## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 20 (2015), paper no. 8, 11 pp.

### Concentration and exact convergence rates for expected Brownian signatures

#### Abstract

The signature of a $d$-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over $\mathbb{R}^{d}$. In this article, we derive the exact rate of convergence for the expected signatures of piecewise linear approximations to Brownian motion. The computation is based on the identification of the set of words whose coefficients are of the leading order, and the convergence is concentrated on this subset of words. Moreover, under the choice of $l^{1}$ tensor norm, we give the explicit value of the leading term constant.

#### Article information

**Source**

Electron. Commun. Probab., Volume 20 (2015), paper no. 8, 11 pp.

**Dates**

Accepted: 28 January 2015

First available in Project Euclid: 7 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1465320935

**Digital Object Identifier**

doi:10.1214/ECP.v20-3636

**Mathematical Reviews number (MathSciNet)**

MR3304414

**Zentralblatt MATH identifier**

1307.60110

**Subjects**

Primary: 60G15: Gaussian processes

**Keywords**

expected signature Brownian motion

**Rights**

This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Ni, Hao; Xu, Weijun. Concentration and exact convergence rates for expected Brownian signatures. Electron. Commun. Probab. 20 (2015), paper no. 8, 11 pp. doi:10.1214/ECP.v20-3636. https://projecteuclid.org/euclid.ecp/1465320935