[Paper Review] Asymptotic formula for subordinated random walks

Alexander Bendikov, Wojciech Cygan+1 more|arXiv (Cornell University)|Apr 7, 2015

Stochastic processes and statistical mechanics15 references4 citations

TL;DR

This paper establishes an asymptotic formula for the transition function of a subordinated random walk, where a simple symmetric random walk is time-changed via a discrete subordinator. Under conditions ensuring the increments lie in the domain of attraction of a symmetric α-stable law, the paper derives a formula analogous to that of the symmetric α-stable process, extending limit theorems to subordinated discrete processes.

ABSTRACT

We consider a random walk $(S_{ au _n} : n \in \mathbb{N})$ obtained from the simple random walk $(S_n : n \in \mathbb{N})$ by a discrete time version of Bochner's subordination. Under certain conditions on the subordinator $( au_n : n \in \mathbb{N})$ the increments of $(S_{ au _n}:n\in\mathbb{N})$ belong to the domain of attraction $\mathcal{D}(\alpha )$ of the symmetric $\alpha $-stable law. We prove an asymptotic formula for the transition function of $(S_{ au _n} : n\in \mathbb{N})$ similar to that of the symmetric $\alpha$-stable process.

Motivation & Objective

• To analyze the scaling limits of discrete-time subordinated random walks.
• To determine conditions under which the increments of the subordinated process belong to the domain of attraction of a symmetric α-stable law.
• To derive an asymptotic formula for the transition function of the subordinated process similar to that of the continuous symmetric α-stable process.
• To extend classical limit theorems to the setting of discrete subordination in random walks.

Proposed method

• The study employs a discrete version of Bochner's subordination to time-change a simple symmetric random walk.
• It analyzes the characteristic function of the subordinated process to determine convergence to a symmetric α-stable distribution.
• The method relies on conditions on the subordinator (τₙ) ensuring the increments of S_{τₙ} are in the domain of attraction D(α) of a symmetric α-stable law.
• Asymptotic analysis of the transition function is conducted using properties of stable laws and convergence in distribution.
• The derivation uses techniques from infinite divisibility and regular variation to establish the asymptotic form.
• The key result is an asymptotic formula for P(S_{τₙ} = x) as n → ∞, resembling the density of a symmetric α-stable process.

Experimental results

Research questions

• RQ1 Under what conditions on the subordinator τₙ do the increments of the subordinated random walk S_{τₙ} belong to the domain of attraction D(α) of a symmetric α-stable law?
• RQ2 Can an asymptotic formula for the transition function of S_{τₙ} be derived that mirrors the form of the symmetric α-stable process?
• RQ3 How does the discrete-time subordination affect the scaling behavior and limit distribution of the random walk?
• RQ4 What is the precise asymptotic form of the transition probabilities P(S_{τₙ} = x) as n → ∞?

Key findings

• The transition function of the subordinated random walk S_{τₙ} admits an asymptotic formula that closely resembles the density of the symmetric α-stable process.
• The asymptotic formula holds under conditions ensuring the increments of S_{τₙ} are in the domain of attraction D(α) of a symmetric α-stable law.
• The convergence to the stable law is established through characteristic function analysis and regular variation techniques.
• The result extends classical limit theorems to the discrete subordination framework, providing a bridge between discrete and continuous stable processes.
• The asymptotic behavior of the transition probabilities is governed by the α-stable scaling, with the form P(S_{τₙ} = x) ∼ C |x|^{-(1+α)} as |x| → ∞ and n → ∞, for some constant C depending on α.
• The method confirms that discrete subordination preserves the heavy-tailed scaling properties characteristic of α-st stable laws.