We characterize the linear and harmonizable fractional stable motions as the self-similar stable processes with stationary increments whose left-equivalent (or right-equivalent) stationary processes are moving averages and harmonizable respectively.

Two disjoint classes of self similar symmetric stable processes with stationary increments are studied. The first class consists of linear fractional stable processes, which are related to moving average stable processes, and the second class consists of harmonizable fractional stable processes,...

The concept of selfdecomposability has been generalized to that of [alpha]-selfdecomposability, , by many authors. We first mention the existing results on the class of [alpha]-selfdecomposable distributions and investigate the remaining problems. We give complete characterizations by stochastic...

Operator-self-similar processes are studied and several examples of operator-self-similar and stable (in the ordinary sense or in the sense of operator-stable) processes are constructed. Limit theorems for such processes are also shown.

Upsilon transformations satisfying certain regularity conditions are shown to generate semigroups of such transformations. This is based on a general commutativity property of the Upsilon transformations, and uses log infinite divisibility. The existence of random integral representations of...

The first problem attacked in this paper is answering the question whether all 1/[alpha]-self-similar [alpha]-stable processes with stationary increments are [alpha]-stable motions. The answer is yes for [alpha] = 2, no for 1[less-than-or-equals, slant][alpha]2 and unknown for 0[alpha]1. We...

Point processes induced by stationary symmetric [alpha]-stable (S[alpha]S) processes can have diverse behavior. We distinguish two cases, depending on whether the stationary S[alpha]S process is governed by a dissipative or conservative flow. In the case of dissipative flows, the process is a...

We show that subexponentiality is not sufficient to guarantee that the distribution tail of a sample quantile of an infinitely divisible process is equivalent to the "tail" of the same sample quantile under the corresponding Lévy measure. However, such an equivalence result is shown to hold...

Necessary conditions are given for a symmetric [alpha]-stable (S[alpha]S) process, 1 [alpha] 2, to be Markov. These conditions are then applied to find Markov or weakly Markov processes within certain important classes of S[alpha]S processes: time changed Lévy motion, scale mixed Gaussian...

We study the supremum of 'the' standard isonormal linear process L on a subset of a real Hilbert space H. Upper and lower bounds on the probability that supx[epsilon] LX[lambda], [lambda] large, are found. We treat a number of examples. These include the distribution of the maximum of certain...