For weighted sums ∑j=1najYj of independent and identically distributed random variables
{Yn,n≥1}, a general weak law of large numbers of the form (∑j=1najYj−νn)/bn→P0 is 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=1najYj indexed 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.