Reducible Markov modulation, pole order, and tail behavior in random growth models
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| Format: | Preprint |
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2024
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| _version_ | 1866910192774414336 |
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| author | Beare, Brendan K. Toda, Alexis Akira |
| author_facet | Beare, Brendan K. Toda, Alexis Akira |
| contents | Recent work on random growth models with light-tailed Markov-modulated additive shocks has shown that irreducible modulation yields tail behavior resembling an exponential distribution. We show that with reducible modulation the tail behavior more generally resembles an Erlang distribution. Our main technical contribution is a theorem on the order of a real pole of the inverse of a holomorphic matrix-valued function with reducible Metzler structure. In a special affine case, the theorem recovers the Rothblum index theorem. Applying this result together with a Tauberian theorem, we characterize the Erlang shape parameter in two models of Markov-modulated random growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reducible Markov modulation, pole order, and tail behavior in random growth models Beare, Brendan K. Toda, Alexis Akira Probability 60J10, 60J27 (Primary) 47A56 (Secondary) Recent work on random growth models with light-tailed Markov-modulated additive shocks has shown that irreducible modulation yields tail behavior resembling an exponential distribution. We show that with reducible modulation the tail behavior more generally resembles an Erlang distribution. Our main technical contribution is a theorem on the order of a real pole of the inverse of a holomorphic matrix-valued function with reducible Metzler structure. In a special affine case, the theorem recovers the Rothblum index theorem. Applying this result together with a Tauberian theorem, we characterize the Erlang shape parameter in two models of Markov-modulated random growth. |
| title | Reducible Markov modulation, pole order, and tail behavior in random growth models |
| topic | Probability 60J10, 60J27 (Primary) 47A56 (Secondary) |
| url | https://arxiv.org/abs/2405.13556 |