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L\'evy processes with respect to the index Whittaker convolution
Probability · Submitted 8 May 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our results demonstrate that a nice theory of L\'evy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator.
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The abstract, sentence by sentence
The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce the class of L\'evy processes with respect to the index Whittaker convolution and study their basic properties. We prove that the square root of the Shiryaev process belongs to our family of L\'evy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to L\'evy's characterization of Brownian motion. Our results demonstrate that a nice theory of L\'evy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. | background 0.98 | 0.71 | 0.07 | asserted 1.00 |
| s1 | In this paper, we introduce the class of L\'evy processes with respect to the index Whittaker convolution and study their basic properties. | contribution 1.00 | 0.93 | 0.09 | asserted 0.94 |
| s2 | We prove that the square root of the Shiryaev process belongs to our family of L\'evy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to L\'evy's characterization of Brownian motion. | result 0.98 | 0.95 | 0.30 | asserted 1.00 |
| s3 | Our results demonstrate that a nice theory of L\'evy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator. | interpretation 0.99 | 0.92 | 0.60 | asserted 0.99 |
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Original abstract
The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce the class of L\'evy processes with respect to the index Whittaker convolution and study their basic properties. We prove that the square root of the Shiryaev process belongs to our family of L\'evy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to L\'evy's characterization of Brownian motion. Our results demonstrate that a nice theory of L\'evy processes with respect to generalized convolutions can be developed even if the usual compactness assumption on the support of the convolution fails, shedding light into the connection between the properties of the convolution algebra and the nature of the singularities of the associated differential operator.
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How the literature uses it
4 arXiv papers cite this in 10 sentences
Papers citing it · 7 sentences without a confident label
Cites it as background 2 papers
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Generalized convolutions were explored with the use of regular variation ( [ref] ) and were used to construct Lévy processes and stochastic integrals ( [ref] , [ref] ).
Fluctuations of extremal Markov chains driven by the Kendall convolution · Introduction · 0.89
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A different approach to generalized convolutions appeared in the theory of harmonic analysis, see e.g. [ref] .
How exceptional is the extremal Kendall and Kendall-type convolution · Motivations · 0.98
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