## Brazilian Journal of Probability and Statistics

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

Gerardo Barrera

#### 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
Accepted: September 2016
First available in Project Euclid: 3 March 2018

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

Digital Object Identifier
doi:10.1214/16-BJPS337

Mathematical Reviews number (MathSciNet)
MR3770869

Zentralblatt MATH identifier
06973954

#### 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

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