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Stein’s method for multivariate Brownian approximations of sums under dependence

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  • Kasprzak, Mikołaj J.

Abstract

We use Stein’s method to obtain a bound on the distance between scaled p-dimensional random walks and a p-dimensional (correlated) Brownian motion. We consider dependence schemes including those in which the summands in scaled sums are weakly dependent and their p components are strongly correlated. As an example application, we prove a functional limit theorem for exceedances in an m-scans process, together with a bound on the rate of convergence. We also find a bound on the rate of convergence of scaled U-statistics to Brownian motion, representing an example of a sum of strongly dependent terms.

Suggested Citation

  • Kasprzak, Mikołaj J., 2020. "Stein’s method for multivariate Brownian approximations of sums under dependence," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4927-4967.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:4927-4967
    DOI: 10.1016/j.spa.2020.02.006
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    References listed on IDEAS

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    1. Hall, Peter, 1979. "On the invariance principle for U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 9(2), pages 163-174, November.
    2. de Jong, Peter, 1990. "A central limit theorem for generalized multilinear forms," Journal of Multivariate Analysis, Elsevier, vol. 34(2), pages 275-289, August.
    3. Yosef Rinott & Vladimir Rotar, 2000. "Normal approximations by Stein's method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(1), pages 15-29.
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