On the clustering of Padé zeros and poles of random power series
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913202875400192 |
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| author | Dostoglou, Stamatis Valettas, Petros |
| author_facet | Dostoglou, Stamatis Valettas, Petros |
| contents | We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé approximant of a random power series $f(z) = \sum_{j=0}^\infty a_j z^j$ for $a_j$ independent, with finite first moment, and Lévy function satisfying $L(a_j , \varepsilon) \leq K\varepsilon$. Under the same assumptions we show that almost surely $f$ has infinitely many zeros in the unit disc, with the unit circle serving as a natural boundary for $f$. For $R_m$ the radius of the largest disc containing at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our setting the poles of the $[m,n]$-Padé approximant almost surely cluster uniformly at the circle of radius $R_m$ as $n \to \infty$ and $m$ stays fixed, and we provide almost sure rates of converge of these $R_m$'s to $1$. We also show that our results on the clustering of the zeros hold for log-concave vectors $(a_j)$ with not necessarily independent coordinates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the clustering of Padé zeros and poles of random power series Dostoglou, Stamatis Valettas, Petros Complex Variables Probability We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé approximant of a random power series $f(z) = \sum_{j=0}^\infty a_j z^j$ for $a_j$ independent, with finite first moment, and Lévy function satisfying $L(a_j , \varepsilon) \leq K\varepsilon$. Under the same assumptions we show that almost surely $f$ has infinitely many zeros in the unit disc, with the unit circle serving as a natural boundary for $f$. For $R_m$ the radius of the largest disc containing at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our setting the poles of the $[m,n]$-Padé approximant almost surely cluster uniformly at the circle of radius $R_m$ as $n \to \infty$ and $m$ stays fixed, and we provide almost sure rates of converge of these $R_m$'s to $1$. We also show that our results on the clustering of the zeros hold for log-concave vectors $(a_j)$ with not necessarily independent coordinates. |
| title | On the clustering of Padé zeros and poles of random power series |
| topic | Complex Variables Probability |
| url | https://arxiv.org/abs/2401.11532 |