Dilatively stable stochastic processes and aggregate similarity

Abstract

Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Igloi (Dilative stability, Ph.D. Thesis, University of Debrecen, Faculty of Informatics, http://www.inf.unideb.hu/valseg/dolgozok/igloi/dissertation.pdf (2008)) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (Fractals in Engineering, New Trends in Theory and Applications, pp 199–218 (2005)). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional Levy processes are given and we point out that certain limit processes in aggregation models are dilatively stable.