## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 1 (1988), 1-57.

### Self-Intersection Gauge for Random Walks and for Brownian Motion

#### Abstract

A class of random fields associated with multiple points of a random walk in the plane is studied. It is proved that these fields converge in distribution to analogous fields measuring self-intersections of the planar Brownian motion. The concluding section contains a survey of literature on intersection local times and their renormalizations. A brief look through the first pages of this section could provide the reader with additional motivation for the present work.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 1 (1988), 1-57.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991884

**Digital Object Identifier**

doi:10.1214/aop/1176991884

**Mathematical Reviews number (MathSciNet)**

MR920254

**Zentralblatt MATH identifier**

0638.60081

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G60: Random fields

Secondary: 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65]

**Keywords**

Self-intersection local times self-intersection gauges the invariance principle multiple points of random walks and of the Brownian motion

#### Citation

Dynkin, E. B. Self-Intersection Gauge for Random Walks and for Brownian Motion. Ann. Probab. 16 (1988), no. 1, 1--57. doi:10.1214/aop/1176991884. https://projecteuclid.org/euclid.aop/1176991884