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) $L_I\left(t\right),t\in [0,1],$ 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 ${\left({\alpha }_n\left(u\right)\right)}_n,\alpha \left(u\right),u\in [0,1],$ be a class of càdlàg functions ranging in $[a,b]\subset (0,2]$ such that the sequence ${\left(\alpha \right)}_n$ tends to $\alpha$ in the uniform metric. Let ${\left(X(k,n)\right)}_{n\in \mathbb{N},k=1,...,2^n}$ be a family of independent and symmetric ${\alpha }_n\left(\frac{k}{2^n}\right)-$stable random variables with unit scale parameter, i.e., $X(k,n)\sim S_{{\alpha }_n\left(\frac{k}{2^n}\right)}(1,0,0)$. Then the sequence of processes

tends in distribution to $L_I\left(u\right)$ in $(D[0,1],d_S),$ where $\lfloor x\rfloor$ is the largest integer smaller than or equal to $x$. In particular, if $\alpha$ satisfies

uniformly for all $x$ as $t\searrow 0,$ then $L_I\left(u\right)$ 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.