## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 23 (2018), paper no. 58, 47 pp.

### Sample path properties of permanental processes

Michael B. Marcus and Jay Rosen

#### Abstract

Let $X_{\alpha }=\{X_{\alpha }(t),t\in{\cal T} \}$, $\alpha >0$, be an $\alpha $-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha }$ is a subgaussian process with respect to the metric \[ \sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2} .\nonumber \] This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $\alpha $-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient Lévy processes that are not necessarily symmetric, or with kernels of the form \[ \widetilde{u} (x,y)= u(x,y)+f(y),\nonumber \] where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.

#### Article information

**Source**

Electron. J. Probab., Volume 23 (2018), paper no. 58, 47 pp.

**Dates**

Received: 3 November 2017

Accepted: 30 May 2018

First available in Project Euclid: 11 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1528704076

**Digital Object Identifier**

doi:10.1214/18-EJP183

**Mathematical Reviews number (MathSciNet)**

MR3814252

**Zentralblatt MATH identifier**

06924670

**Subjects**

Primary: 60K99: None of the above, but in this section 60G15: Gaussian processes 60G17: Sample path properties 60G99: None of the above, but in this section

**Keywords**

permanental processes subgaussian processes moduli of continuity of permanental processes permanental processes at infinity

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Marcus, Michael B.; Rosen, Jay. Sample path properties of permanental processes. Electron. J. Probab. 23 (2018), paper no. 58, 47 pp. doi:10.1214/18-EJP183. https://projecteuclid.org/euclid.ejp/1528704076