Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying L\'{e}vy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.Comment: Published in at http://dx.doi.org/10.1214/13-AOP899 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Empirical Likelihood Estimation of Levy Processes (Revised in March 2005)

We propose a new parameter estimation procedure for the Levy processes and the class of infinitely divisible distribution. We shall show that the empirical likelihood method gives an easy way to estimate the key parameters of the infinitely divisible distributions including the class of stable distributions as a special case. The maximum empirical likelihood estimator by using the empirical characteristic functions gives the consistency, the asymptotic normality, and the asymptotic efficiency for the key parameters when the number of restrictions on the empirical characteristic functions is large. Test procedures can be also developed. Some extensions to the estimating equations problem with the infinitely divisible distributions are discussed.