On the weak law of large numbers for normed weighted sums of I.I.D. random variables

Abstract

For weighted sums∑j=1najYjof independent and identically distributed random variables{Yn,n≥1}, a general weak law of large numbers of the form(∑j=1najYj−νn)/bn→P0is established where{νn,n≥1}and{bn,n≥1}are statable constants. The hypotheses involve both the behavior of the tail of the distribution of|Y1|and the growth behaviors of the constants{an,n≥1}and{bn,n≥1}. Moreover, a weak law is proved for weighted sums∑j=1najYjindexed by random variables{Tn,n≥1}. An example is presented wherein the weak law holds but the strong law fails thereby generalizing a classical example.