Random Processes of the Form X _n+1 = a _n X _n + b _n (mod p)where b _n takes on a Single Value

Summary

This paper studies random processes of the form X _n +1 = a _n X _n + b _n (mod p) where b _n has only one possible value and a _n is o or 1 with probability l/2 each. For values of p satisfying certain constraints imposed by a _n and b _n , X _n gets close to uniformly distributed on Z/pZ for large enough n. This paper explores how large n needs to be as a function of p. Adapting techniques used by Chung, Diaconis, and Graham and techniques previously developed by the author, this paper shows that if n > c ₁ log p log log p then X _n gets close to uniformly distributed as p approaches infinity and that if n > c ₂ log p then X _n approaches the uniform distribution for almost all p satisfying the constraints. Furthermore, if a = 2, this paper shows that for certain p, if X _n gets close to uniformly distributed, then n > C₃ log p log log p.