On the Range of a Regenerative Sequence.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Consider a system which evolves randomly in time the trajectory of such a system traces a path through space. If one partitions space into a disjoint collection of subsets, one can study the number of subsets visisted by the trajectory up to a certain instant this paper shows that, under certain conditions, the number of subsets never grows linearly in time, regardless of the partition used. On the other hand, the precise order of growth which can be arbitrarily close to linear order does depend on the way in which space is partitioned. These results are obtained for regenerative random processes. Such processes describe systems which, when viewed on a certain random time scale, evolve in an independent and identically distributed fashion. Virtually any ergodic discrete-time Markov chain has this property.
- Statistics and Probability