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Strong approximations for nonconventional sums and almost sure limit theorems

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  • Kifer, Yuri

Abstract

We improve, first, a strong invariance principle from Kifer (2013) [10] for nonconventional sums of the form ∑n=1[Nt]F(X(n),X(2n),…,X(ℓn)) (normalized by 1/N) where X(n),n≥0’s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties and F satisfies some regularity conditions. Applying this result we obtain next a version of the law of iterated logarithm for such sums, as well as an almost sure central limit theorem. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems.

Suggested Citation

  • Kifer, Yuri, 2013. "Strong approximations for nonconventional sums and almost sure limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2286-2302.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:6:p:2286-2302
    DOI: 10.1016/j.spa.2013.02.009
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    References listed on IDEAS

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    1. Heck, Matthias K., 1998. "The principle of large deviations for the almost everywhere central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 61-75, August.
    2. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, March.
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    Cited by:

    1. Hafouta, Yeor & Kifer, Yuri, 2016. "Berry–Esseen type estimates for nonconventional sums," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2430-2464.

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