## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 2 (1993), 710-720.

### Random Processes of the Form $X_{n+1} = a_nX_n + b_n (\mod p)$

#### Abstract

This paper considers random processes of the form $X_{n + 1} = a_nX_n + b_n (\operatorname{mod} p)$, where $X_0 = 0$ and the sequences $a_n$ and $b_n$ are independent with $a_n$ identically distributed for $n = 0, 1, 2, \ldots$ and $b_n$ identically distributed for $n = 0, 1, 2, \ldots$. Chung, Diaconis and Graham studied such processes where $a_n = 2$ always; this paper considers more general distributions for $a_n$ and $b_n$. The question is how long does it take these processes to get close to the uniform distribution? If $a_n$ is a distribution on $\mathbf{Z}^+$ which does not vary with $p$ and $b_n$ is a distribution on $\mathbf{Z}$ which also does not vary with $p$, an upper bound on this time is $O((\log p)^2)$ with appropriate restrictions on $p$ unless $a_n = 1$ always, $b_n = 0$ always or $a_n$ and $b_n$ can each take on only one value. This paper uses a recursive relation involving the discrete Fourier transform to find the bound. Under more restrictive conditions for $a_n$ and $b_n$, this paper finds that a generalization of the technique of Chung, Diaconis and Graham shows that $O(\log p \log \log p)$ steps suffice.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 2 (1993), 710-720.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989264

**Digital Object Identifier**

doi:10.1214/aop/1176989264

**Mathematical Reviews number (MathSciNet)**

MR1217562

**Zentralblatt MATH identifier**

0776.60012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Secondary: 60J15

**Keywords**

Random processes Fourier transform convergence uniform distribution upper bound lemma recursion

#### Citation

Hildebrand, Martin. Random Processes of the Form $X_{n+1} = a_nX_n + b_n (\mod p)$. Ann. Probab. 21 (1993), no. 2, 710--720. doi:10.1214/aop/1176989264. https://projecteuclid.org/euclid.aop/1176989264