Abstract

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment. We establish nonuniform Berry-Esseen bounds for the process $(Z_n)$, which refine the Berry-Esseen bound of Grama et al. [Stochastic Process. Appl., 127(4), 1255-1281, 2017]. We also discuss an application of our result to constructing confidence intervals for the criticality parameter.

Keywords: Branching processes, Random environment, Nonuniform Berry-Esseen bounds

MSC(2010): 60J80, 60K37, 60F05, 62E20

doi: 10.1360/012011-XXX

1. Introduction

Branching processes in random environments have been extensively studied since the pioneering work of Smith and Wilkinson [1]. Consider a branching process $(Z_n){n\geq 0}$ in an i.i.d. random environment $\xi = (\xi_0, \xi_1, \ldots)$. The process evolves such that, given the environment $\xi_n$, the offspring distribution at generation $n$ is $p(\xi_n) = {p_k(\xi_n) = P(X$ are i.i.d. random variables representing the number of offspring of the $i$-th individual in generation $n$.} = k \mid \xi_n) : k \in \mathbb{N}}$, where $(X_{n,i})_{i\geq 1

Let $m_n = E_\xi X_{n,i}$ denote the conditional mean offspring number in generation $n$, and define $\Pi_n = \prod_{i=0}^{n-1} m_i$ with $\Pi_0 = 1$. The process $W_n = Z_n / \Pi_n$ forms a nonnegative martingale with respect to the filtration $\mathcal{F}n = \sigma{\xi, X, 0 \leq k \leq n-1, i \geq 1}$. Under suitable conditions, $W_n$ converges almost surely to a limit $W$ with $E[W] \leq 1$.

The asymptotic behavior of $Z_n$ is governed by the random walk $S_n = \log \Pi_n = \sum_{i=0}^{n-1} X_i$, where $X_i = \log m_i$. Let $\mu = E[X]$ and $\sigma^2 = \operatorname{Var}(X)$. The process is classified as subcritical ($\mu < 0$), critical ($\mu = 0$), or supercritical ($\mu > 0$). In the supercritical case, $\mu > 0$ and the process grows exponentially at rate $\mu$.

Previous work has established central limit theorems and Berry-Esseen bounds for $\log Z_n$. Grama et al. [9] proved a uniform Berry-Esseen bound:
$$
\sup_{x \in \mathbb{R}} \left| P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq \frac{C}{\sqrt{n}},
$$
under moment conditions $E[X^{2+\delta}] < \infty$ and $E[Z_1^p] < \infty$ for some $p > 1$ and $\delta \in (0,1]$. However, uniform bounds do not capture the precise rate of convergence in the tails.

Nonuniform Berry-Esseen bounds provide refined estimates that depend on $x$, typically of the form:
$$
\left| P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq \frac{C}{(1+|x|^{1+\delta'})},
$$
for $\delta' \in (0,\delta)$. Such bounds are crucial for constructing confidence intervals and understanding moderate deviations.

2. Model and Main Results

2.1 Branching Process in Random Environment

Let $\xi = (\xi_0, \xi_1, \ldots)$ be an i.i.d. sequence of random variables representing the environment. The branching process $(Z_n){n\geq 0}$ is defined recursively by:
$$
Z_0 = 1, \quad Z, \quad n \geq 0,} = \sum_{i=1}^{Z_n} X_{n,i
$$
where, conditioned on $\xi$, the random variables $(X_{n,i}){i\geq 1}$ are i.i.d. with distribution $p(\xi_n)$. The conditional mean offspring number is $m_n = E\xi X_{n,i}$, and we define:
$$
\Pi_0 = 1, \quad \Pi_n = \prod_{i=0}^{n-1} m_i, \quad n \geq 1.
$$
The normalized process $W_n = Z_n / \Pi_n$ is a nonnegative martingale converging a.s. to $W$ with $E[W] \leq 1$.

Let $X_i = \log m_i$ and define the random walk:
$$
S_0 = 0, \quad S_n = \log \Pi_n = \sum_{i=0}^{n-1} X_i, \quad n \geq 1.
$$
Assume $\mu = E[X] \in (0,\infty)$ and $\sigma^2 = \operatorname{Var}(X) \in (0,\infty)$. The criticality parameter $\mu$ determines the exponential growth rate of $Z_n$.

We impose the following moment conditions:

• (A1) There exists $\delta \in (0,1]$ such that $E[|X|^{2+\delta}] < \infty$.
• (A2) There exists $p > 1$ such that $E[Z_1^p] < \infty$.
• (A3) There exists $\lambda_0 > 0$ such that $E[e^{\lambda_0 X}] = E[m_0^{\lambda_0}] < \infty$ (Cramér's condition).
• (A4) There exists $p > 1$ such that $E[Z_1^p] < \infty$.

2.2 Main Results

Our first result establishes a nonuniform Berry-Esseen bound under conditions (A1) and (A2).

Theorem 1. Assume (A1) and (A2) hold. Then for any $\delta' \in (0,\delta)$, there exists a constant $C > 0$ such that for all $x \in \mathbb{R}$ and $n \geq 1$,
$$
\left| P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq \frac{C}{1+|x|^{1+\delta'}}. \tag{2.3}
$$

This refines the uniform bound of Grama et al. [9] by providing $x$-dependent convergence rates. The bound (2.3) is particularly useful for moderate deviations where $|x| = o(\sqrt{n})$.

Under stronger exponential moment conditions, we obtain an even sharper bound:

Theorem 2. Assume (A3) and (A4) hold. Then there exist constants $C, c > 0$ such that for all $x \in \mathbb{R}$ and $n \geq 1$,
$$
\left| P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq C(1+x^2)\exp\left(-\frac{cx^2}{1+|x|/\sqrt{n}}\right). \tag{2.4}
$$

This bound provides exponential decay in the tails and is valid for $|x| = o(\sqrt{n})$. The result improves upon Theorem 1.1 in Grama et al. [9] by giving an explicit nonuniform estimate.

