arXiv subjects

Maximillian Newman

Publications and source records attributed to Maximillian Newman.

4 recordsLinked to original sources

arXiv2026

Rigidity of infinite exchangeable sequences with Gaussian marginals

We study infinite exchangeable sequences with Gaussian one-dimensional marginals. We formulate the conjecture that joint Gaussianity of a single pair of coordinates forces the entire sequence to be a Gaussian process. Although this conjecture remains open, we prove that joint Gaussianity of the first four coordinates is sufficient. We also establish the corresponding two-point criterion under the additional assumption that the directing measure is almost surely infinitely divisible.

math.PR↗

arXiv2026

The quenched coalescent for structured diploid populations with large migrations and uneven offspring distributions

In this work we describe a new model for the evolution of a diploid structured population backwards in time that allows for large migrations and uneven offspring distributions. The model generalizes both the mean-field model of Birkner et al. [\textit{Electron. J. Probab.} 23: 1-44 (2018)] and the haploid structured model of M\"{o}hle [\textit{Theor. Popul. Biol.} 2024 Apr:156:103-116]. We show convergence, with mild conditions on the joint distribution of offspring frequencies and migrations, of gene genealogies conditional on the pedigree to a time-inhomogeneous coalescent process driven by a Poisson point process $\Psi$ that records the timing and scale of large migrations and uneven offspring distributions. This quenched scaling limit demonstrates a significant difference in the predictions of the classical annealed theory of structured coalescent processes. In particular, the annealed and quenched scaling limits coincide if and only if these large migrations and uneven offspring distributions are absent. The proof proceeds by the method of moments and utilizes coupling techniques from the theory of random walks in random environments. Several examples are given and their quenched scaling limits established.

math.PR↗

arXiv2025

Quenched coalescent for diploid population models with selfing and overlapping generations

We introduce a general diploid population model with self-fertilization and possible overlapping generations, and study the genealogy of a sample of $n$ genes as the population size $N$ tends to infinity. Unlike traditional approach in coalescent theory which considers the unconditional (annealed) law of the gene genealogies averaged over the population pedigree, here we study the conditional (quenched) law of gene genealogies given the pedigree. We focus on the case of high selfing probability and obtain that this conditional law converges to a random probability measure, given by the random law of a system of coalescing random walks on an exchangeable fragmentation-coalescence process of \cite{berestycki04}. This system contains the system of coalescing random walks on the ancestral recombination graph as a special case, and it sheds new light on the site-frequency spectrum (SFS) of genetic data by specifying how SFS depends on the pedigree. The convergence result is proved by means of a general characterization of weak convergence for random measures on the Skorokhod space with paths taking values in a locally compact Polish space.

math.PR↗

arXiv2024

Conditional gene genealogies given the population pedigree for a diploid Moran model with selfing

We introduce a stochastic model of a population with overlapping generations and arbitrary levels of self-fertilization versus outcrossing. We study how the global graph of reproductive relationships, or population pedigree, influences the genealogical relationships of a sample of two gene copies at a genetic locus. Specifically, we consider a diploid Moran model with constant population size $N$ over time, in which a proportion of offspring are produced by selfing. We show that the conditional distribution of the pairwise coalescence time at a single locus given the random pedigree converges to a limit law as $N$ tends to infinity. The distribution of coalescence times obtained in this way predicts variation among unlinked loci in a sample of individuals. Traditional coalescent analyses implicitly average over pedigrees and generally make different predictions. We describe three different behaviors in the limit depending on the relative strengths, from large to small, of selfing versus outcrossing: partial selfing, limited outcrossing, and negligible outcrossing. In the case of partial selfing, coalescence times are related to the Kingman coalescent, similar to what is found in traditional analyses. In the case of limited outcrossing, the retained pedigree information forms a random graph, with coalescence times given by the meeting times of random walks on this graph. In the case of negligible outcrossing, which represents complete or nearly complete selfing, coalescence times are determined entirely by the fixed times to common ancestry of diploid individuals in the pedigree.

q-bio.PE↗