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On Weak Invariance Principles for Partial Sums

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Abstract

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.

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Acknowledgments

I would like to thank the anonymous reviewer for the many comments and suggestions. The generous help has been of major benefit.

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  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Moritz Jirak

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  1. Moritz Jirak
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Jirak, M. On Weak Invariance Principles for Partial Sums. J Theor Probab 30, 703–728 (2017). https://doi.org/10.1007/s10959-016-0670-z

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  • DOI: https://doi.org/10.1007/s10959-016-0670-z

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