Moment-optimal Gamma approximation of Bessel Process exit times
Abstract
We construct finite Gamma convolutions approximating the first hitting time of $1$ by a Bessel process of index ν>−1 started at the origin. The scale measure in the classical exponential representation of this law is obtained by squaring the positive half of a dilated modified Lommel polynomial orthogonality measure. Its Jacobi recurrence yields an N-factor Gamma convolution that uniquely matches the first 2N cumulants, the maximal number attainable in this class. From an exact Gaussian quadrature remainder we derive supergeometric convergence of the Laplace transforms on compact subsets of a common slit plane and geometric convergence of the distribution functions uniformly on the real line. The approximants have total Gamma shape N(N+ν+1), and their Laplace transforms and negative moments decrease to those of the exit law.
Related
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BibTeX
@misc{clark2026momentoptimal,
title = {{Moment-optimal Gamma approximation of Bessel Process exit times}},
author = {Steven P. Clark},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19233},
eprint = {2610.19233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19233},
}
RIS
TY - UNPB
TI - Moment-optimal Gamma approximation of Bessel Process exit times
AU - Steven P. Clark
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19233
UR - https://arxiv.org/abs/2610.19233
AB - We construct finite Gamma convolutions approximating the first hitting time of $1$ by a Bessel process of index ν>−1 started at the origin. The scale measure in the classical exponential representation of this law is obtained by squaring the positive half of a dilated modified Lommel polynomial orthogonality measure. Its Jacobi recurrence yields an N-factor Gamma convolution that uniquely matches the first 2N cumulants, the maximal number attainable in this class. From an exact Gaussian quadrature remainder we derive supergeometric convergence of the Laplace transforms on compact subsets of a common slit plane and geometric convergence of the distribution functions uniformly on the real line. The approximants have total Gamma shape N(N+ν+1), and their Laplace transforms and negative moments decrease to those of the exit law.
ER -