Recurrence of multidimensional affine recursions in the critical case
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910559641796608 |
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| author | Aoun, Richard Brofferio, Sara Peigné, Marc |
| author_facet | Aoun, Richard Brofferio, Sara Peigné, Marc |
| contents | We prove, under different natural hypotheses, that the random multidimensional affine recursion $X_n=A_nX_{n-1}+B_n\in\mathbb{R}^d, n \geq 1,$ is recurrent in the critical case. In particular we cover the cases where the matrices $A_n$ are similarities, invertible, rank 1 or with non negative coefficients.
These results are a consequence of a criterion of recurrence for a large class of affine recursions on $\mathbb R^d$, based on some moment assumptions of the so-called ``reverse norm control random variable". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03853 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Recurrence of multidimensional affine recursions in the critical case Aoun, Richard Brofferio, Sara Peigné, Marc Probability 60J80, 60F17, 60K37 We prove, under different natural hypotheses, that the random multidimensional affine recursion $X_n=A_nX_{n-1}+B_n\in\mathbb{R}^d, n \geq 1,$ is recurrent in the critical case. In particular we cover the cases where the matrices $A_n$ are similarities, invertible, rank 1 or with non negative coefficients. These results are a consequence of a criterion of recurrence for a large class of affine recursions on $\mathbb R^d$, based on some moment assumptions of the so-called ``reverse norm control random variable". |
| title | Recurrence of multidimensional affine recursions in the critical case |
| topic | Probability 60J80, 60F17, 60K37 |
| url | https://arxiv.org/abs/2408.03853 |