Hausdorff Dimension of Operator Stable Sample Paths

The Hausdorff dimension of the sample paths of a stochastic process with stationary independent operator stable increments is computed. With probability one, every sample path has the same dimension, depending on the real parts of the eigenvalues of the operator stable exponent.     

A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented. (Received 31 January 2000; in revised form 5 April 2000)     

Parameter estimation of selfsimilarity exponents

The characteristic feature of operator selfsimilar stochastic processes is that a linear rescaling in time is equal in the sense of distributions to a linear operator rescaling in space, which in turn is characterized by the selfsimilarity exponent. The growth behaviour of such processes in any radial direction is determined by the real parts of the eigenvalues of the selfsimilarity exponent. We extend an estimation method of Meerschaert and Scheffler [M.M. Meerschaert, H.-P. Scheffler, Moment estimator for random vectors with heavy tails, J. Multivariate Anal. 71 (1999) 145-159, M.M. Meerschaert, H.-P. Scheffler, Portfolio modeling with heavy tailed random vectors, in: S.T. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance, Elsevier Science B.V., Amsterdam, 2003, pp. 595-640] to be applicable for estimating the real parts of the eigenvalues of the selfsimilarity exponent and corresponding spectral directions given by the eigenvectors. More generally, the results are applied to operator semi-selfsimilar processes, which obey a weaker scaling property, and to certain Ornstein-Uhlenbeck type processes connected to operator semi-selfsimilar processes via Lamperti's transformation.     

Stable and Semistable Hemigroups: Domains of Attraction and Self-Decomposability

A hemigroup of probability measures builds the set of distributions of the increments of an independent increment process. Decomposability properties of the hemigroup lead to stable respectively semistable hemigroups and enable us to show that such hemigroups appear as limits of certain functional limit theorems for operator-normed independent (non-identically) distributed random vectors. Regular variation properties of the norming operators show that the functional limit theorems are closely related to limits of infinitesimal triangular arrays of independent random vectors, i.e., to operator-self-decomposable respectively operator-semi-self-decomposable laws.     

Explicit representation of roots on <Emphasis Type="Italic">p</Emphasis>-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

This note generalizes known results concerning the existence of roots and embedding one-parameter subgroups on p-adic solenoids. An explicit representation of the roots leads to the construction of two distinct rational embedding one-parameter subgroups. The results contribute to enlighten the group structure of solenoids and to point out difficulties arising in the context of the embedding problem in probability theory. As a consequence, the uniqueness of embedding of infinitely divisible probability measures on p-adic solenoids is solved under a certain natural condition.     

Hausdorff Dimension of Operator Stable Sample Paths

The Hausdorff dimension of the sample paths of a stochastic process with stationary independent operator stable increments is computed. With probability one, every sample path has the same dimension, depending on the real parts of the eigenvalues of the operator stable exponent.Received May 28, 2002; in revised form October 2, 2002 Published online May 15, 2003     

Almost sure limit theorems of mantissa type for semistable domains of attraction

A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods lead to previously unknown versions of semistable almost sure limit theorems. This research has been carried out while the author was staying at the University of Debrecen, Hungary, with the kind support of Deutsche Forschungsgemeinschaft.     

A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented.