Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

Dilatively stable stochastic processes and aggregate similarity

“…Since all of these processes are self-similar, they are naturally dilatively stable as well. All in all, each of the limit processes in Theorems 2.1 and 2.2 in [15] is dilatively stable.…”

“…Note that, by Theorems 2.1 and 2.2 in [15], the limit process of the appropriately rescaled system (3.1) under slow and fast growth condition, i.e., c = 0 or c = ∞ in (3.2), or under iterated aggregation lim n→∞ lim N →∞ or lim N →∞ lim n→∞ is either fractional Brownian motion, or a linear time multiple of a stable random variable, or a variance mixture of Brownian motion with a stable mixing variable. Since all of these processes are self-similar, they are naturally dilatively stable as well.…”

“…. , N}, with some given mixing density, recently investigated by Pilipaysjautė and Surgailis [15]. Under an intermediate growth condition…”

“…Here the process Z β is infinitely divisible by Proposition 3.1 in [15] and determined by the characteristic exponent of (Z β (t 1 ), . .…”

“…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 of stochastic processes that do not fit into the scope of self-similarity, but are very natural from the point of view of scaling limit theorems and various types of invariance properties in probability theory, besides Iglói [7], see, e.g., Biermé et al [2], Kaj [9] and Pilipaysjautė and Surgailis [15]. However, there is no common framework for these kinds of processes.…”