Maximal generalization of Baum-Katz theorem and optimality of sequential tests
نویسنده
چکیده
Baum-Katz theorem asserts that the Cesàro means of i.i.d. increments distributed like X r-converge if and only if |X| is integrable. We generalize this, and we unify other results, by proving that the following equivalence holds, if and only if G is moderate: the Cesàro means G-converge if and only if G(La) is integrable for every a if and only if |X| G(|X|) is integrable. Here, La is the last time when the deviation of the Cesàro mean from its limit exceeds a, and G-convergence is the analogue of r-convergence. This solves a question about the asymptotic optimality of Wald’s sequential tests.
منابع مشابه
A Remark on Baum-katz Type Theorem for Φ-mixing Sequences of Random Variables
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