Stationary Increments Research Articles

On Lamperti transformation and AR(1) type characterisations of discrete random fields

In this article we characterise discrete time stationary fields by difference equations involving stationary increment fields and self-similar fields. This gives connections between stationary fields, stationary increment fields and, through Lamperti transformation, self-similar fields. Our contribution is a natural generalisation of recently proved results covering the case of stationary processes.

Limit behavior in high-dimensional regime for Wishart tensors with Rosenblatt entries

In this paper, we consider an initial [Formula: see text] random matrix with non-Gaussian correlated entries on each row and independent entries from one row to another. The correlation on the rows is given by the correlation of the increments of the Rosenblatt process, which is a non-Gaussian self-similar process with stationary increments, living in the second Wiener chaos. To this initial matrix, we associate a Wishart tensor of length [Formula: see text]. We study the limit behavior in distribution of this Wishart tensor in the high-dimensional regime, i.e. when [Formula: see text] are large enough. We prove that the vector corresponding to the [Formula: see text]-Wishart tensor converges to an [Formula: see text]-dimensional non-Gaussian vector, with Rosenblatt random variables on its hyper-diagonals and zeros outside the hyper-diagonals.

A unifying representation of path integrals for fractional Brownian motions

Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian Gaussian stochastic process with long-ranged correlations between the increments, parametrised by the so-called Hurst exponent H; depending on its value the process can be sub-diffusive (0<H<1/2) , diffusive (H=1/2) or super-diffusive (1/2<H<1) . There exist three alternative equally often used definitions of fBm—due to Lévy and due to Mandelbrot and van Ness (MvN), which differ by the interval on which the time variable is formally defined. Respectively, the covariance functions of these fBms have different functional forms. Moreover, the MvN fBms have stationary increments, while for the Lévy fBm this is not the case. One may therefore be tempted to conclude that these are, in fact, different processes which only accidentally bear the same name. Recently determined explicit path integral representations also appear to have very different functional forms, which only reinforces the latter conclusion. Here we develop a unifying equivalent path integral representation of all three fBms in terms of Riemann–Liouville fractional integrals, which links the fBms and proves that they indeed belong to the same family. We show that the action in such a representation involves the fractional integral of the same form and order (dependent on whether H<1/2 or H>1/2 ) for all three cases, and differs only by the integration limits.

Spectral density estimation for random processes with stationary increments

AbstractSpectral density analysis plays an important role in studying a stationary random process on a real line. In this paper, we extend this discussion for the random process with stationary increments. We investigate the properties of the method of moments structure function estimation, and propose a nonparametric spectral density function estimator. Our numerical results show that the proposed spectral density estimator performs comparable with the parametric counterpart when the underlying process is assumed to be band‐limited. Additionally, this method is applied to analyze US Housing Starts Data, where the hidden periodicities are detected, providing consistent conclusions with previous economic studies.

Limiting Ramp Rate of Wind Power Output using a Battery Based on the Variance Gamma Process

A battery system is designed to reduce violations of ramping limit due to severe and long ramps of wind power output. The power rating, battery capacity, and operational policies of the battery are decided based on the assumption that the wind power output follows the Variance Gamma (V.G.) process over short time interval. The histogram of the ramp rates of wind power follows the symmetric Laplace distribution. Furthermore, ramp rates have independent and stationary increments, and their variances are proportional to the time interval. Consequently, the wind power output can be modelled as a simple form of the V.G. process. Battery operational oplicies based on the V.G. process are tested using statistics representing the sum of all wind power outputs from wind farms in the Electric Reliability Council of Texas (ERCOT). It is found that the battery can essentially eliminate violations of ramping limits due to severe ramp events. Furthermore, in this operation, it is discovered that the power rating influences the ramp limiting performance of the battery more than battery storage rating.

On Gaussian Markov processes and Polya processes

In previous work we characterized Gaussian Markov processes with stationary increments and showed that they arise as asymptotic approximations for stochastic point processes with a random rate such as Polya processes, which can be useful to model over-dispersion and path-dependent behavior in service system arrival processes. Here we provide additional insight into these stochastic processes.

An Improved Model of Wavelet Leader Covariance for Estimating Multifractal Properties

Complex systems often produce multifractal signals defined by stationary increments that exhibit power-law scaling properties. The Legendre transform of the domain-dependent scaling function that defines the power law is known as the multifractal spectrum. The multifractal spectrum can also be defined by a power-series expansion of the scaling function and in practice the first two leading coefficients of that series are estimated from the discrete wavelet transform of the signal. To quantify, validate, and compare simulations of complex systems with data collected empirically from the actual system, practitioners require methods for approximating the variance associated with estimates of these coefficients. In this work, we generalize a previously developed semi-parametric statistical model for the values extracted from a discrete multi-scale wavelet transform to include both within-scale and between-scale covariance dependencies. We employ multiplicative cascades to simulate multifractals with known parameters to illustrate the necessity for this generalization and to test the precision of our improved model. The combined within- and between-scale model of covariance results in a more accurate estimate of the expected variance of the coefficients extracted from an empirical data set.

Filtering problem for sequences with periodically stationary multi-seasonal increments with spectral densities allowing canonical factorizations

We consider a stochastic sequence $\xi(m)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. The filtering problem is solved for this type of sequences based on observations with a periodically stationary noise. When spectral densities are known and allow the canonical factorizations, we derive the mean square error and the spectral characteristics of the optimal estimate of the functional $A{\xi}=\sum_{k=0}^{\infty}{a}(k) {\xi}(-k)$. Formulas that determine the least favourable spectral densities and the minimax (robust) spectralcharacteristics of the optimal linear estimate of the functional are proposed in the case where the spectral densities are not known, but some sets of admissible spectral densities are given.

