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Wasserstein-$1$ distance and nonuniform Berry-Esseen bound for a supercritical branching process in a random environment

Hao Wu, Xiequan Fan, Zhiqiang Gao, Yinna Ye

TL;DR

The paper analyzes a supercritical branching process in a random environment and proves an optimal $W_1$-distance rate for the centered log-population, extending prior results to the case where $X=\log m_0$ has a $2+\delta$ moment. It also establishes an exponential nonuniform Berry-Esseen bound under stronger moment (Cramér/Bernstein-type) conditions, with explicit tail controls. The results are then translated into practical interval estimation for the criticality parameter $\mu$ and for the population size $Z_n$, leveraging a decomposition $\log Z_n = S_n + \log W_n$ and martingale techniques. Overall, the work provides sharp probabilistic approximations for $\log Z_n$ and has direct applications to confidence interval construction in branching processes with random environments.

Abstract

Let $ (Z_{n})_{n\geq 0} $ be a supercritical branching process in an independent and identically distributed random environment. We establish an optimal convergence rate in the Wasserstein-$1$ distance for the process $ (Z_{n})_{n\geq 0} $, which completes a result of Grama et al. [Stochastic Process. Appl., 127(4), 1255-1281, 2017]. Moreover, an exponential nonuniform Berry-Esseen bound is also given. At last, some applications of the main results to the confidence interval estimation for the criticality parameter and the population size $Z_n$ are discussed.

Wasserstein-$1$ distance and nonuniform Berry-Esseen bound for a supercritical branching process in a random environment

TL;DR

The paper analyzes a supercritical branching process in a random environment and proves an optimal -distance rate for the centered log-population, extending prior results to the case where has a moment. It also establishes an exponential nonuniform Berry-Esseen bound under stronger moment (Cramér/Bernstein-type) conditions, with explicit tail controls. The results are then translated into practical interval estimation for the criticality parameter and for the population size , leveraging a decomposition and martingale techniques. Overall, the work provides sharp probabilistic approximations for and has direct applications to confidence interval construction in branching processes with random environments.

Abstract

Let be a supercritical branching process in an independent and identically distributed random environment. We establish an optimal convergence rate in the Wasserstein- distance for the process , which completes a result of Grama et al. [Stochastic Process. Appl., 127(4), 1255-1281, 2017]. Moreover, an exponential nonuniform Berry-Esseen bound is also given. At last, some applications of the main results to the confidence interval estimation for the criticality parameter and the population size are discussed.
Paper Structure (10 sections, 12 theorems, 64 equations)

This paper contains 10 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Application to Interval Estimation
  4. Confidence Intervals for $\mu$
  5. Confidence Intervals for $Z_{n}$
  6. Proof of Main Results
  7. Preliminary Lemmas for Theorem \ref{['theorem2.1']}
  8. Proof of Theorem \ref{['theorem2.1']}
  9. Preliminary Lemmas for Theorem \ref{['theorem2.2']}
  10. Proof of Theorem \ref{['theorem2.2']}

Key Result

Theorem 2.1

Suppose that the conditions (A1) and (A2) are satisfied. Then Moreover, the same inequality holds when $\frac{\log Z_{n}-n\mu}{\sigma\sqrt{n}}$ is replaced by $-\frac{\log Z_{n}-n\mu}{\sigma\sqrt{n}}.$

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • ...and 2 more