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Limit theorems and lack thereof for a multilayer random walk mimicking human mobility

Alessandra Bianchi, Marco Lenci, Françoise Pène

TL;DR

This work analyzes a continuous-time multilayer random walk on $\mathbb{R}^d \times \mathbb{N}$ with level-dependent in-layer dynamics and probabilistic level transitions. It proves a functional central limit theorem for the horizontal coordinate under an integrability condition, connecting diffusion to a time-change driven Brownian limit with covariance $\Sigma$ when $\bar v<\infty$ and $m<\infty$. In regimes where the integrability fails, the authors uncover a dichotomy: the appropriate scaling is $n^{1/\alpha}$ with $\alpha = \log(p_\downarrow/p_\uparrow)/\log \Lambda$, but convergence can fail in general; under certain circumstances (Gaussian increments) a symmetric $\alpha$-stable limit can arise if a certain function $c_\theta$ is constant, while non-constancy of $c_\theta$ leads to non-convergence. A central analytical tool is the analysis of the conditional variance $V_n$ and the random variable $Z=V_{\tau_0}$, whose characteristic function satisfies a fixed-point equation that yields a detailed small-argument asymptotic via contraction mappings. The paper further develops a computer-assisted framework to certify non-convergence and non-attraction to stable laws in a broad class of parameter choices, illustrating a form of strong anomalous diffusion in this multilayer mobility model.

Abstract

We introduce a continuous-time random walk model on an infinite multilayer structure inspired by transportation networks. Each layer is a copy of $\mathbb{R}^d$, indexed by a non-negative integer. A walker moves within a layer by means of an inertial displacement whose speed is a deterministic function of the layer index and whose direction and duration are random, but with a timescale that depends on the layer. After each inertial displacement, the walker may randomly shift level, up or down, independently of its past. The multilayer structure is hierarchical, in the sense that the speed is a nondecreasing function of the layer index. Our primary focus is on the diffusive properties of the system. Under a natural condition on the parameters of the model, we establish a functional central limit theorem for the $\mathbb{R}^d$-coordinate of the process. By contrast, in a class of examples where this condition is violated, we are able to determine the correct scaling of the process while proving that no limit theorem holds.

Limit theorems and lack thereof for a multilayer random walk mimicking human mobility

TL;DR

This work analyzes a continuous-time multilayer random walk on with level-dependent in-layer dynamics and probabilistic level transitions. It proves a functional central limit theorem for the horizontal coordinate under an integrability condition, connecting diffusion to a time-change driven Brownian limit with covariance when and . In regimes where the integrability fails, the authors uncover a dichotomy: the appropriate scaling is with , but convergence can fail in general; under certain circumstances (Gaussian increments) a symmetric -stable limit can arise if a certain function is constant, while non-constancy of leads to non-convergence. A central analytical tool is the analysis of the conditional variance and the random variable , whose characteristic function satisfies a fixed-point equation that yields a detailed small-argument asymptotic via contraction mappings. The paper further develops a computer-assisted framework to certify non-convergence and non-attraction to stable laws in a broad class of parameter choices, illustrating a form of strong anomalous diffusion in this multilayer mobility model.

Abstract

We introduce a continuous-time random walk model on an infinite multilayer structure inspired by transportation networks. Each layer is a copy of , indexed by a non-negative integer. A walker moves within a layer by means of an inertial displacement whose speed is a deterministic function of the layer index and whose direction and duration are random, but with a timescale that depends on the layer. After each inertial displacement, the walker may randomly shift level, up or down, independently of its past. The multilayer structure is hierarchical, in the sense that the speed is a nondecreasing function of the layer index. Our primary focus is on the diffusive properties of the system. Under a natural condition on the parameters of the model, we establish a functional central limit theorem for the -coordinate of the process. By contrast, in a class of examples where this condition is violated, we are able to determine the correct scaling of the process while proving that no limit theorem holds.

Paper Structure

This paper contains 15 sections, 12 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Proof of Theorem \ref{['th:firstcase']}
  4. Proofs of Theorems \ref{['THMcase2']} and \ref{['THMcase2-convergence']}
  5. Strategy of the proofs
  6. The conditional variance Vn
  7. The characteristic function of Z
  8. Proof of Proposition \ref{['asymptphi']}
  9. Final arguments
  10. Z is not in the domain of attraction of a stable random variable
  11. Case $k=1$
  12. Case $k=2$
  13. Case $k=3$
  14. Cases $\Lambda=2$
  15. Cases $\Lambda=3$

Key Result

Theorem 2.1

Assume For any distribution $\nu$ on $\mathbb{N}$ (representing the initial distribution of the Markov chain $L$) and any $T>0$, where $m :={\mathbb E}_\mu[\mathcal{T}(1)] = {\mathbb E}[ |\xi_0| ] \sum_{\ell=0}^\infty \mu_\ell \, \sigma_\ell < \infty$ and $( B^\Sigma_t )_{t\ge 0}$ denotes Brownian motion with covariance matrix $\Sigma$.

Figures (2)

  • Figure 1: A trajectory of the process $W_t$, in the case $d=2$.
  • Figure 2: A realization of $L_j$, for $j=0, \ldots, \tau_0$. The figure illustrates the definitions \ref{['def-tau1k']} and Eqn. \ref{['EqZ']} (with $\mathcal{N}=4$).

Theorems & Definitions (26)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3: Scaling
  • Theorem 2.4: Convergence
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3
  • Corollary 4.4
  • ...and 16 more