Laws of Large Numbers for Random Variables with Arbitrarily Different and Finite Expectations Via Regression Method

Abstract

The main argument of this paper is the study of the asymptotic behaviour for the sequence $\{ \frac{1} {n}\sum _{j=1}^{n}Y _{ j}: n\geqslant 1\}$ when $\{Y _{j}: j\geqslant 1\}$ are independent or pairwise uncorrelated real random variable having possibly different and finite expectations $\{E(Y _{j}): j\geqslant 1\}$. The usual method of replacing each Y _j with the difference (Y _j − E(Y _j )) is not interesting for our purposes: in fact the differences (Y _j − E(Y _j )) work in the opposite direction giving random variables with the same null expectation and convergence of

$$\displaystyle{ \frac{1} {n}\sum _{j=1}^{n}(Y _{ j} - E(Y _{j}))}$$

to zero, in the general case, gives no information about the convergence of $\frac{1} {n}\sum _{j=1}^{n}Y _{ j}$. Thus the procedure here adopted is directly based on the sequence $\frac{1} {n}\sum _{j=1}^{n}Y _{ j}$ where the expectations $\{E(Y _{j}): j\geqslant 1\}$ define a general sequence of finite real values satisfying no regularity assumptions, i.e. all E(Y _j )’s are arbitrarily different values and the sequence {E(Y _j )} is not a convergent one. The first three sections below deal with a general and intuitive presentation of the method, whereas the remaining sections contain the rigorous proofs.