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) LI(t),t[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 (αn(u))n,α(u),u[0,1], be a class of càdlàg functions ranging in [a,b](0,2] such that the sequence (α)n tends to α in the uniform metric. Let X(k,n)n,k=1,...,2n be a family of independent and symmetric αn(k2n)-stable random variables with unit scale parameter, i.e., X(k,n)Sαn(k2n)(1,0,0). Then the sequence of processes

L I ( n ) ( u ) = k = 1 2 n u 1 2 n 1 / α n ( k 2 n ) X ( k , n ) , u [ 0 , 1 ] , (6)

tends in distribution to LI(u) in (D[0,1],dS), where x is the largest integer smaller than or equal to x. In particular, if α satisfies

α ( x ) - α ( x + t ) ln t 0 (7)

uniformly for all x as t0, then LI(u) 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.