多项式平方类有理核中的最小变异性
Least Variability in a Polynomial-Square Class of Rational Kernels
AI总结：
该研究解决多项式平方类有理核的极值问题，通过构造得到最小方差的最优解，其方差二次衰减优于Erlang基准，可由雅可比矩阵谱计算。
AI中文摘要：
我们针对正随机化核求解一个极值问题，这类核通过指数阻尼多项式的平方得到，能将随机时间尽可能紧密地集中在确定性目标周围。该构造源自集中矩阵指数文献，自动保证非负性，使核具有带单个重实极点的拉普拉斯变换，且当多项式为纯幂时包含经典Erlang随机器作为特例。半多项式无需为实根，可具有非实共轭零点，但所有最优解被证明是实根。将核归一化为单位均值后，多项式次数为m时的最小方差等于拉盖尔多项式L_{m+2}相邻零点间的最小相对间隙，每个最优解通过删除一对达到该最小值的零点得到。该刻画给出了最小方差、归一化密度及所有最优解的闭式形式，最优解可由对称三对角雅可比矩阵的谱计算。在大阶极限下，最小方差随核的阶数呈二次衰减，远优于Erlang基准的线性衰减率，且被删除的零点对定位于归一化位置2，极限绝对间距为2π。
英文摘要：
We solve an extremal problem for positive randomization kernels that concentrate a random time as tightly as possible around a deterministic target, within the class obtained by exponentially damping the square of a polynomial. The construction, borrowed from the concentrated matrix-exponential literature, automatically guarantees nonnegativity, gives the kernel a Laplace transform with a single repeated real pole, and contains the classical Erlang randomizer as the special case in which the polynomial is a pure power. The half-polynomial need not be real-rooted and so may have nonreal conjugate zeros, yet every optimizer is proved to be real-rooted. After normalizing the kernel to unit mean, the minimum variance at polynomial degree $m$ turns out to equal the smallest relative gap between adjacent zeros of the Laguerre polynomial $L_{m+2}$, with every optimizer obtained by deleting a pair attaining this minimum. This characterization yields the minimal variance, the normalized density, and every optimizer in closed form. The optimizer can be computed from the spectrum of a symmetric tridiagonal Jacobi matrix. In the large-order limit, the minimal variance decays quadratically in the order of the kernel, a marked improvement over the linear decay rate of the Erlang benchmark, and the deleted pair of zeros localizes at normalized location $2$, with limiting absolute separation $2π$.
