\left(k\right)$, where g and h satisfy some mild conditions. We investigate the almost sure behavior of such transformations. We also take a close look at the connection between the method of summation (that is the pair of functions (g,h)) and the coefficients that measure dependence...