## Electronic Communications in Probability

### 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

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

Rights

#### 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

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