[Paper Review] Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

Anatoly Manita|arXiv (Cornell University)|Sep 9, 2014

Network Time Synchronization Technologies57 references3 citations

TL;DR

This paper introduces a class of high-dimensional, non-Markovian stochastic synchronization models driven by Lévy processes with synchronizing jumps. It establishes that symmetric systems achieve asymptotic synchronization in distribution, and under conditions where the Lévy process is in a stable law's domain of attraction, intrinsic spatial scales emerge—specifically, differences between components converge to a limit when scaled by $ b_N = (N-1)^{1/\alpha} $, with limiting laws including Linnik distributions.

ABSTRACT

We propose stochastic $N$-component synchronization models $(x_{1}(t),...,x_{N}(t))$, $x_{j}\in\mathbb{R}^{d}$, $t\in\mathbb{R}_{+}$, whose dynamics is described by Levy processes and synchronizing jumps. We prove that symmetric models reach synchronization in a stochastic sense: differences between components $d_{kj}^{(N)}(t)=x_{k}(t)-x_{j}(t)$ have limits in distribution as $t ightarrow\infty$. We give conditions of existence of natural (intrinsic) space scales for large synchronized systems, i.e., we are looking for such sequences $\{b_{N}\}$ that distribution of $d_{kj}^{(N)}(\infty)/b_{N}$ converges to some limit as $N ightarrow\infty$. It appears that such sequence exists if the Levy process enters a domain of attraction of some stable law. For Markovian synchronization models based on $α$-stable Levy processes this results holds for any finite $N$ in the precise form with $b_{N}=(N-1)^{1/α}$. For non-Markovian models similar results hold only in the asymptotic sense. The class of limiting laws includes the Linnik distributions. We also discuss generalizations of these theorems to the case of non-uniform matrix-based intrinsic scales. The central point of our proofs is a representation of characteristic functions of $d_{kj}^{(N)}(t)$ via probability distribution of a superposition of $N$ independent renewal processes.

Motivation & Objective

• To develop a general framework for non-Markovian, high-dimensional stochastic synchronization systems with Lévy-driven dynamics.
• To establish conditions under which such systems achieve long-time synchronization in distribution.
• To identify intrinsic spatial scales $ b_N $ such that the rescaled differences between components converge as $ N \to \infty $.
• To extend results from Markovian $ \alpha $-stable models to non-Markovian settings via renewal process superposition and characteristic function analysis.
• To explore the role of operator stable laws and matrix-based intrinsic scales in the asymptotic behavior of synchronized systems.

Proposed method

• Modeling component dynamics as independent Lévy processes interrupted by random synchronizing jumps at non-i.i.d. epochs.
• Representing the characteristic function of component differences via the superposition of $ N $ independent renewal processes.
• Applying Laplace transforms and generating functions to analyze the asymptotic behavior of characteristic functions.
• Using key renewal theorem approximations and bounds on characteristic function differences to derive convergence results.
• Introducing matrix-based intrinsic scales using Jurek coordinates to generalize scalar scaling in non-symmetric settings.
• Establishing convergence to Linnik distributions under domain-of-attraction conditions for stable laws.

Experimental results

Research questions

• RQ1 Under what conditions do non-Markovian, Lévy-driven synchronization models achieve asymptotic synchronization in distribution?
• RQ2 What intrinsic spatial scales $ b_N $ exist such that the rescaled differences $ d_{kj}^{(N)}(\infty)/b_N $ converge as $ N \to \infty $?
• RQ3 How do the limiting distributions of component differences depend on the Lévy process characteristics, particularly in the domain of attraction of a stable law?
• RQ4 Can the results for Markovian $ \alpha $-stable models be extended to non-Markovian models with general inter-event interval distributions?
• RQ5 What is the role of matrix-based scaling and Jurek coordinates in generalizing intrinsic scales beyond scalar forms?

Key findings

• Symmetric $ N $-component systems driven by Lévy processes achieve synchronization in distribution: $ d_{kj}^{(N)}(t) $ has a limit in distribution as $ t \to \infty $.
• Intrinsic scales $ b_N = (N-1)^{1/\alpha} $ exist for $ \alpha $-stable Lévy processes, yielding convergence to a non-degenerate limit as $ N \to \infty $.
• For non-Markovian models, intrinsic scaling holds asymptotically, with convergence of $ \chi_N(\infty; \lambda) $ to a characteristic function of the form $ \frac{1}{1 + \theta_{1,N} \boldsymbol{\eta}(\lambda)} $, where $ \theta_{1,N} \sim mN/\varkappa $.
• The limiting laws include Linnik distributions when the Lévy process is in the domain of attraction of a stable law.
• Matrix-based intrinsic scales via Jurek coordinates generalize scalar scaling, allowing for non-uniform scaling in high-dimensional systems.
• The error in approximating the limiting characteristic function is bounded by $ \mathcal{O}(N l_N) $, which vanishes as $ N \to \infty $, ensuring convergence.