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Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema

Xiaoyu Liu, Zhe Wang

TL;DR

This work analyzes a one-dimensional self-interacting random walk with weight function $w(n)$ obeying $\frac{1}{w(n)}=1+2^p B n^{-p}+O(n^{-1-\kappa})$ for $p\in(0,\tfrac{1}{2}]$, establishing a diffusion-scale limit to a Brownian motion perturbed at extrema (BMPE) with equal perturbation parameters $\gamma$. The authors implement a decomposition of the walk into a martingale and a drift, proving the martingale part converges to Brownian motion while the drift converges to $\gamma$ times the walk’s running range, via generalized Ray-Knight theorems and a mesoscopic analysis using generalized Polya urns and branching-like processes that model local times and up/down-crossings. A key technical contribution is handling non-absolute summability of local drifts for $p\le\tfrac{1}{2}$ by establishing typical-event control and conditional-mean approximations, which together yield process-level convergence to the BMPE. The results deepen the understanding of universality in BMPE limits for asymptotically free SIRWs and provide a robust framework (urns, BLPs, and filtration-based martingale arguments) potentially applicable to related non-Markov walks with edge-local-time interactions.

Abstract

We consider a family of one-dimensional self interacting walks whose dynamics characterized by a monotone weight function $w$ on $\mathbb{N}\cup \{0\}$. The weight function takes the form $w(n) = (1 + 2^p Bn^{-p} + O(n^{-1-κ}))^{-1}$, for some $B \in \mathbb{R} $, $κ>0$ and $p\in (0,1]$. Our main model parameter is $p$, and for $p\in (0,1/2]$ we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range $(0,1]$. Our method depends on the generalized Ray-Knight theorems ([T96], [KMP23]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker.

Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema

TL;DR

This work analyzes a one-dimensional self-interacting random walk with weight function obeying for , establishing a diffusion-scale limit to a Brownian motion perturbed at extrema (BMPE) with equal perturbation parameters . The authors implement a decomposition of the walk into a martingale and a drift, proving the martingale part converges to Brownian motion while the drift converges to times the walk’s running range, via generalized Ray-Knight theorems and a mesoscopic analysis using generalized Polya urns and branching-like processes that model local times and up/down-crossings. A key technical contribution is handling non-absolute summability of local drifts for by establishing typical-event control and conditional-mean approximations, which together yield process-level convergence to the BMPE. The results deepen the understanding of universality in BMPE limits for asymptotically free SIRWs and provide a robust framework (urns, BLPs, and filtration-based martingale arguments) potentially applicable to related non-Markov walks with edge-local-time interactions.

Abstract

We consider a family of one-dimensional self interacting walks whose dynamics characterized by a monotone weight function on . The weight function takes the form , for some , and . Our main model parameter is , and for we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range . Our method depends on the generalized Ray-Knight theorems ([T96], [KMP23]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker.
Paper Structure (15 sections, 17 theorems, 101 equations)

This paper contains 15 sections, 17 theorems, 101 equations.

Table of Contents

  1. Introduction and Main Result
  2. Main Result
  3. Organization of the Paper
  4. Notation
  5. Functional Limit Theorem for AF-SIRW: Proof of Theorem \ref{['th: main']}
  6. Generalized Pólya Urn, Branching-Like Processes
  7. Generalized Pólya Urn
  8. Branching-Like Processes
  9. Filtrations of BLPs, and Approximation of Accumulated Drifts
  10. Preliminary Results
  11. Approximations of Accumulated Drift
  12. Typical Events
  13. Control of Martingale Terms, Lemma 2.1
  14. Convergence of Conditional Expectation, Lemma 2.3
  15. Approximation of Local Drifts by Conditional Means, Lemma 2.4

Key Result

Theorem 1.1

Let $w(.)$ be monotone and satisfy eq: asymptotics of w for $p\in (\frac{1}{2},1]$, and $\kappa >0$. Consider the SIRW $(X_k)_{k\geq 0}$ defined in eq: dynamic with $X_0 =0$. We have the following process-level convergence as $n$ goes to infinity, in the standard Skorohod topology on $D([0,\infty) ).$

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.1: KMP23, Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Remark 3.1
  • Remark 3.2
  • ...and 19 more