Least Variability in a Polynomial-Square Class of Rational Kernels

Abstract

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 $2\pi$.