A discrete-time Matsumoto-Yor theorem
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909302532341760 |
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| author | Herent, Charlie |
| author_facet | Herent, Charlie |
| contents | We study a random walk on the subgroup of lower triangular matrices of SL$_2$, with i.i.d. increments. We prove that the process of the lower corner of the random walk satisfies a Rogers-Pitman criterion to be a Markov chain if and only if the increments are distributed according to a Generalized Inverse Gaussian (GIG) law on their diagonals. For this, we prove a new characterization of these laws. We prove a discrete-time version of the Dufresne identity. We show how to recover the Matsumoto-Yor theorem by taking the continuous limit of the random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01044 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A discrete-time Matsumoto-Yor theorem Herent, Charlie Probability We study a random walk on the subgroup of lower triangular matrices of SL$_2$, with i.i.d. increments. We prove that the process of the lower corner of the random walk satisfies a Rogers-Pitman criterion to be a Markov chain if and only if the increments are distributed according to a Generalized Inverse Gaussian (GIG) law on their diagonals. For this, we prove a new characterization of these laws. We prove a discrete-time version of the Dufresne identity. We show how to recover the Matsumoto-Yor theorem by taking the continuous limit of the random walk. |
| title | A discrete-time Matsumoto-Yor theorem |
| topic | Probability |
| url | https://arxiv.org/abs/2409.01044 |