ON THE STRUCTURE AND THE HAUSDORFF DIMENSION OF THE SUPPORT OF A CLASS OF DISTRIBUTION FUNCTIONS INDUCED BY ERGODIC SEQUENCES

摘要

In this note we examine the distribution function of the random variable Z = Σ, Xi D-1 when the random variables X1,X2••• form a stationary ergodici-1 sequence and each X. takes on integer values running from 0 to D-l. We show that the distribution function will be a step function, or a continuous and purely singular function, or else a linear function of its argument. The form of the distribution function is related to the entropy rate of the random sequence. We discuss this relationship and also the relationship of entropy rate to the concept of Hausdorff dimension.