随机三角多项式零点分布的渐近性质

Andrew Granville; Igor Wigman

随机三角多项式零点分布的渐近性质

概率论 2010-03-24 v2 经典分析与常微分方程

作者: Andrew Granville , Igor Wigman

摘要

我们研究了当 $N\to\infty$ 时， $N$ 次随机三角多项式零点个数 $Z_{N}$ 的渐近分布。已知当 $N$ 趋于无穷时，零点期望数渐近于 $\frac{2}{\sqrt{3}}\cdot N$ 。Bogomolny、Bohigas和Leboeuf曾预测其方差的渐近形式为 $cN$ ，其中 $c>0$ 。我们证明了 $\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ 收敛于标准高斯分布。此外，我们发现类似结论也适用于短区间内的零点个数。

关键词

随机三角多项式零点分布渐近正态性方差短区间

引用

@article{arxiv.0809.1848,
  title  = {The distribution of the zeroes of random trigonometric polynomials},
  author = {Andrew Granville and Igor Wigman},
  journal= {arXiv preprint arXiv:0809.1848},
  year   = {2010}
}

备注

51 pages. We cut the size of the paper to better suit publication. In particular, all the results of empirical experiments were cut off. Some standard results in probability and stochastic processes were also omitted. Numerous typos and mistakes were corrected following the suggestions of referees. This paper was accepted for publication in the American Journal of Mathematics.

随机三角多项式零点分布的渐近性质

摘要

关键词

引用

备注

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