## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 32, Number 1 (2018), 188-199.

### Abrupt convergence for a family of Ornstein–Uhlenbeck processes

#### Abstract

We consider a family of Ornstein–Uhlenbeck processes. Under some suitable assumptions on the behaviour of the drift and diffusion coefficients, we prove profile cut-off phenomenon with respect to the total variation distance in the sense of the definition given by Barrera and Ycart [*ALEA Lat. Am. J. Probab. Math. Stat.* **11** (2014) 445–458]. We compute explicitly the cut-off time, the window time, and the profile function. Moreover, we prove that the average process satisfies a profile cut-off phenomenon with respect to the total variation distance. Also, a sample of $N$ Ornstein–Uhlenbeck processes has a window cut-off with respect to the total variation distance in the sense of the definition given by Barrera and Ycart [*ALEA Lat. Am. J. Probab. Math. Stat.* **11** (2014) 445–458]. The cut-off time and the cut-off window for the average process and for the sampling process are the same.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 32, Number 1 (2018), 188-199.

**Dates**

Received: December 2015

Accepted: September 2016

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1520046140

**Digital Object Identifier**

doi:10.1214/16-BJPS337

**Mathematical Reviews number (MathSciNet)**

MR3770869

**Zentralblatt MATH identifier**

06973954

**Keywords**

Cut-off phenomenon total variation distance Ornstein–Uhlenbeck processes

#### Citation

Barrera, Gerardo. Abrupt convergence for a family of Ornstein–Uhlenbeck processes. Braz. J. Probab. Stat. 32 (2018), no. 1, 188--199. doi:10.1214/16-BJPS337. https://projecteuclid.org/euclid.bjps/1520046140