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Alphabets of Acyclic Invariant Structures

Janicki R., Kleijn J., Koutny M., Mikulski Ł.

Fundamenta Informaticae

2017

Vol. 154, nr 1/4

207--224

A step trace is an equivalence class of step sequences, where the equivalence is determined by dependencies between pairs of actions expressed as potential simultaneity and sequentialisability. Step traces can be represented by invariant structures with two relations: mutual exclusion and (possibly cyclic) weak causality. An important issue concerning invariant structures is to decide whether an invariant structure represents a step trace over a given step alphabet. For the general case this problem has been solved and an effective decision procedure has been proposed. In this paper, we restrict the class of order structures being considered with the aim of achieving a better characterisation. Requiring that the weak causality relation is acyclic, makes it possible to solve the problem in a purely local way, by considering pairs of events, rather than whole structures.

Dependence structure of stable R-GARCH processes

Nowicka-Zagrajek J., Weron A.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

371--380

In this paper we investigate properties of R-GARCH processes with positive strictly stable innovations. We derive the unconditional distributions and analyze the dependence structure. This analysis is carried out by means of the measure of dependence - the codifference - which extends the behavior of the covariance function to situations where the covariance function is no longer defined. In the case of R-GARCH (1,1,0) process we determine the exact asymptotic behavior.