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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.08623 |
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| _version_ | 1866914649985777664 |
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| author | Kraus, Isabelle Michelen, Marcus O'Rourke, Sean |
| author_facet | Kraus, Isabelle Michelen, Marcus O'Rourke, Sean |
| contents | Let $μ$ and $ν$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $μ$ and $ν$, respectively. Under assumptions on the measures $μ$ and $ν$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $μ$ and $ν$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $μ$ and $ν$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_08623 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Sums of random polynomials with differing degrees Kraus, Isabelle Michelen, Marcus O'Rourke, Sean Probability Let $μ$ and $ν$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $μ$ and $ν$, respectively. Under assumptions on the measures $μ$ and $ν$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $μ$ and $ν$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $μ$ and $ν$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment. |
| title | Sums of random polynomials with differing degrees |
| topic | Probability |
| url | https://arxiv.org/abs/2110.08623 |