A limit theorem for the total progeny distribution of multi-type branching processes

TL;DR

The paper studies the total progeny of a multi-type branching process with labeled offspring across $m$ types and proves that the probability of observing a given total progeny vector decays exponentially at rate $\Gamma(\boldsymbol{\rho})$, where $\boldsymbol{\rho}$ is the limiting relative frequency and $\Gamma$ is the Legendre-type transform $\Gamma(\boldsymbol{\rho}) = \sup_{\boldsymbol{\lambda}}\{\boldsymbol{\lambda}\cdot\boldsymbol{\rho} - \sum_{k=1}^m \rho_k \log \mathbb{E}[e^{\boldsymbol{\lambda}\cdot \mathbf{X}_k}]\}$. Under mild moment and support assumptions, the authors show that conditioning on large total size $N$ yields a limiting composition $\boldsymbol{\rho}^* = \arg\min_{\Delta_m} \Gamma(\boldsymbol{\rho})$, and, in the special case where the mean offspring matrix is right stochastic, $\boldsymbol{\rho}^*$ coincides with the principal eigenvector. The main technique blends arborescent Lagrange inversion, a measure-tilting argument, and a multivariate local limit theorem to derive matching exponential upper and lower bounds for $\mathbb{P}(\mathbf{T}=\mathbf{n})$, with implications for random graphs and coagulation processes. This yields a precise large-deviation-type description of the total-progeny distribution beyond classical LDP frameworks. Formulas involve $\Gamma(\boldsymbol{\rho})$ and its Legendre transform, providing a concrete rate function for the tail behavior.

Abstract

A multi-type branching process is defined as a random tree with labeled vertices, where each vertex produces offspring independently according to the same multivariate probability distribution. We demonstrate that in realizations of the multi-type branching process, the relative frequencies of the different types in the whole tree converge to a fixed ratio, while the probability distribution for the total size of the process decays exponentially. The results hold under the assumption that all moments of the offspring distributions exist. The proof uses a combination of the arborescent Lagrange inversion formula, a measure tilting argument, and a local limit theorem. We illustrate our concentration result by showing applications to random graphs and multi-component coagulation processes.

Figures (1)

• Figure 1: An example of the tree used in the proof for $m = 5$.

Theorems & Definitions (8)

• Theorem 2.3
• Lemma 3.1: good_1955
• Lemma 3.2: Lagrange-Good inversion, good_1960bergeron_1998
• Remark 3.3
• Lemma 3.4: Arborescent Lagrange inversion, bender_1998bousquet_2003
• proof
• Lemma 3.5
• proof