Uniform quadratic variation for Gaussian 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...

Weak Variation of Gaussian Processes

Optimal and suboptimal quadratic forms for noncentered Gaussian processes.

Individual random trajectories of stochastic processes are often analyzed by using quadratic forms such as time averaged (TA) mean square displacement (MSD) or velocity auto-correlation function (VACF). The appropriate quadratic form is expected to have a narrow probability distribution in order to reduce statistical uncertainty of a single measurement. We consider the problem of finding the op...

Large deviations for quadratic functionals of Gaussian processes

The Large Deviation Principle is derived for several unbounded additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to 1 T ∫ T 0 X t dt is the Fenchel-Legendre transform of L(y) = − 1 4π ∫∞ −∞ log(1−4πyf(s))ds, where Xt is a continuous time process with the bounded spectral density f(s). Similar results in the discrete-time version are obt...

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...