Construction and convergence results for stable webs

TL;DR

This paper introduces a new metric for collections of aged paths, establishes criteria for compactness, and proves the convergence of coalescing random walks to stable webs in the domain of attraction of stable laws.

Contribution

It develops a new metric and compactness criteria for stable webs and proves their convergence from coalescing random walks in the stable law domain.

Findings

Distribution of stable webs is tight under the new metric.

Normalized coalescing random walks converge to stable webs.

Path properties are analyzed in the Brownian case.

Abstract

We introduce a new metric for collections of aged paths and a robust set of criteria for compactness for a set of collection of aged paths in the topology corresponding to this metric. We show that the distribution of stable webs ($1< \alpha \leq 2$) made up of collections of stable paths is tight in this topology. We then show the weak convergence of appropriately normalized systems of coalescing random walks in the domain of attraction of stable laws for $1 < \alpha \leq 2$ under this metric to the corresponding stable web. We obtain some path results in the brownian case.