Convergence to Stable Limits for Ratios of Trimmed L\'evy Processes and their Jumps

#### Y. F. Ipsen, P. Kevei, R. Maller

2018, v.24, Issue 4, 539-562

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.

Keywords: L\'evy process; large jumps of L\'evy process; trimmed L\'evy process; stable process; trimmed subordinator; domain of attraction of stable laws; conditional distributions of L\'evy processes; small time convergence of L\'evy processes; generalised Poisson\tire Dirichlet laws