Convergence to Stable Limits for Ratios of Trimmed Lévy Processes and their Jumps

Ipsen, Y. F. and Kevei, P. and Maller, R. A. (2018) Convergence to Stable Limits for Ratios of Trimmed Lévy Processes and their Jumps. Markov Processes and Related Fields, 24 (4). pp. 539-562. ISSN 1024-2953

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Abstract

We derive characteristic function identities for conditional distributions of an $r$-trimmed L\'evy process given its $r$ largest jumps up to a designated time $t$. Assuming the underlying L\'evy process is in the domain of attraction of a stable process as $t\dto 0$, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in \cite{KeveiMason2014} and produces new discrete distributions on the infinite simplex in the limit.

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