On the empirical spectral distribution of matrix perpetuities
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914616253087744 |
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| author | Kołodziejek, Bartosz Szpojankowski, Kamil |
| author_facet | Kołodziejek, Bartosz Szpojankowski, Kamil |
| contents | We study matrix perpetuities, that is, solutions to affine fixed-point equations of the form \[ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent}, \] with particular emphasis on the empirical spectral distribution of the solution. We first establish existence and uniqueness results by relating the problem to classical vector perpetuities, and then develop tools that preserve the matrix structure under orthogonal invariance. For positive semidefinite, orthogonally invariant models, we obtain power-law tail asymptotics for the expected empirical spectral distribution and show that the tail is governed by the largest eigenvalue. We also prove that, in the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly, as the dimension tends to infinity, to the distribution of the corresponding free perpetuity. Our results are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_31054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the empirical spectral distribution of matrix perpetuities Kołodziejek, Bartosz Szpojankowski, Kamil Probability Primary 60B20, Secondary 60H25, 46L54, 60G70 We study matrix perpetuities, that is, solutions to affine fixed-point equations of the form \[ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent}, \] with particular emphasis on the empirical spectral distribution of the solution. We first establish existence and uniqueness results by relating the problem to classical vector perpetuities, and then develop tools that preserve the matrix structure under orthogonal invariance. For positive semidefinite, orthogonally invariant models, we obtain power-law tail asymptotics for the expected empirical spectral distribution and show that the tail is governed by the largest eigenvalue. We also prove that, in the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly, as the dimension tends to infinity, to the distribution of the corresponding free perpetuity. Our results are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available. |
| title | On the empirical spectral distribution of matrix perpetuities |
| topic | Probability Primary 60B20, Secondary 60H25, 46L54, 60G70 |
| url | https://arxiv.org/abs/2605.31054 |