Abstract
Let \(\rho \) be a probability measure on \(\varGamma :=\mathrm {SL}_d(\mathbb {Z})\). Consider the random walk defined by \(\rho \) on the torus \(\mathbb {T}^d= \mathbb {R}^d/\mathbb {Z}^d\): for any \(x\in \mathbb {T}^d\), the walk starting at x is defined by
where \((g_n)\in \varGamma ^\mathbb {N}\) is chosen with the law \(\rho ^{\otimes \mathbb {N}}\). Bourgain, Furmann, Lindenstrauss and Mozes proved that under an assumption on the group generated by the support of \(\rho \), the random walk starting at any irrational point equidistributes in the torus. In this article, we study the Functional Central Limit Theorem and the almost sure Functional Central Limit Theorem for this walk starting at some points having good diophantine properties.
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Notes
There is \(\varepsilon \in \mathbb {R}_+^*\) such that
$$\begin{aligned} \int _{\mathrm {SL}_d(\mathbb {Z})} \Vert g\Vert ^\varepsilon \mathrm{d}\rho (g) \text { is finite}. \end{aligned}$$The fact that \(\lambda _1\) exists and is non-negative and the uniqueness of \(\overline{\nu }\) come from a theorem of Guivarc’h and Raugi [15].
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We would like to thank the anonymous referee for his careful reading and all his remarks that helped to simplify this paper and make it clearer.
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Boyer, JB. Almost sure functional central limit theorem for the linear random walk on the torus. Probab. Theory Relat. Fields 173, 651–696 (2019). https://doi.org/10.1007/s00440-018-0871-8
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DOI: https://doi.org/10.1007/s00440-018-0871-8