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Probabilistic Proof of Conditional Limit Theorem for Critical Galton--Waston Process

Jiayan Guo, Wenming Hong

TL;DR

This paper provides a probabilistic proof of the conditional limit theorem for a critical Galton–Watson process with finite variance under conditioning on non-extinction. It identifies the limiting conditional distribution of Z_{nt}/n as the sum of two independent exponentials U_t and V_t, whose rates are $2/(t(1-t)\sigma^{2})$ and $2/(t\sigma^{2})$, and links these components to the left and right parts of Geiger's spine. A central contribution is the introduction and analysis of the reduced Geiger's tree, in which split times converge to nested uniform variables and two-descent properties hold, yielding asymptotic independence between the left and right spine contributions. Together, these results give a constructive, probabilistic origin for the Spitzer–Lamperti–Ney limit and deepen the understanding of conditioned critical branching structures along the spine.

Abstract

Let $\{Z_{n}\}_{n\geq0}$ be a critical Galton--Waston branching process with finite variance $σ^{2}$. Spitzer (unpublished), Lamperti and Ney (1968) proved that for any fixed $0<t<1$, $$\mathscr{L}\left(\frac{Z_{nt}}{n}\Big|Z_{n}>0\right)\overset{\text{d}}{\rightarrow}U_{t}+V_{t}$$ as $n\rightarrow\infty$, where $U_{t}$ and $V_{t}$ are independent random variables having exponential distributions with parameters $2/(t(1-t)σ^{2})$ and $2/(tσ^{2})$ respectively. The proof is short and elegent based on the Laplace transform. In this paper, we will specify where the two exponential random variables come from explicitly, in terms of the Geiger's conditioned tree. Actually, $U_{t}$ and $V_{t}$ are resulted from the ``left'' and ``right'' parts of the ``spine'' of the Geiger's tree at generation $[nt]$. To this end, more details and intrinsic properties about the Geiger's conditioned tree will be investigated, which are interesting in its own right as well.

Probabilistic Proof of Conditional Limit Theorem for Critical Galton--Waston Process

TL;DR

This paper provides a probabilistic proof of the conditional limit theorem for a critical Galton–Watson process with finite variance under conditioning on non-extinction. It identifies the limiting conditional distribution of Z_{nt}/n as the sum of two independent exponentials U_t and V_t, whose rates are and , and links these components to the left and right parts of Geiger's spine. A central contribution is the introduction and analysis of the reduced Geiger's tree, in which split times converge to nested uniform variables and two-descent properties hold, yielding asymptotic independence between the left and right spine contributions. Together, these results give a constructive, probabilistic origin for the Spitzer–Lamperti–Ney limit and deepen the understanding of conditioned critical branching structures along the spine.

Abstract

Let be a critical Galton--Waston branching process with finite variance . Spitzer (unpublished), Lamperti and Ney (1968) proved that for any fixed , as , where and are independent random variables having exponential distributions with parameters and respectively. The proof is short and elegent based on the Laplace transform. In this paper, we will specify where the two exponential random variables come from explicitly, in terms of the Geiger's conditioned tree. Actually, and are resulted from the ``left'' and ``right'' parts of the ``spine'' of the Geiger's tree at generation . To this end, more details and intrinsic properties about the Geiger's conditioned tree will be investigated, which are interesting in its own right as well.
Paper Structure (19 sections, 28 theorems, 163 equations, 2 figures)

This paper contains 19 sections, 28 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geiger's Conditioned Tree
  3. Main Result
  4. Reduced Geiger's Tree
  5. Properties of Reduced Geiger's Tree
  6. Branching Mechanism
  7. Left Part: Branching Process in Varying Environment
  8. Right Part: Critical Galton--Waston Process
  9. Split Times of the Spine in Reduced Geiger's Tree
  10. Proof of Theorem \ref{['th-main']}
  11. Exponential Limit of Left Part
  12. Identification of Limiting Distribution
  13. Convergence: Proof of Theorem \ref{['th-main']} (i)
  14. Exponential Limit of Right Part
  15. Identification of Limiting Distribution
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose $\alpha=1$ and $\sigma^{2}<\infty$, then for any fixed $0<t<1$, as $n\rightarrow\infty$, where $U_{t}$ and $V_{t}$ are independent random variables having exponential distributions with parameters $2/(t(1-t)\sigma^{2})$ and $2/(t\sigma^{2})$ respectively.

Figures (2)

  • Figure 1: The Geiger's conditioned tree $\tilde{T}_{n}$. Notice that $\{s_{i}\}$ and $\{(V_{i}, X_{i})\}$ are consistent with the notations in the construction of Geiger's tree, being in reverse order from $n$ to $1$, while the order of numbering in generation is from $1$ to $n$.
  • Figure 2: The reduced Geiger's tree. Only particles having non-empty descendants at $[nt]$ are left.

Theorems & Definitions (54)

  • Theorem 1.1: Athreya, Lamperti
  • Lemma 1.2: Geiger99
  • Theorem 1.3
  • Remark 1
  • Theorem 1.4
  • Remark 2
  • Remark 3
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.1: nested uniform random variables
  • ...and 44 more