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On a Multivariate Strong Renewal Theorem

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  • Zhiyi Chi

    (University of Connecticut)

Abstract

This paper takes the so-called probabilistic approach to the strong renewal theorem (SRT) for multivariate distributions in the domain of attraction of a stable law. A version of the SRT is obtained that allows any kind of lattice–nonlattice composition of a distribution. A general bound is derived to control the so-called small-n contribution, which arises from random walk paths that have a relatively small number of steps but make large cumulative moves. The asymptotic negligibility of the small-n contribution is essential to the SRT. Applications of the SRT are given, including some that provide a unified treatment to known results but with substantially weaker assumptions.

Suggested Citation

  • Zhiyi Chi, 2018. "On a Multivariate Strong Renewal Theorem," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1235-1272, September.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:3:d:10.1007_s10959-017-0754-4
    DOI: 10.1007/s10959-017-0754-4
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    References listed on IDEAS

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    1. Doney, R. A., 1991. "A bivariate local limit theorem," Journal of Multivariate Analysis, Elsevier, vol. 36(1), pages 95-102, January.
    2. Resnick, Sidney & Greenwood, Priscilla, 1979. "A bivariate stable characterization and domains of attraction," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 206-221, June.
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