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On equitable ratios of Dubins-Freedman type

Author

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  • Isaac, Richard

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.

Suggested Citation

  • Isaac, Richard, 1999. "On equitable ratios of Dubins-Freedman type," Statistics & Probability Letters, Elsevier, vol. 42(1), pages 1-6, March.
    • Handle: RePEc:eee:stapro:v:42:y:1999:i:1:p:1-6
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    Cited by:

    1. Yang, Weiguo & Yang, Xue, 2008. "A note on strong limit theorems for arbitrary stochastic sequences," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2018-2023, October.

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