Section: New Results
Functional central limit theorem for multistable Lévy motions
Participants : Xiequan Fan, Jacques Lévy Véhel.
We prove a functional central limit theorem (FCLT) for the independent-increments multistable Lévy motions (MsLM) as well as of integrals with respect to these processes, using weighted sums of independent random variables. In particular, we prove that multistable Lévy motions are stochastic Hölder continuous and strongly localisable.
Theorem 0.1 Let be a class of càdlàg functions ranging in such that the sequence tends to in the uniform metric. Let be a family of independent and symmetric stable random variables with unit scale parameter, i.e., . Then the sequence of processes
tends in distribution to in where is the largest integer smaller than or equal to . In particular, if satisfies
uniformly for all as then is localisable at all times.
We have defined integrals of MsLM, and given criteria for convergence,independence, stochastic Hölder continuity and strong localisability of such integrals.