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 Lm+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.
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BibTeX
@misc{battagliola2026least,
title = {{Least Variability in a Polynomial-Square Class of Rational Kernels}},
author = {Maria Laura Battagliola and Oscar Peralta},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29143},
eprint = {2608.29143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29143},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Least Variability in a Polynomial-Square Class of Rational Kernels
AU - Maria Laura Battagliola
AU - Oscar Peralta
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29143
UR - https://arxiv.org/abs/2608.29143
AB - 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.
ER -