On the weak law of large numbers for normed weighted sums of I.I.D. random variables - André Adler; Andrew Rosalsky (International Journal of Mathematics and Mathematical Sciences)
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.