On discrete stochastic processes generated by deterministic sequences and multiplication machines

Abstract We consider a discrete stochastic process X = (X0, X1, …) 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 logb 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 αn, 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.