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Strong approximations for nonconventional sums and almost sure limit theorems. (English) Zbl 1290.60034

Summary: We improve, first, a strong invariance principle from [the author, Probab. Theory Relat. Fields 155, No. 1–2, 463–486 (2013; Zbl 1271.60047)] for nonconventional sums of the form \(\sum_ {n=1}^{[Nt]}F(X(n),X(2n),\dots,X(\ell n))\) (normalized by \(1/\sqrt N\)), where \(X(n)\), \(n\geq 0\)’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.

MSC:

60F15 Strong limit theorems
60G15 Gaussian processes
60G42 Martingales with discrete parameter
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
60F17 Functional limit theorems; invariance principles

Citations:

Zbl 1271.60047
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References:

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