论文标题
随机三角多项式的平均零数的新渐近数，具有强烈依赖的高斯系数
New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients
论文作者
论文摘要
我们考虑表格\ [f_n（t）：= \ frac {1} {\ sqrt {n}} \ sum_ {k = 1}^{n} a_k \ cos（t） $（b_k）_ {k \ geq 1} $是两个具有相同相关函数$ρ的独立固定高斯进程$ρ：k \ mapsto \ cos（kα）$，带有$α\ geq 0 $。我们表明，在独立或依赖依赖的系数的情况下，保持的真实零的预期数量的渐近数与通用的$ \ frac {2} {\ sqrt {3}} $不同。更确切地说，对于所有$ \ varepsilon> 0 $，对于所有$ \ ell \ in（\ sqrt {2}，2] $，存在$α\ geq 0 $和$ n \ geq 1 $足够大，以便$ \左| \ frac {\ mathbb {e} \ left [\ mathcal {n}（f_n，[0,2π]）\ right]} {n} - \ el \ ell \ ell \ right | \ leq \ leq \ leq \ leq \ varepsilon，$ \ varepsilon，$ \ \ \ \ \ \ m nere $ \ mathcal n（f_n（f_n（f_n（f_n）） $ f_n $在间隔$ [0,2π]中，此结果提供了第一个示例，其中预期的零零不会收敛，因为$ n $通过表现出从$ \ sqrt {2} $到2到2的整个可能的限制来进行无限。
We consider random trigonometric polynomials of the form \[ f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $ρ: k \mapsto \cos(kα)$, with $α\geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2}{\sqrt{3}}$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon>0$, for all $\ell \in (\sqrt{2},2]$, there exists $α\geq 0$ and $n\geq 1$ large enough such that $$\left|\frac{\mathbb{E}\left[\mathcal{N}(f_n,[0,2π])\right]}{n}-\ell\right|\leq \varepsilon,$$ where $\mathcal N(f_n,[0,2π])$ denotes the number of real zeros of the function $f_n$ in the interval $[0,2π]$. Therefore, this result provides the first example where the expected number of real zeros do not converge as $n$ goes to infinity by exhibiting a whole range of possible limits ranging from $\sqrt{2}$ to 2.
