Random Sums of Weighted Orthogonal Polynomials in ${\mathbb C}^d$
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| Format: | Preprint |
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2024
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| author | Bloom, T. Dauvergne, D. Levenberg, N. |
| author_facet | Bloom, T. Dauvergne, D. Levenberg, N. |
| contents | We consider random polynomials of the form $G_n(z):= \sum_{|α|\leq n} ξ^{(n)}_αp_{n,α}(z)$ where $\{ξ^{(n)}_α\}_{|α|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,α}\}_{|α|\leq n}$ form a basis for $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\mathbb C}^d$. In particular, this includes the setting where $\{p_{n,α}\}$ are orthonormal in a space $L^2(e^{-2n Q} τ)$, where $τ$ is a compactly supported Bernstein-Markov measure and $Q$ is a continuous weight function. Under an optimal moment condition on the random variables $\{ξ^{(n)}_α\}$, in dimension $d=1$ we prove convergence in probability of the zero measure to the weighted equilibrium measure, and in dimension $d \ge 2$ we prove convergence of zero currents. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random Sums of Weighted Orthogonal Polynomials in ${\mathbb C}^d$ Bloom, T. Dauvergne, D. Levenberg, N. Probability Complex Variables 60B10 (Primary) 32U35, 60G57, 30C15 (Secondary) We consider random polynomials of the form $G_n(z):= \sum_{|α|\leq n} ξ^{(n)}_αp_{n,α}(z)$ where $\{ξ^{(n)}_α\}_{|α|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,α}\}_{|α|\leq n}$ form a basis for $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\mathbb C}^d$. In particular, this includes the setting where $\{p_{n,α}\}$ are orthonormal in a space $L^2(e^{-2n Q} τ)$, where $τ$ is a compactly supported Bernstein-Markov measure and $Q$ is a continuous weight function. Under an optimal moment condition on the random variables $\{ξ^{(n)}_α\}$, in dimension $d=1$ we prove convergence in probability of the zero measure to the weighted equilibrium measure, and in dimension $d \ge 2$ we prove convergence of zero currents. |
| title | Random Sums of Weighted Orthogonal Polynomials in ${\mathbb C}^d$ |
| topic | Probability Complex Variables 60B10 (Primary) 32U35, 60G57, 30C15 (Secondary) |
| url | https://arxiv.org/abs/2412.11969 |