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Strong Gaussian approximation for cumulative processes

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  • Bashtova, Elena
  • Shashkin, Alexey

Abstract

We establish the optimal rates of strong approximation by Wiener process for vector-valued cumulative processes. The Komlós–Major–Tusnády bounds are given both in the case when exponential moments exist and for the power moments case. Applications to strong invariance principle for stopped sums and birth and death processes are provided. As a tool we use a maximal inequality for sums over random intervals which is of independent interest.

Suggested Citation

  • Bashtova, Elena & Shashkin, Alexey, 2022. "Strong Gaussian approximation for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1-18.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:1-18
    DOI: 10.1016/j.spa.2022.04.003
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    References listed on IDEAS

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    1. Gauthier Dierickx & Uwe Einmahl, 2018. "A General Darling–Erdős Theorem in Euclidean Space," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1142-1165, June.
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    4. Asmussen, Søren, 1991. "Ladder heights and the Markov-modulated M/G/1 queue," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 313-326, April.
    5. Glynn, Peter W. & Whitt, Ward, 1993. "Limit theorems for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 299-314, September.
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