Abstract
Let \(\mu \) be a probability measure on \(\textrm{GL}_d(\mathbb {R})\) and denote by \(S_n:= g_n \cdots g_1\) the associated random matrix product, where \(g_j\)’s are i.i.d.’s with law \(\mu \). We study statistical properties of random variables of the form
where \(x \in \mathbb {P}^{d-1}\), \(\sigma \) is the norm cocycle and u belongs to a class of admissible functions on \(\mathbb {P}^{d-1}\) with values in \(\mathbb {R}\cup \{\pm \infty \}\). Assuming that \(\mu \) has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on \(\mathbb {R}\) and Hölder continuous target functions on \(\mathbb {P}^{d-1}\). As particular cases, we obtain new limit theorems for \(\sigma (S_n,x)\) and for the coefficients of \(S_n\).
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Notes
The notation \(H_y\) was used before to denote a hyperplane in \(\mathbb {P}^{d-1}\). There should be no confusion with a real-valued function H on \(\mathbb R\) above.
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We would like to thank the anonymous referee for the remarks which helped us improve the presentation.
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This work was supported by the NUS and MOE Grants AcRF Tier 1 R-146-000-319-114 and MOE-T2EP20120-0010. L. Kaufmann was supported by the Institute for Basic Science (IBS-R032-D1).
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Dinh, TC., Kaufmann, L. & Wu, H. Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices. J Geom Anal 33, 76 (2023). https://doi.org/10.1007/s12220-022-01127-3
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DOI: https://doi.org/10.1007/s12220-022-01127-3