[Paper Review] Calculus on random integral mappings $I^{h,r}_{(a,b]}$ and their domains

Zbigniew J. Jurek|arXiv (Cornell University)|Oct 22, 2012

advanced mathematical theories12 references3 citations

TL;DR

This paper establishes that random integral mappings $I^{h,r}_{(a,b]}$ are isomorphisms between convolution semigroups of infinitely divisible measures on real separable Banach spaces. It characterizes their domains in multiple ways, proves that compositions of such mappings are equivalent to single random integrals, and shows that while forward mappings preserve the random integral form, their inverses do not. The results generalize and unify prior work on random integral representations of selfdecomposable and infinitely divisible distributions.

ABSTRACT

It is proved that the random integral mappings (some type of functionals of Lévy processes) are always isomorphisms between convolution semigroups of infinitely divisible measures. However, the inverse mappings are no longer of the random integral form. Domains are characterized in may ways. Compositions (iterated integrals) can be expressed as a single random integral mapping. Finally, all obtained results are illustrated by examples.

Motivation & Objective

• To establish a general framework for random integral mappings $I^{h,r}_{(a,b]}$ as isomorphisms between convolution semigroups of infinitely divisible measures.
• To characterize the domains of these mappings in multiple equivalent ways, including through Lévy-Khintchine representations and image measures.
• To show that compositions of such mappings can be expressed as a single random integral mapping.
• To clarify that while forward mappings preserve the random integral form, their inverses do not.
• To unify and extend prior results on random integral representations of selfdecomposable and infinitely divisible distributions.

Proposed method

• The paper uses the Lévy-Khintchine formula to represent the characteristic functional of infinitely divisible measures on Banach spaces.
• It defines the random integral mapping $I^{h,r}_{(a,b]}(\nu)$ as the distribution of $\int_{(a,b]} h(t)\,dY_\nu(r(t))$, where $Y_\nu$ is a Lévy process with $\mathcal{L}(Y_\nu(1)) = \nu$.
• Image measures and tensor product techniques are employed to analyze the structure of the mappings and their domains.
• The paper applies properties of Fourier transforms and weak convergence to study the behavior of these mappings under composition and inversion.
• It leverages known results on background driving Lévy processes (BDLP) and random integral representations to generalize to broader classes of functions $h$ and $r$.
• Theoretical results are illustrated with examples involving time-changed Lévy processes and specific choices of $h$ and $r$, such as exponential and arcsine time changes.

Experimental results

Research questions

• RQ1 Under what conditions is the random integral mapping $I^{h,r}_{(a,b]}$ an isomorphism between convolution semigroups of infinitely divisible measures?
• RQ2 How can the domain of $I^{h,r}_{(a,b]}$ be characterized in multiple equivalent ways, especially in terms of Lévy triplets and spectral measures?
• RQ3 Can compositions of multiple random integral mappings be reduced to a single random integral mapping, and if so, under what conditions on $h$ and $r$?
• RQ4 Why are the inverse mappings of $I^{h,r}_{(a,b]}$ not of the same random integral form, despite the forward mappings being isomorphisms?
• RQ5 How do known random integral representations (e.g., for selfdecomposable or stable laws) fit into this generalized framework?

Key findings

• The mapping $I^{h,r}_{(a,b]}$ is an isomorphism between convolution semigroups of infinitely divisible measures, preserving the semigroup structure.
• The domain of $I^{h,r}_{(a,b]}$ is characterized via the Lévy-Khintchine representation, image measures, and weak convergence of approximating sequences.
• Compositions of such mappings satisfy $I^{h_1,r_1}_{(a,b]} \circ I^{h_2,r_2}_{(c,d]} = I^{h,r}_{(a,b]}$ under appropriate conditions on $h$ and $r$, with explicit formulas derived.
• The inverse of $I^{h,r}_{(a,b]}$ is not representable as a random integral of the same form, indicating a fundamental asymmetry in the mapping.
• The results generalize and unify earlier constructions, such as those of Jurek-Vervaat (1983) for selfdecomposable measures and Sato (2006) for time-changed Lévy processes.
• Specific examples include $I^{t,\Gamma(\alpha;t)}_{(0,\infty)} \circ I^{e^{-s},s}_{(0,\infty)} = I^{t,\int_t^\infty s^{-1}\Gamma(\alpha;s)ds}_{(0,\infty)}$, demonstrating composition closure.