Gaussian and non-Gaussian processes of zero power variation

Gaussian and non-Gaussian processes of zero power variation

This paper considers the class of stochastic processes X defined on [0, T ] by X (t) = ∫ T 0 G (t, s) dM (s) where M is a square-integrable martingale and G is a deterministic kernel. When M is Brownian motion, X is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let m be an odd integer. Under the assumption that t...

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...

Weak Variation of Gaussian Processes

Covariances of Zero Crossings in Gaussian Processes

For a zero-mean Gaussian process, the covariances of zero crossings can be expressed as the sum of quadrivariate normal orthant probabilities. In this paper, we demonstrate the evaluation of zero crossing covariances using one-dimensional integrals. Furthermore, we provide asymptotics of zero crossing covariances for large time lags and derive bounds and approximations. Based on these results, ...

Power variation for Gaussian processes with stationary increments ∗

We develop the asymptotic theory for the realised power variation of the processes X = φ • G, where G is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G and certain regularity condition on the path of the process φ we prove the convergence in probability for the properly normalised realised power varia...