A necessary and suucient condition is given to reduce a non-stationary random process fZ(t) : t 2 T Rg to stationarity via a bijective diieren-tiable time deformation so that its correlation function r(t; t 0) depends only on the diierence (t 0)?(t) through a stationary correlation function R: r(t; t 0) = R(((t 0) ? (t)).

Multivariate COGARCH(1,1) processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent timevarying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the COGARCH(1,1) process, we analyze its probabilistic propertie...

We study the permutation complexity of finite-state stationary stochastic processes based on a duality between values and orderings between values. First, we establish a duality between the set of all words of a fixed length and the set of all permutations of the same length. Second, on this basis, we give an elementary alternative proof of the equality between the permutation entropy rate and ...

This note presents four independent sets of open problems. The first set suggests an extension of the limit theory for positive recurrent renewal processes to the null recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread-out. The third concerns the coupling characterization of setwise convergence of distributions o...

We study long strange intervals in a linear stationary stochastic process with regularly varying tails. It turns out that the length of the longest strange interval grows, as a function of the sample size, at diierent rates in diierent parts of the parameter space. We argue that this phenomenon may be viewed in a fruitful way as a phase transition between short and long range dependence. We pro...

By making use of the Langevin equation with a kernel, it was shown that the Feynman measure e−S can be realized in a restricted sense in a diffusive stochastic process, which diverges and has no equilibrium, for bottomless systems. In this paper, the dependence on the initial conditions and the temporal behavior are analyzed for 0-dim bottomless systems. Furthermore, it is shown that it is poss...