Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913631551094784 |
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| author | Mei, Jianzhang Liu, Quansheng |
| author_facet | Mei, Jianzhang Liu, Quansheng |
| contents | Consider the multivariate smoothing transform fixed-point equation: $η=$ law of $ \sum_{i=1}^N A_i Z_i$, where $N \geq 0$ is a random integer, $(A_i)_{i \geq 1}$ are $d \times d$ random nonnegative matrices, $(Z_i)_{i \geq 1}$ is a sequence of $\mathbb{R}_+^d$-valued random variables independent of $(N, A_1, A_2, \cdots)$, and all $Z_i$ have the same law $η$. For each fixed point $η$, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21173 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform Mei, Jianzhang Liu, Quansheng Probability Consider the multivariate smoothing transform fixed-point equation: $η=$ law of $ \sum_{i=1}^N A_i Z_i$, where $N \geq 0$ is a random integer, $(A_i)_{i \geq 1}$ are $d \times d$ random nonnegative matrices, $(Z_i)_{i \geq 1}$ is a sequence of $\mathbb{R}_+^d$-valued random variables independent of $(N, A_1, A_2, \cdots)$, and all $Z_i$ have the same law $η$. For each fixed point $η$, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments. |
| title | Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform |
| topic | Probability |
| url | https://arxiv.org/abs/2412.21173 |