## Bernoulli

- Bernoulli
- Volume 15, Number 1 (2009), 195-222.

### Discrete approximation of a stable self-similar stationary increments process

C. Dombry and N. Guillotin-Plantard

#### Abstract

The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known about the context in which such processes can arise. To our knowledge, discretization and convergence theorems are available only in the case of stable Lévy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema first introduced by Cohen and Samorodnitsky, which we consider in a more general setting. Strong relationships with Kesten and Spitzer’s random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process.

#### Article information

**Source**

Bernoulli, Volume 15, Number 1 (2009), 195-222.

**Dates**

First available in Project Euclid: 3 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1233669888

**Digital Object Identifier**

doi:10.3150/08-BEJ147

**Mathematical Reviews number (MathSciNet)**

MR2546804

**Zentralblatt MATH identifier**

1214.60020

**Keywords**

random scenery random walk self-similarity stable process

#### Citation

Dombry, C.; Guillotin-Plantard, N. Discrete approximation of a stable self-similar stationary increments process. Bernoulli 15 (2009), no. 1, 195--222. doi:10.3150/08-BEJ147. https://projecteuclid.org/euclid.bj/1233669888