On equitable ratios of Dubins-Freedman type

Richard Isaac

Statistics & Probability Letters, 1999, vol. 42, issue 1, 1-6

Abstract: Let Xk be non-negative random variables with positive, finite expectation, an increasing sequence of [sigma]-fields to which the Xk are adapted, and . It is shown how a martingale form of Kronecker's Lemma and the theorem of Abel and Dini on series can quickly yield a number of sufficient conditions ensuring that [summation operator]Xk/[summation operator]pk-->1 on certain sets. In this case we say that the ratios are equitable on the sets. A number of new and old results are seen to be immediate consequences, including a martingale proof of the strong law of large numbers. The paper concludes with a few simple examples illustrating non-equitable behavior.

Keywords: Conditional; expectation; a.s.; convergence; Martingales; Strong; laws (search for similar items in EconPapers)
Date: 1999
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Citations: View citations in EconPapers (1)

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