Marginal Distributions of Self-Similar Processes with Stationary Increments

Vervaat was a visitor from Katholieke Universiteit. Technical report dedicated to Professor John Lamperti in recognition of his pioneering work in this field.Let X= (Xt)t more than or equal to 0 to be a real-valued stochastic process which is self-similar with exponent H>0 and has stationary increments. Several results about the marginal distribution of X1 are given. For H inequal to 1, there is a universal bound, depending only on H, on the concentration function of logXsuper+sub1. For all H>0, X1 cannot have any atoms except in certain trivial cases. Some lower bounds are given for the tails of the distribution of X1 in case H>1. Finally, some results are given concerning the connectedness of the support of X1.Supported by the Natural Sciences ad Engineering Research Council of Canada, School of ORIE, Center of Applied Mathematics at Cornell University, NATO Science Fellowship from the Netherlands Organization for the Advancement of Pure Research (ZWO) and Fulbright-Hays travel grant