[Paper Review] CLT for the capacity of the range of stable random walks

Wojciech Cygan, Nikola Sandrić+1 more|arXiv (Cornell University)|Apr 11, 2019

Stochastic processes and statistical mechanics18 references4 citations

TL;DR

This paper establishes a central limit theorem (CLT) for the capacity of the range of $d$-dimensional symmetric $α$-stable random walks under the condition $d/\alpha > 5/2$. The authors derive the asymptotic normality of the capacity by analyzing the variance of the range capacity and applying the Lindeberg-Feller CLT, proving convergence to a normal distribution after appropriate centering and scaling.

ABSTRACT

In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $α$-stable random walks with the index satisfying $d > 5α/2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.

Motivation & Objective

• To establish a central limit theorem (CLT) for the capacity of the range process of $d$-dimensional symmetric $\alpha$-stable Lévy processes.
• To analyze the long-time behavior of the capacity of the range set $\mathcal{R}_n = \{S_0, \dots, S_n\}$, where $\{S_n\}$ is a stable random walk.
• To determine the conditions under which the normalized capacity $\mathrm{Cap}(\mathcal{R}_n)$ converges in distribution to a normal random variable.
• To control the variance of the capacity process and verify Lindeberg-type conditions for the CLT to hold.

Proposed method

• Define the capacity of a set $A \subseteq \mathbb{Z}^d$ as $\mathrm{Cap}(A) = \sum_{x \in A} \mathbb{P}_x(T_A^+ = \infty)$, where $T_A^+$ is the first return time to $A$.
• Use the decomposition $\overline{\mathcal{C}}_n = \sum_{i=1}^{2^L} \overline{\mathcal{C}}^{(i)}_{n/2^L} + \mathcal{E}(n)$, splitting the range into $2^L$ blocks to control error terms.
• Apply the Lindeberg-Feller central limit theorem to the sum of i.i.d. block capacities $\overline{\mathcal{C}}^{(i)}_{n/2^L}$, verifying the variance and Lindeberg conditions.
• Control the error term $\mathbb{E}[|\mathcal{E}(n)|]$ via bounds on $h_d(n)$, the expected capacity of a block, using Lemma 3.2 and Lemma 5.4.
• Use the Cauchy-Schwarz inequality and Chebyshev’s inequality to bound the Lindeberg condition, ensuring asymptotic negligibility of large deviations.
• Choose $L = \lfloor \log_2(n^{\Delta/2}) \rfloor$ for $5/2 < d/\alpha < 3$ and $L = \lfloor \log_2(n^{1/4}) \rfloor$ for $d/\alpha \geq 3$ to ensure $\mathbb{E}[|\mathcal{E}(n)|]/\sqrt{n} \to 0$.

Experimental results

Research questions

• RQ1 Under what conditions does the capacity of the range of a $d$-dimensional symmetric $\alpha$-stable random walk satisfy a central limit theorem?
• RQ2 How does the variance of the capacity of the range process behave asymptotically as $n \to \infty$?
• RQ3 What is the role of strong transience ($d/\alpha > 2$) and the domain of attraction condition ($\alpha$-stable limit) in ensuring the CLT?
• RQ4 Can the Lindeberg-Feller condition be verified for the block-decomposed capacity process under the given assumptions?
• RQ5 How does the error in block decomposition $\mathcal{E}(n)$ decay relative to $\sqrt{n}$, and what choice of $L$ ensures convergence?

Key findings

• The capacity of the range $\mathrm{Cap}(\mathcal{R}_n)$ satisfies a central limit theorem under the condition $d/\alpha > 5/2$, with convergence to a normal distribution after proper centering and scaling.
• The asymptotic variance of the normalized capacity $\overline{\mathcal{C}}_n / \sqrt{n}$ converges to $\sigma_d^2 > 0$, ensuring a non-degenerate limit distribution.
• The error term $\mathbb{E}[|\mathcal{E}(n)|]$ decays faster than $\sqrt{n}$, specifically $\mathbb{E}[|\mathcal{E}(n)|]/\sqrt{n} \to 0$, when $L$ is chosen as $\lfloor \log_2(n^{\Delta/2}) \rfloor$ for $5/2 < d/\alpha < 3$ or $\lfloor \log_2(n^{1/4}) \rfloor$ for $d/\alpha \geq 3$.
• The Lindeberg condition is verified via moment bounds and tail probability estimates, ensuring the sum of block capacities satisfies the CLT.
• The proof relies on the strong transience of the walk ($d/\alpha > 2$) and the domain of attraction to an $\alpha$-stable law, which ensures the required regular variation of the scaling function.
• The result holds under assumptions (A1)–(A4), including aperiodicity, symmetry, strong transience, and the presence of one-step loops ($p_{1}(0) > 0$).