In this paper we establish a relationship between convergence in probability and almost surely for sums of independent random variables. It turns out that whenever there is a relatively stable weak law of large numbers, there is a corresponding strong law. Our goal is to explore whether or not there exist constants that asymptotically behave like our partial sums. Previous results seem to indicate, in the i.i.d. case, that whenever the tails of the distribution at hand are regularly varying with exponent minus one and P{X < - x} = o(P{X> x}), then one can always find constants so that the weighted and normalized partial sums converge to one almost surely. However, a few extreme cases until now had offered evidence to the contrary. Herein, we show that even in those cases almost sure stability can be obtained.
