| Summary: | This brief monograph is an in-depth study of the infinite divisibility and self-decomposability properties of central and noncentral Student<U+0019>s distributions, represented as variance and mean-variance mixtures of multivariate Gaussian distributions with the reciprocal gamma mixing distribution. These results allow us to define and analyse Student-Lv̌y processes as Thorin subordinated Gaussian Lv̌y processes. A broad class of one-dimensional, strictly stationary diffusions with the Student<U+0019>s t-marginal distribution are defined as the unique weak solution for the stochastic differential equation. Using the independently scattered random measures generated by the bi-variate centred Student-Lv̌y process, and stochastic integration theory, a univariate, strictly stationary process with the centred Student<U+0019>s t- marginals and the arbitrary correlation structure are defined. As a promising direction for future work in constructing and analysing new multivariate Student-Lv̌y type processes, the notion of Lv̌y copulas and the related analogue of Sklar<U+0019>s theorem are explained.
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