Abstract
A symmetric stable random process has many integral representations. Among these, the so-called minimal representations play a fundamental role, as described in the chapter. Minimal representations are characterized by a rigidity property that allows relating stable processes with an invariance property to nonsingular flows and their functionals. Various types of nonsingular flows (dissipative, conservative, periodic, fixed and others) are also discussed. They underlie the decompositions of stable processes derived in the following chapter.
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Notes
- 1.
A measure m 1 is absolutely continuous with respect to a measure m 2, written m 1 ≪ m 2, if for every measurable set A, m 2(A) = 0 implies m 1(A) = 0.
- 2.
A process {X(t)} t ∈ T is separable in probability if there is a countable subset T 0 ⊂ T such that for every t ∈ T, X(t) is the limit in probability of X(t k ) for some t k ∈ T. By Remark 2 in Samorodnitsky and Taqqu [56], p. 153, separability in probability is equivalent to the so-called S condition. The minimality result of Hardin [20], in fact, assumes that the S condition holds.
- 3.
The fact that (2.8) holds for all t is critical.
- 4.
Indeed, if F(t, u) = f(t − u) − h(u)f(t −ϕ(u)), then for all \(t \in \mathbb{R}\), F(t, u) = 0 a.e. du. By Lemma 1.1, (i), we have F(t, u) = 0 a.e. dtdu.
- 5.
Recall that this is part of Definition 2.1.
- 6.
Note also that minimality follows directly from Example 3.3 since the stable Lévy motion is stationary increments moving average.
- 7.
One argues first that it is enough to consider the relation (2.3) for a countable number of pairs (u 1, v 1), (u 2, v 2), … and the relation (2.4) for a countable number of points t = u 1, v 1, u 2, v 2, …. Setting \(F = (f_{u_{1}}/f_{v_{1}},f_{u_{2}}/f_{v_{2}},\ldots )\), which is now a map with values in \(Y:= (\mathbb{R} \cup \{ \partial \})^{\infty }\), the relation (2.3) becomes \(F^{-1}(\mathcal{}B(Y )) = \mathcal{}B(S)\) mod m (that is, for every \(A \in F^{-1}(\mathcal{}B(Y ))\), there is \(B \in \mathcal{}B(S)\) so that m(A△B) = 0), and (2.4) can be written as F(ϕ(s)) = F(s) a.e. m. Then, arguing by contradiction and quoting Rosiński [48] in slightly adapted form, the idea of the proof is: “Since \(F^{-1}(\mathcal{}B(Y ))\neq \mathcal{}B(S)\), the partition {F −1{y}: y ∈ Y } contains sufficiently many sets consisting of more than just one point. On each such set one can define an isomorphism different from the identity. Then a function ϕ is obtained by pasting together such isomorphisms.”
- 8.
That is, for any s 1 ≠ s 2 in S, there is a set K n such that s ∈ K n and s 2 ∉ K n for some n.
- 9.
- 10.
Given two normed vector spaces V and W, a linear isometry is a map U: V → W which is linear (that is, U(af + g) = aU(f) + U(g) for any f, g ∈ V and \(a \in \mathbb{R}\)) and preserves the norms (that is, ∥U(f)∥ = ∥f∥). Note also that the isometry U 0 in (2.20) actually acts on the linear span span(F) and, more generally, on the closure of the linear span in L α(S, m). We write just F for notational simplicity and the understanding that the values of U 0 on the span are determined by those on F from the definition of linear isometry.
- 11.
As we will see below in (2.30), the mapping \(T : S \rightarrow \widetilde{ S}\) will play the role of \(\phi ^{-1} :\widetilde{ S} \rightarrow S\). The map U, on the other hand, is defined as in (2.22) as the image in \(\widetilde{S}\) of a function f in S. We first consider indicator functions f = 1 A , A ∈ B(S). The map T maps A into the image in \(\widetilde{S}\) of 1 A , excluding the points where this image vanishes.
- 12.
The terms “fixed” and “identity” are used interchangeably and labeled by the letter F.
- 13.
We use F for fixed or identity. We use D for dissipative and C for conservative, so we cannot use C for cyclic. We use L instead to refer to cycLic.
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Pipiras, V., Taqqu, M.S. (2017). Minimality, Rigidity, and Flows. In: Stable Non-Gaussian Self-Similar Processes with Stationary Increments. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-62331-3_2
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