No abstract

No abstract

Abstract. We show that for a large class of positive weights including the ones that are eventually monotone decreasing and those that are eventually monotone increasing but vary regularly, if the averages of random variables converge in some sense, then their corresponding weighted averages also converge in the same sense. We will also replace the sufficient conditions in the fundamental result of Jamison, Pruitt, and Orey for i.i.d. random variables that make their work more transparent.Let {X i : i ≥ 1} be an arbitrary sequence of random variables and let the weights {w i : i ≥ 1} be a sequence of positive numbers. Define the total weight by W n = n i=1 w i and the weighted sum by S n = n i=1 w i X i . It is desirable to know under what conditions on the weights and/or the finite-dimensional distributions of the underlying random variables the weighted averages S n /W n , n ≥ 1, converge in some sense. Since we already have theorems concerning the convergence of regular averages, n i=1 X i /n, n ≥ 1, it is natural to ask for conditions on the weights which are decoupled from the random variables so that the convergence of weighted averages would follow in the same sense. Surprisingly, a large class of weights, including the ones that are monotone decreasing and monotone increasing but varying regularly (see Feller [5], p. 447), satisfy our requirement. Also, since for i.i.d. random variables the condition EX 1 = 0 is both necessary and sufficient for the strong law, n i=1 X i /n a.e.

No abstract