Baum-Katz定理的极大推广与序贯检验的最优性
Maximal generalization of Baum-Katz theorem and optimality of sequential tests
摘要：
Baum-Katz定理断言i.i.d.像X r-一样分布的增量收敛当且仅当|X| ^{r+1}是可积的。我们推广了这一点，并统一了其他结果，通过证明以下等价成立，当且仅当G是适度的：Cesaro意味着G-收敛当且仅当G（L（a））是可积的每一个a当且仅当|X| .G（|X|）是可积的。这里，L（a）是Cesaro平均值与其极限的偏差超过a的最后一次，G-收敛是r-收敛的类似物。这解决了Wald的序贯检验的渐近最优性问题。
英文摘要：
Baum-Katz theorem asserts that the Cesaro means of i.i.d. increments distributed like X r-converge if and only if |X|^{r+1} 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 Cesaro means G-converge if and only if G(L(a)) is integrable for every a if and only if |X|.G(|X|) is integrable. Here, L(a) is the last time when the deviation of the Cesaro 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.
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