Dilatively stable stochastic processes and aggregate similarity

Dilatively stable processes generalize the class of infinitely divisible self-similar processes. We reformulate and extend the definition of dilative stability introduced by Iglói (2008) using characteristic functions. We also generalize the concept of aggregate similarity introduced by Kaj (2005). It turns out that these two notions are essentially the same for infinitely divisible processes. Examples of dilatively stable generalized fractional Lévy processes are given and we point out that certain limit processes in aggregation models are dilatively stable. 1 Dilative stability and aggregate similarity Self-similarity is a scaling property of stochastic processes. It was Lamperti’s paper [13] which called the attention to the significance of this property (named there semi-stability). As a generalization, Iglói [7, Definition 2.1.3 and Theorem 2.2.1] introduced a more general scaling property of certain infinitely divisible processes called dilative stability. Iglói [7, Examples 2.1.5 -7] already provided some important (non self-similar) dilatively stable processes, such as non-Gaussian moving-average fractional Lévy motions. Roughly speaking, non-Gaussian fractional Lévy motions are not self-similar but they belong to a wider class of processes, to the class of dilatively stable processes, which underlines the importance of dilative stability. To put dilative stability into a more general context, we note that there are many other known examples 2010 Mathematics Subject Classifications : 60G18, 60G22, 39B05.