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

For weighted sums ∑ j = 1 n a j Y j of independent and identically distributed random variables { Y n , n ≥ 1 } , a general weak law of large numbers of the form ( ∑ j = 1 n a j Y j − ν n ) / b n → P 0 is established where { ν n , n ≥ 1 } and { b n , n ≥ 1 } are statable constants. The hypotheses involve both the behavior of the tail of the distribution of | Y 1 | and the growth behaviors of the constants { a n , n ≥ 1 } and { b n , n ≥ 1 } . Moreover, a weak law is proved for weighted sums ∑ j = 1 n a j Y j indexed by random variables { T n , n ≥ 1 } . An example is presented wherein the weak law holds but the strong law fails thereby generalizing a classical example.