Explicit Stable Strong Laws of Large Numbers

Abstract

In this article it is shown, through a very interesting class of random variables, how one may go about explicitly obtaining constants in order to obtain a stable strong law of large numbers. The question at hand is, not when we can find constants a_n and b_n so that our sequence of i. i.d. random variables obeys this type of strong law of large numbers, but how one goes about constructing these constants so that ${\sum }_{\hspace{0.17em}k=1}^{\hspace{0.17em}n}\hspace{0.17em}\hspace{0.17em}{a}_k{X}_k\hspace{0.17em}⁄b_n\to 1$ almost surely, even though {X, X_n} are i.i.d. with either $EX=0\hspace{0.17em}\hspace{0.17em}\mathrm{or}\hspace{0.17em}\hspace{0.17em}E\left|X\right|=\infty .$ There are three possible cases. We exhibit all three via a particular family of random variables.