Makoto Maejima

Weighted sums of I.I.D. Random variables attracted to integrals of stable processes

SummaryUnder the assumption of the convergence of partial sum processesZn(t) of i.i.d. random variables to an α-stable Lévy processZ(σ)(t), 0<α≦2, the convergence of weighted sums ∫fn(u)dZn(u) to ∫f(u)dZ(α)(u) is studied. The general convergence result is then applied to examine the domain of attraction of the fractional stable process.

Wiener integrals with respect to the hermite process and a non-central limit theorem

Abstract We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.

Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of Rd-valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.

Operator-self-similar stable processes

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.

Asymptotic Behaviour of Compound Distributions

We improve on some results of SUNDT (1982) on the asymptotic behaviour of compound negative binomial distributions.

Two classes of self-similar stable processes with stationary increments

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, which are connected to harmonizable stationary stable processes. The domain of attraction of the harmonizable fractional stable processes is also discussed.

AN INTRODUCTION TO THE THEORY OF SELF-SIMILAR STOCHASTIC PROCESSES

Self-similar processes such as fractional Brownian motion are stochastic processes that are invariant in distribution under suitable scaling of time and space. These processes can typically be used to model random phenomena with long-range dependence. Naturally, these processes are closely related to the notion of renormalization in statistical and high energy physics. They are also increasingly important in many other fields of application, as there are economics and finance. This paper starts with some basic aspects on self-similar processes and discusses several topics from the point of view of probability theory.

Self-Similar Stable Processes with Stationary Increments

Let T be (-∞, ∞), [0, ∞) or [0, 1]. A real- or complex-valued stochastic process X = (X(t)) t∈T is said to be H-self-similar (H-ss) if all finite-dimensional distributions of (X(ct)) and (c H X(t)) are the same for every c > 0 and to have stationary increments (si) if the finite-dimensional distributions of (X(t + b) - X(b)) do not depend on b ∈ T. A real-valued process X = (X(t)) t∈T is said to be symmetric α-stable (SαS), 0 < α ≤ 2, if any linear combination \( \sum\nolimits_{k = 1}^n {{a_k}} X\left( {{t_k}} \right)isS\alpha S\) is SαS.

A self-similar process with nowhere bounded sample paths

On considere un processus auto-similaire a parametre H et on montre la non-existence de trajectoires bornees pour certaines valeurs de α et β

Distributions of selfsimilar and semi-selfsimilar processes with independent increments

Relationships between marginal and joint distributions of selfsimilar processes with independent increments are shown in terms of the Urbanik-Sato-type nested subclasses of the class of selfdecomposable distributions. Similar results are also shown for semi-selfsimilar processes with independent increments.