Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.23647

Lower deviation probabilities for supercritical multi-type Galton--Watson processes

Tan Jiangrui
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper provides a detailed analysis of the lower deviation probability properties for a $d$-type ($d>1$) Galton--Watson (GW) process $\{\textbf{Z}_n=(Z_n^{(i)})_{1\le i\le d};n\ge0\}$ in both Schröder and Böttcher cases. We establish explicit decay rates for the following probabilities: $$\mathbb{P}(\textbf{Z}_n=\textbf{k}_n),~ \mathbb{P}(|\textbf{Z}_n|\le k_n), ~\mathbb{P}(Z^{(i)}_n=k_n)~~\text{and}~~\mathbb{P}(Z^{(i)}_n\le k_n), 1\le i \le d,$$ respectively, where $\textbf{k}_n\in\mathbb{Z}_+^d$, $|\textbf{k}_n|=\mathrm{o}(c_n)$, $k_n=\mathrm{o}(c_n)$ and $c_n$ characterizes the growth rate of $\textbf{Z}_n$. These results extend the single-type lower deviation theorems of Fleischmann and Wachtel (Ann. Inst. Henri Poincaré Probab. Statist.\textbf{43} (2007) 233-255), thereby paving the way for analysis of precise decay rates of large deviations in multi-type GW processes.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tan Jiangrui. 2025-06-30. Lower deviation probabilities for supercritical multi-type Galton--Watson processes. https://arxiv.org/abs/2506.23647

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The extremal process of a cascading family of branching Brownian motion

We study the asymptotic behaviour of the extremal process of a cascading family of branching Brownian motions. This is a particle system on the real line such that each particle has a type in addition to his position. Particles of type $1$ move on the real line according to Brownian motions and branch at rate $1$ into two children of type $1$. Furthermore, at rate $α$, they give birth to children too of type $2$. Particles of type $2$ move according to standard Brownian motion and branch at rate $1$, but cannot give birth to descendants of type $1$. We obtain the asymptotic behaviour of the extremal process of particles of type $2$.

math.PR↗
arXiv2026

On the Wasserstein distance between a hyperuniform point process and its mean

We study the existence of bounds on the expected $p$-Wasserstein distance between a random measure and its mean under the assumption that the $p$-th centered moments of the counting statistics are controlled uniformly in space. The average Wasserstein transport cost is shown to be bounded from above and from below by some multiples of the number of points. $D$-dimensional versions of those results are also obtained. As a corollary, we prove that for any value of $p\geq 1$ the Ginibre point process can be seen as a perturbed lattice with identically distributed perturbations with a finite $p$-th moment.

math.PR↗
arXiv2026

Breuer-Major Theorems for Hilbert Space-Valued Random Variables

Let $\{X_k\}_{k\in\mathbb Z}$ be a stationary Gaussian process with values in a separable Hilbert space $\mathcal H_1$, and let $G:\mathcal H_1\to\mathcal H_2$ be a measurable map into another separable Hilbert space $\mathcal H_2$. We derive a central limit theorem for the centered normalized partial sums of the Hilbert space-valued subordinated process $\{G[X_k]\}_{k\in\mathbb Z}$. Our result holds under either of two sets of sufficient conditions, formulated in terms of the transformation $G$ and the temporal and cross-sectional dependence structure of $\{X_k\}_{k\in\mathbb Z}$. These conditions coincide in finite dimensions but lead to genuinely different phenomena in the infinite-dimensional setting. The proof relies on the recently developed Fourth Moment Theorem on Hilbert spaces, leveraging tools from the infinite-dimensional Malliavin-Stein framework. We also provide continuous-time and quantitative versions of the central limit theorem. In a series of examples, we recover and strengthen limit theorems for a wide array of statistics relevant in functional data analysis, and present, as an application of our result, a novel limit theorem in the framework of neural operators.

math.PR↗