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

• Main Authors: André Adler, Andrew Rosalsky
• Format: Article
• Language: English
• Published: Hindawi Limited 1991-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: weighted sums of independent and identically distributed random variables; weak law of large numbers; convergence in probability; random indices; strong law of large numbers; almost certain convergence.
• Online Access: http://dx.doi.org/10.1155/S0161171291000182
• ISSN: 0161-1712; 1687-0425

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.