arXiv subjects

Helmut Pitters

Publications and source records attributed to Helmut Pitters.

3 recordsLinked to original sources

arXiv2022

Mod-$\varphi$ convergence of Stirling distributions and limit theorems for zeros of their generating functions

We study mod-$\varphi$ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when $n$ balls are allocated equiprobably and independently among $\theta$ boxes in the regime when $\theta$ grows linearly with $n$.

math.PR↗

arXiv2019

The number of cycles in a random permutation and the number of segregating sites jointly converge to the Brownian sheet

Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next, consider a sample drawn from a large, neutral population of haploid individuals subject to mutation under the infinitely many sites model of Kimura whose genealogy is governed by Kingman's coalescent. Let $S(n, t)$ count the number of segregating sites in a sample of size $\lfloor n^{t_2}\rfloor$ when mutations arrive at rate $t_1/2$. Our main result comprises two different couplings of the above models for all parameters $n\geq 2,$ $t\in [0, 1]^2$ such that in both couplings one has weak convergence of processes as $n\to\infty$ \begin{align*} \left\{\frac{(K(n, s), S(n, t))-(s_1s_2, t_1t_2)\log n}{\sqrt{\log n}}, s, t\in [0, 1]^2\right\}\to\{(\mathscr B(s), \mathscr B(t)), s, t\in [0, 1]^2\}, \end{align*} where $\mathscr B$ is a one-dimensional Brownian sheet. This generalises and unifies a number of well-known results.

math.PR↗

arXiv2015

Absorption time and tree length of the Kingman coalescent and the Gumbel distribution

Formulas are provided for the cumulants and the moments of the time $T$ back to the most recent common ancestor of the Kingman coalescent. It is shown that both the $j$th cumulant and the $j$th moment of $T$ are linear combinations of the values $\zeta(2m)$, $m\in\{0,\ldots,\lfloor j/2\rfloor\}$, of the Riemann zeta function $\zeta$ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length $L_n$ of the Kingman coalescent restricted to a sample of size $n$ is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.

math.PR↗