CONVERGENCE AND NONCONVERGENCE OF SCALED SELF-INTERACTING RANDOM WALKS TO BROWNIAN MOTION PERTURBED AT EXTREMA

• 成果类型: Article
• 署名作者: Kosygina, Elena; Mountford, Thomas; Peterson, Jonathon
• 署名单位: City University of New York (CUNY) System; Baruch College (CUNY); Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Purdue University System; Purdue University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1629
• 发表日期: 2023
• 页码: 1684-1728
• 关键词: excited random-walks; bond repulsion

摘要

We use generalized Ray-Knight theorems, introduced by B. Toth in 1996, together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRW) to Brownian motions perturbed at extrema (BMPE). Toth's work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Toth has shown, in particular, that the distribution function of a scaled SIRW observed at independent geometric times converges to that of a BMPE indicated by the generalized Ray-Knight theorem for this SIRW. The question of weak convergence of one-dimensional distributions of scaled SIRW remained open. In this paper, on the one hand, we prove a full func-tional limit theorem for a large class of asymptotically free SIRWs, which in-cludes the asymptotically free walks considered by Toth. On the other hand, we show that rescaled polynomially self-repelling SIRWs do not converge to the BMPE predicted by the corresponding generalized Ray-Knight theorems and hence do not converge to any BMPE.