On discrete stochastic processes generated by deterministic sequences and multiplication machines

Abstract

We consider a discrete stochastic process X = ( X 0, X 1, …) with finite state space {0, 1, …, b − 1}, which carries the random asymptotic behaviour of the relative frequency in which the digits appear in the expansion in base b of a linear recurrent sequence of real numbers. If ϱ denotes the dominant root of the characteristic polynomial associated with the linear recurrence relation, by a classical result, the stochastic process X does not depend on the recurrence relation whenever ϱ > 1 and log b ϱ is irrational. We prove that this stochastic process X has asymptotically independent values and is asymptotically identically distributed, with asymptotic distribution of equal probability to every state. We also show that in the case of β-expansions of a linear recurrent sequence of real numbers, the corresponding stochastic process X is asymptotically identically distributed, but in the case β > 1 is not a integer, it does not have asymptotically independent values. The speed of convergence to equilibrium is shown to be exponential. Moreover, in the case of the sequence α n , we show an explicit relationship between multiplication by α (as a multiplication machine in the β-shift) and the irrational rotation. We finish with a remark that some of these results are satisfied for other sequences of real numbers.