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 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.

An integral representation of dilatively stable processes with independent increments

“…From this point of view, dilatively stable processes naturally appear as the class of limit processes in certain aggregation models as shown in Theorem 3.1 of [12]. Examples of dilatively stable limit processes in aggregation schemes appear in [11,19], see section 3 in [1] for a detailed analysis. In this paper we will restrict our considerations to additive processes (X t ) t∈T which are defined as in [21] by the following conditions:…”

“…In particular, additionally assuming weak right-continuity of the infinitely divisible process X, dilative stability of X is equivalent to the notion of aggregate-similarity introduced by Kaj [11], see Proposition 1.5 in [1]. From this point of view, dilatively stable processes naturally appear as the class of limit processes in certain aggregation models as shown in Theorem 3.1 of [12].…”

Dilatively semistable stochastic processes

“…To give a more advanced example we now turn to the class of generalized fractional Lévy processes, extending section 2 of [1]. Let (L (1) t ) t≥0 be a centered Lévy process without Gaussian component, whose Lévy measure φ fulfills {|x|>1}…”

“…Let T be either R, [0, ∞) or (0, ∞). Following [1] a stochastic process (X t ) t∈T on R is called (α, δ)-dilatively stable for some parameters α, δ ∈ R if all its finite-dimensional marginal distributions are infinitely divisible and the scaling relation ψ T t 1 ,...,T t k (θ 1 , . .…”

“…, X s 1/δ t k ), whereas for δ = 0 dilative stability coincides with selfsimilarity. Note that Kaj [5] introduced a weaker scaling relation called aggregate-similarity, which has been extended in Definition 1.4 of [1] such that dilative stability and aggregate similarity essentially define the same property if one additionally assumes infinite divisibility and weak right-continuity of the finite-dimensional marginal distributions; see Proposition 1.5 in [1] for details.…”