由OPUC生成的复随机多项式零点个数的方差
The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC
摘要：
令$\{φ_k\}_{k=0}^\infty $是关于概率测度$ μ$的单位圆上的正交多项式序列（OPUC）。本文研究了$$ P_n(z)=\sum_{k=0}^{n}η_kφ_k(z), $$形式的随机线性组合的零点个数的方差，其中$\{η_k\}_{k=0}^n $是复值随机变量。在每个$η_k$的分布满足分数矩和对数矩的某些一致界限的假设下，对于当$\{φ_k\}$在Ullman-Stahl-Totik意义上是正则的或者使得正交性度量$μ$满足$dμ(θ)=w(θ)dθ$的情况，其中$w(θ)=v(θ)\prod_{j=1}^J|θ- θ_j|^{α_j}$，具有$v(θ)\geq c>0$、$θ,θ_j\in [0,2π)$和$α_j>0$，给出了$P_n$在与单位圆相交的扇区中零点个数的方差的定量估计。当$\{φ_k\}$是Nevai类的实值且$\{η_k\}$是i.i.d.~复值标准高斯分布，证明了不含单位圆的环中$P_n$零点个数方差的极限值公式。
英文摘要：
Let $\{φ_k\}_{k=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ μ$. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=\sum_{k=0}^{n}η_kφ_k(z), $$ where $\{η_k\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $η_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $\{φ_k\}$ are regular in the sense of Ullman-Stahl-Totik or are such that the measure of orthogonality $μ$ satisfies $dμ(θ)=w(θ)dθ$ where $w(θ)=v(θ)\prod_{j=1}^J|θ- θ_j|^{α_j}$, with $v(θ)\geq c>0$, $θ,θ_j\in [0,2π)$, and $α_j>0$, we give a quantitative estimate on the the variance of the number of zeros of $P_n$ in sectors that intersect the unit circle. When $\{φ_k\}$ are real-valued on the real-line from the Nevai class and $\{η_k\}$ are i.i.d.~complex-valued standard Gaussian, we prove a formula for the limiting value of variance of the number of zeros of $P_n$ in annuli that do not contain the unit circle.
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