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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Probabilistic and deterministic averaging

Authors: N. H. Bingham and Charles M. Goldie
Journal: Trans. Amer. Math. Soc. 269 (1982), 453-480
MSC: Primary 60F15; Secondary 40G05, 60K05
MathSciNet review: 637702
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Abstract: Let $ \{ {S_n}\} $ be a random walk whose step distribution has positive mean $ \mu $ and an absolutely continuous component. For any bounded measurable function $ f$, a Marcinkiewicz-Zygmund strong law in an $ r$-quick version (a 'Lai strong law') is proved for $ f({S_n})$, assuming existence of a suitable higher moment of the step distribution. This is extended to show $ {n^{ - \alpha }}\{ \sum\nolimits_1^n {f({S_k})} - \int_0^n {f(\mu t)dt\} \to 0} $ ($ r$-quickly). These results remain true when the step distribution is lattice, provided $ f$ is constant between lattice points. Certain intermediate results on renewal theory, mixing, local limit theory, ladder height, and a strong law of Lai for mixing random variables are of independent interest.

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Keywords: Backward recurrence time, Cesàro convergence, ladder height, laws of large numbers, local limit theorem, martingale, renewal theory, $ r$-quick, strong mixing, variation norm
Article copyright: © Copyright 1982 American Mathematical Society