Integer-valued self-similar processes

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Abstract
Let IN0 denote the set of nonnegative integers. We consider IN0-valued analogues of self-similar processes by defining Unvalued fractions of IN0-valued processes. These fractions are defined in terms of sums of independent Markov branching processes, in such a way that the one-dimensional marginals coincide with the IN0-valued multiples of IN0-valued random variables as introduced in [10] and [3]; this requirement still leaves room for several definitions of an 0-valued fraction, and a sensible choice has to be made. The relation with branching processes has two aspects. On the one hand, results from the theory of these processes can be used to prove analogues of classical theorems, on the other hand new results about branching processes are suggested by translating analogues of classical results in terms of branching processes. In this way, analogues are derived of the basic properties of classically self-similar processes such as obtained by Lamperti [5], and a simple relationship is established between the IN0-valued self-similar processes and the classically self-similar processes. Translation of this main result in terms of branching processes yields a slightly surprising limit theorem.

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