Self-similar Gaussian Processes Research Articles

Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks

Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system come at two levels. On the central limit scale n1/2 one sees initial fluctuations transported along characteristics and no dynamical noise. The second order of fluctuations comes from the particle current across the characteristic. For a system made up of independent random walks we show that the second-order fluctuations appear at scale n1/4 and converge to a certain self-similar Gaussian process. If the system is in equilibrium, this limiting process specializes to fractional Brownian motion with Hurst parameter 1/4. This contrasts with asymmetric exclusion and Hammersley’s process whose second-order fluctuations appear at scale n1/3, as has been discovered through related combinatorial growth models.

Generalized Ornstein Uhlenbeck processes and associated self-similar processes

We consider three types of generalized Ornstein–Uhlenbeck processes: the stationary process obtained from the Lamperti transformation of fractional Brownian motion, the process with stretched exponential covariance and the process obtained from the solution of the fractional Langevin equation. These stationary Gaussian processes have many common properties, such as the fact that their local covariances share a similar structure and they exhibit identical spectral densities at large frequency limit. In addition, the generalized Ornstein–Uhlenbeck processes can be shown to be local stationary representations of fractional Brownian motion. Two new self-similar Gaussian processes, in addition to fractional Brownian motion, are obtained by applying the (inverse) Lamperti transformation to the generalized Ornstein–Uhlenbeck processes. We study some of the properties of these self-similar processes such as the long-range dependence. We give a simulation of their sample paths based on numerical Karhunan–Loeve expansion.

Self-similar Gaussian processes for modeling anomalous diffusion

We study some Gaussian models for anomalous diffusion, which include the time-rescaled Brownian motion, two types of fractional Brownian motion, and models associated with fractional Brownian motion based on the generalized Langevin equation. Gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean-square displacement to vary with t(alpha), 0<alpha<2. However, these processes have different properties, thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism. Although the two versions of fractional Brownian motion and time-rescaled Brownian motion all have the same probability distribution function, the Slepian theorem can be used to compare their first passage time distributions, which are different. Finally, in order to model anomalous diffusion with a variable exponent alpha(t) it is necessary to consider the multifractional extensions of these Gaussian processes.

Identification of a Locally Self-similar Gaussian Process by Using Convex Rearrangements

We propose a new approach for identifying a locally self-similar Gaussian process. The method is based on the asymptotic behavior of convex rearrangement obtained by Davydov and Thilly (2002). Some simulations illustrate the behavior of the resulting estimates in the particular case of the fractional Brownian motion.

Strong approximation of fractional Brownian motion by moving averages of simple random walks

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to Mandelbrot and van Ness (SIAM Rev. 10 (1968) 422) as a self-similar Gaussian process W ( H) ( t) with stationary increments. Here self-similarity means that (a −HW (H)(at) : t⩾0) = d (W (H)(t) : t⩾0) , where H∈(0,1) is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by Révész (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990) and then by Szabados (Studia Sci. Math. Hung. 31 (1996) 249–297). This approach is quite natural and elementary, and as such, can be extended to more general situations. Based on this, here we use moving averages of a suitable nested sequence of simple random walks that almost surely uniformly converge to fractional Brownian motion on compacts when H∈( 1 4 ,1) . The rate of convergence proved in this case is O(N − min(H−1/4,1/4) log N) , where N is the number of steps used for the approximation. If the more accurate (but also more intricate) Komlós et al. (1975,1976) approximation is used instead to embed random walks into ordinary Brownian motion, then the same type of moving averages almost surely uniformly converge to fractional Brownian motion on compacts for any H∈(0,1). Moreover, the convergence rate is conjectured to be the best possible O(N −H log N) , though only O(N − min(H,1/2) log N) is proved here.

Self-similar communication models and very heavy tails

Several studies of file sizes either being downloaded or stored in the World Wide Web have commented that tails can be so heavy that not only are variances infinite, but so are means. Motivated by this fact, we study the infinite node Poisson model under the assumption that transmission times are heavy tailed with infinite mean. The model is unstable but we are able to provide growth rates. Self-similar but nonstationary Gaussian process approximations are provided for the number of active sources, cumulative input, buffer content and time to buffer overflow.

Extremes of a certain class of Gaussian processes

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend − ct β for some constants c, β>0 and a variance t 2 H . We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t 0 where the related boundary function ( u+ ct β )/ t H is minimal. We consider the case that H< β.

On the law of the iterated logarithm for Gaussian processes

We present some optimal conditions for the compact law of the iterated logarithm of a sequence of jointly Gaussian processes in different situations. We also discuss the local law of the iterated logarithm for Gaussian processes indexed by arbitrary index sets, in particular for self-similar Gaussian processes. We apply these results to obtain the law of the iterated logarithm for compositions of Gaussian processes.

Rates of clustering for some Gaussian self-similar processes

The analogue of Strassen's functional law of the iterated logarithm in known for many Gaussian processes which have suitable scaling properties, and here we establish rates at which this convergence takes place. We provide a new proof of the best upper bound for the convergence toK by suitably normalized Brownian motion, and then continue with this method to get similar bounds for the Brownian sheet and other self-similar Gaussian processes. The previous method, which produced these results for Brownian motion in ℝ1, was highly dependent on many special properties unavailable when dealing with other Gaussian processes.