Extremes of reflecting Gaussian processes on discrete grid

For {X(t),t∈Gδ} a centered Gaussian process with variance σ2 and stationary increments on a discrete grid Gδ={0,δ,2δ,...}, where δ>0, we investigate the stationary reflected processQδ,X(t)=sups∈[t,∞)∩Gδ⁡(X(s)−X(t)−c(s−t)),t∈Gδ with c>0. We derive the exact asymptotics of P(supt∈[0,T]∩Gδ⁡Qδ,X(t)>u) and P(inft∈[0,T]∩Gδ⁡Qδ,X(t)>u), as u→∞, with T>0. It appears that φ=limu→∞⁡σ2(u)u determines the asymptotics, leading to three qualitatively different scenarios: φ=0, φ∈(0,∞) and φ=∞.

A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes

We establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.

Correlation Structure of Time-Changed Generalized Mixed Fractional Brownian Motion

The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. In this paper, we investigate the long-time behavior of gmfBm when it is time-changed by a tempered stable subordinator or a gamma process. As a main result, we show that the time-changed process exhibits a long-range dependence property under some conditions on the Hurst indices. The time-changed gmfBm can be used to model natural phenomena that exhibit long-range dependence, even when the underlying process is not itself long-range dependent.

Estimation problem for continuous time stochastic processes with periodically correlated increments

We deal with the problem of optimal estimation of the linear functionals constructed from unobserved values of a continuous time stochastic process with periodically correlated increments based on past observations of this process. To solve the problem, we construct a corresponding to the process sequence of stochastic functions which forms an infinite dimensional vector stationary increment sequence. In the case of known spectral density of the stationary increment sequence, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

Nuclear magnetic resonance signal decay in the presence of a background gradient: Normal and anomalous diffusion.

A novel way for calculating the diffusion-weighted nuclear magnetic resonance (NMR) attenuation signal expression in the presence of a background gradient is developed. This method is easily applicable to NMR-attenuated signals arising from any pulse field gradient sequence experiments. Here, we provide detailed calculations for the classical pulsed gradient stimulated echo and the pulsed gradient spin echo, as the particular cases. Within this general theoretical framework, devised for Gaussian processes with stationary increments, we recover and extend the previous Stejskal-Tanner results in the case of normal diffusion and we furnish a new expression in the case of anomalous diffusion.

The overdamped generalized Langevin equation with Hermite noise

We consider the generalized Langevin equation driven by a fractional non-Gaussian noise, the so-called Hermite process. We prove the existence and the uniqueness of the solution for this equation as well as various properties. An useful link between the solution to the Langevin equation with Hermite noise and a class of self-similar stochastic processes with stationary increments is given in our work. Based on this relationship, we deduce the regularity of the sample paths of the solution and the behavior of its p-variation. Some applications concerning the statistical estimation of the noise intensity parameter or the Euler method for the generalized Langevin equation are also discussed.

Kolmogorov bounds in the CLT of the LSE for Gaussian Ornstein Uhlenbeck processes

In this paper, we consider the Ornstein–Uhlenbeck (OU) process defined as solution to the equation [Formula: see text], [Formula: see text], where [Formula: see text] is a Gaussian process with stationary increments, whereas [Formula: see text] is unknown parameter to be estimated. We provide an upper bound in Kolmogorov distance for normal approximation of the least squares estimator [Formula: see text] of the drift parameter [Formula: see text] on the basis of the continuous observation [Formula: see text], as [Formula: see text]. Our approach is based on some novel estimates involving a combination of Malliavin calculus and Stein’s method for normal approximation. We apply our result to fractional OU processes of the first kind, and improved the upper bound of the Kolmogorov distance for the LSE [Formula: see text] provided by [Y. Chen, N. Kuang and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes, Stoch. Dyn. 20(4) (2020) 2050023; Y. Chen and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes with the hurst parameter [Formula: see text], Commun. Stat. Theory Methods 50(13) (2021) 2996–3013], respectively, in the cases [Formula: see text] and [Formula: see text]. We also apply our approach to fractional OU processes of the second kind.

Simultaneous ruin probability for multivariate Gaussian risk model

Let Z(t)=(Z1(t),…,Zd(t))⊤,t∈R where Zi(t),t∈R, i=1,…,d are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For X(t)=AZ(t),t∈R, where A is a nonsingular d×d real-valued matrix, u,c∈Rd and T>0 we derive tight bounds for P∃t∈[0,T]:∩i=1d{Xi(t)−cit>ui}and find exact asymptotics as (u1,…,ud)⊤=(ua1,…,uad)⊤ for any (a1,…,ad)⊤∈Rd∖(−∞,0]d and u→∞.

Lévy models for collapse of the wave function

Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.

Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields

We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.

A new perspective of molecular diffusion by nuclear magnetic resonance

The diffusion-weighted NMR signal acquired using Pulse Field Gradient (PFG) techniques, allows for extrapolating microstructural information from porous materials and biological tissues. In recent years there has been a multiplication of diffusion models expressed by parametric functions to fit the experimental data. However, clear-cut criteria for the model selection are lacking. In this paper, we develop a theoretical framework for the interpretation of NMR attenuation signals in the case of Gaussian systems with stationary increments. The full expression of the Stejskal–Tanner formula for normal diffusing systems is devised, together with its extension to the domain of anomalous diffusion. The range of applicability of the relevant parametric functions to fit the PFG data can be fully determined by means of appropriate checks to ascertain the correctness of the fit. Furthermore, the exact expression for diffusion weighted NMR signals pertaining to Brownian yet non-Gaussian processes is also derived, accompanied by the proper check to establish its contextual relevance. The analysis provided is particularly useful in the context of medical MRI and clinical practise where the hardware limitations do not allow the use of narrow pulse gradients.

An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.