Theorem 3 (Confidence Interval). Under (A1) and (A2), let $\kappa_n \in (0,1)$ satisfy $|\log \kappa_n| = o(\log n)$. Then for the interval:
$$
\left[ \frac{\log Z_n}{\sqrt{n}} - \sigma\Phi^{-1}(1-\kappa_n/2), \frac{\log Z_n}{\sqrt{n}} + \sigma\Phi^{-1}(1-\kappa_n/2) \right],
$$
we have:
$$
P\left( \mu \in \left[ \frac{\log Z_n}{n} - \frac{\sigma}{\sqrt{n}}\Phi^{-1}(1-\kappa_n), \frac{\log Z_n}{n} + \frac{\sigma}{\sqrt{n}}\Phi^{-1}(1-\kappa_n) \right] \right) = 1 - \kappa_n + o(1).
$$

This provides a practical method for constructing asymptotic confidence intervals for the criticality parameter $\mu$.

3. Preliminary Lemmas

We establish several technical lemmas concerning the martingale limit $W$ and its logarithm.

Lemma 2. Under (A1) and (A2), for any $q \in (1,1+\delta)$, we have $E[|\log W|^q] < \infty$ and $\sup_n E[|\log W_n|^q] < \infty$.

Proof. Using the decomposition $\log Z_n = S_n + \log W_n$ from (2.1) and moment estimates for $W_n$, we obtain the uniform bound:
$$
E[|\log W_n|^q] \leq C E[W] + E[|\log W|^q \mathbf{1}_{{W \leq 1}}] < \infty.
$$

Lemma 3. Under (A1) and (A2), there exists $\gamma \in (0,1)$ such that:
$$
E[|\log W_n - \log W|] \leq C\gamma^n.
$$

This exponential convergence is crucial for controlling the remainder term in the decomposition of $\log Z_n$.

Lemma 4. Under (A1) and (A2), for any $x \in \mathbb{R}$ and $\delta' \in (0,\delta)$:
$$
\left| P\left(\frac{S_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq \frac{C}{1+|x|^{1+\delta'}}. \tag{4.18}
$$

The proof follows from classical nonuniform Berry-Esseen bounds for i.i.d. sums (see Bikelis [11] and Chen & Shao [12]).

Lemma 5. Under (A1) and (A2), for any $x > 0$:
$$
P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \geq x\right) \leq C\exp\left(-\frac{cx^2}{1+|x|/\sqrt{n}}\right). \tag{4.19}
$$

4. Proof of Main Results

The key identity is the decomposition:
$$
\log Z_n = S_n + \log W_n, \tag{2.1}
$$
where $S_n = \sum_{i=0}^{n-1} X_i$ is the random walk of environmental means and $W_n = Z_n / \Pi_n$ is the normalized population size.

4.1 Proof of Theorem 1

We analyze the probability:
$$
P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) = P\left(\frac{S_n - n\mu}{\sigma\sqrt{n}} + \frac{\log W_n}{\sigma\sqrt{n}} \leq x\right).
$$

Let $m = \lfloor n/2 \rfloor$ and define $V_n = \log W_n$. Then:
$$
P\left(\frac{S_n - n\mu}{\sigma\sqrt{n}} \leq x, \frac{\log Z_n - n\mu}{\sigma\sqrt{n}} > x\right) \leq P(|V_n - V_m| > \alpha_n) + P\left(Y_n \leq x + \alpha_n, Y_n > x\right),
$$
where $Y_n = \frac{S_n - n\mu}{\sigma\sqrt{n}}$ and $\alpha_n = n^{-1/2}$.

Using Lemma 3, we bound:
$$
P(|V_n - V_m| > \alpha_n) \leq \frac{E|V_n - V_m|}{\alpha_n} \leq C\gamma^m \sqrt{n} \leq \frac{C}{1+|x|^{1+\delta'}}.
$$

The main term is controlled by Lemma 4, yielding:
$$
\left| P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq \frac{C}{1+|x|^{1+\delta'}}.
$$

4.2 Proof of Theorem 2

Under the Cramér condition (A3), we apply Nagaev's inequality to obtain exponential tail bounds for $S_n$. Combining with the decomposition (2.1) and Lemma 5, we derive for $|x| \leq n^{1/4}$:
$$
\left| P\left(\frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \leq x\right) - \Phi(x) \right| \leq C(1+x^2)\exp\left(-\frac{cx^2}{1+|x|/\sqrt{n}}\right).
$$

For $|x| > n^{1/4}$, the bound follows from Lemma 5 and standard Gaussian tail estimates:
$$
1 - \Phi(x) \leq \frac{e^{-x^2/2}}{\sqrt{2\pi}(1+x)}.
$$

4.3 Proof of Theorem 3

The confidence interval construction uses the quantile function $\Phi^{-1}(p)$. For $p \to 0$, we have the asymptotic expansion:
$$
\Phi^{-1}(1-p) = -\sqrt{2\log(1/p)} + O(\log\log(1/p)).
$$

Applying Theorem 1 with $\kappa_n \to 0$ slowly enough that $|\log \kappa_n| = o(\log n)$, we obtain:
$$
P\left( \frac{\log Z_n - n\mu}{\sigma\sqrt{n}} \in \left[-\Phi^{-1}(1-\kappa_n), \Phi^{-1}(1-\kappa_n)\right] \right) = 1 - \kappa_n + o(1).
$$

This yields the claimed confidence interval for $\mu$.

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