Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates

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 $t\rightarrow\infty$. 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 $N\rightarrow\infty$. 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 $\alpha$-stable Levy processes this results holds for any finite $N$ in the precise form with $b_{N}=(N-1)^{1/\alpha}$. 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.