Representations of Gaussian processes by Wiener processes

The Rate of Entropy for Gaussian Processes

In this paper, we show that in order to obtain the Tsallis entropy rate for stochastic processes, we can use the limit of conditional entropy, as it was done for the case of Shannon and Renyi entropy rates. Using that we can obtain Tsallis entropy rate for stationary Gaussian processes. Finally, we derive the relation between Renyi, Shannon and Tsallis entropy rates for stationary Gaussian proc...

Representations of Gaussian processes with stationary increments

Gaussian Processes: the Wiener Isometry and the Kolmogorov-chentsov Theorem

We will be most interested in Gaussian processes indexed by an interval J ⇢ R, usually J = [0,1) or J = [0, T ] for some 0 < T < 1. Under weak hypotheses there will always exist centered Gaussian processes with specified covariance functions, but it is not generally the case that such processes will have versions with continuous sample paths. Our primary objective in this chapter will be to dev...

Cylindrical Wiener processes

In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this defin...

Gaussian Processes

Gaussian processes can be used as a supervised learning technique for classification as well as regression. An example of a classification task would be to recognize handwritten digits, whereas an example of a regression problem would be to learn the inverse dynamics of a robot arm (Fig. 1(a)). For the latter, the task is to obtain a mapping from the state of the arm (given by the positions, ve...