Type A Distributions: Infinitely Divisible Distributions Related to Arcsine Density

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Two transformations A₁ and A₂ of Lévy measures on Rᵈ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of A₁ and A₂ are determined and it is shown that they have the same range. Infinitely divisible distributions on Rᵈ with Lévy measures being in the common range are called type A distributions and expressed as the law of a stochastic integral ∫₀¹ (2⁻¹πt)dXₜ with respect to Lévy process {Xₜ}. This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type G distributions are the image of type A distributions under a mapping defined by an appropriate stochastic integral. A₂ is identified as an Upsilon transformation, while A₁ is shown to be not.