Self-similar Gaussian Processes Research Articles

Gaussian Volterra processes: Asymptotic growth and statistical estimation

The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volterra process under consideration. We construct a strongly consistent estimator and investigate its asymptotic properties. Namely, we prove that it has the Cauchy asymptotic distribution.

Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised kth order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised kth order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyse various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.

Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

We study the fluctuations of the area under a self-similar Gaussian process x(τ) with Hurst exponent H > 0 (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate r. Typical fluctuations of A(t) scale as for large t and on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations of A(t). In the long-time limit t → ∞, we find that the full distribution of the area takes the form with anomalous exponents α = 1/(2H + 2) and β = (2H + 3)/(4H + 4) in the regime of moderately large fluctuations, and a different anomalous scaling form in the regime of very large fluctuations. The associated rate functions Φ(y) and Ψ(w) depend on H and are found exactly. Remarkably, Φ(y) has a singularity that we interpret as a first-order dynamical condensation transition, while Ψ(w) exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of Φ(y) around the origin y = 0 correctly describes the typical, Gaussian fluctuations of A(t). Despite these anomalous scalings, we find that all of the cumulants of the distribution grow linearly in time, , in the long-time limit. For the case of reset Brownian motion (corresponding to H = 1/2), we develop a recursive scheme to calculate the coefficients c n exactly and use it to calculate the first six nonvanishing cumulants.

Asymptotics of the Cross-Variation of Young Integrals with Respect to a General Self-Similar Gaussian Process

We show in this work that the limit in law of the cross-variation of processes having the form of Young integral with respect to a general self-similar centered Gaussian process of order β ∈ (1/2, 3/4] is normal according to the values of β. We apply our results to two self-similar Gaussian processes: the subfractional Brownian motion and the bifractional Brownian motion.

Oscillating Gaussian processes

In this article we introduce and study oscillating Gaussian processes defined by X_t = alpha _+ Y_t mathbf{1}_{Y_t >0} + alpha _- Y_tmathbf{1}_{Y_t<0}, where alpha _+,alpha _->0 are free parameters and Y is either stationary or self-similar Gaussian process. We study the basic properties of X and we consider estimation of the model parameters. In particular, we show that the moment estimators converge in L^p and are, when suitably normalised, asymptotically normal.

Volatility estimation of general Gaussian Ornstein–Uhlenbeck process

In this article we study the asymptotic behavior of the realized quadratic variation of a process ∫0tusdGsH, where u is a Hölder continuous process with order β>1−H and GH is a self-similar Gaussian process with parameter H∈(0,3∕4). We prove almost sure convergence uniformly in time and a stable weak convergence for the realized quadratic variation. As an application, we construct strongly consistent estimator for the integrated volatility parameter in Ornstein–Uhlenbeck process driven by GH.

Large deviations for functionals of some self-similar Gaussian processes

ABSTRACT We prove large deviation principles for , where X is a d-dimensional self-similar Gaussian process and takes the form of the Dirac delta function , with , or with . In particular, large deviations are obtained for the functionals of d-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.

Continuous Breuer–Major theorem: Tightness and nonstationarity

Let $Y=(Y(t))_{t\geq 0}$ be a zero-mean Gaussian stationary process with covariance function $\rho :\mathbb{R}\to \mathbb{R}$ satisfying $\rho (0)=1$. Let $f:\mathbb{R}\to \mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_{\mathbb{R}}|\rho (s)|^{d}\,ds<\infty $, then the celebrated Breuer–Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_{\varepsilon }:=\sqrt{\varepsilon }\int_{0}^{\cdot /\varepsilon }f(Y(s))\,ds$ converge to those of $\sigma W$ as $\varepsilon \to 0$, where $W$ is a standard Brownian motion and $\sigma $ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes. The goal of this paper is twofold. First, we investigate the tightness in the Breuer–Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [J. Multivariate Anal. 80 (2002) 191–216] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^{p}$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities. Second, and motivated by a problem of geometrical nature, we extend the continuous Breuer–Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Brownian motion concludes the paper.

Bifractional Brownian motion for [formula omitted] and [formula omitted

Bifractional Brownian motion on R+ is a two parameter centered Gaussian process with covariance function: RH,K(t,s)=12Kt2H+s2HK−|t−s|2HK,s,t≥0.This process has been originally introduced by Houdré and Villa in Houdré and Villa (2002) in for the range of parameters H∈(0,1] and K∈(0,1]. Since then, the range of parameters, for which RH,K is known to be nonnegative definite has been somewhat extended, but the full range is still not known. We give an elementary proof that RH,K is nonnegative definite for parameters H,K satisfying H>1 and 0<2HK≤1. We show that RH,K can be decomposed into a sum of two nonnegative definite functions. As a side product we obtain a decomposition of the fractional Brownian motion with Hurst parameter H<12 into a sum of time rescaled Brownian motion and another independent self-similar Gaussian process. We also discuss some simple properties of bifractional Brownian motion with H>1.

Transfer principle for $n$th order fractional Brownian motion with applications to prediction and equivalence in law

The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the $n$the order fractional Brownian motion and also a representation formula for all the Gaussian processes that are equivalent in law to the $n$th order fractional Brownian motion.

Non-Gaussian behavior of reflected fractional Brownian motion

A possible mechanism leading to anomalous diffusion is the presence of long-range correlations in time between the displacements of the particles. Fractional Brownian motion, a non-Markovian self-similar Gaussian process with stationary increments, is a prototypical model for this situation. Here, we extend the previous results found for unbiased reflected fractional Brownian motion (Wada et al 2018 Phys. Rev. E 97 020102) to the biased case by means of Monte Carlo simulations and scaling arguments. We demonstrate that the interplay between the reflecting wall and the correlations leads to highly non-Gaussian probability densities of the particle position x close to the reflecting wall. Specifically, the probability density develops a power-law singularity with if the correlations are positive (persistent) and if the correlations are negative (antipersistent). We also analyze the behavior of the large-x tail of the stationary probability density reached for bias towards the wall, the average displacements of the walker, and the first-passage time, i.e. the time it takes for the walker reach position x for the first time.

Statistical inference for Vasicek-type model driven by self-similar Gaussian processes

In this paper, we consider the drift parameters estimation problem for the Vasicek-type model defined aswhere a < 0 and are considered as unknown drift parameters and Gt is a self-similar Gaussian process with index . We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distributions of our estimators of a and of b based on the observation as . Our approach extend the result of Xiao and Yu (2017) for the case when G is a fractional Brownian motion with Hurst parameter . We also discuss the cases of sub-fractional Browian motion and bi-fractional Brownian motion. The conclusion can also be extended to more general self-similarity processes, such as Hermite processes.

Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes

We derive the asymptotic behavior for an additive functional of two independent self-similar Gaussian processes when their intersection local time exists, using the method of moments.

On a covariance structure of some subset of self-similar Gaussian processes

We introduce a class of self-similar Gaussian processes and provide sufficient and necessary conditions for a member of the class to admit a unique small scale limit in the Skorokhod space. The class includes several well known processes. An example of application to the problem of estimation is given.

Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

We study the asymptotic behavior of the $$\nu $$ -symmetric Riemann sums for functionals of a self-similar centered Gaussian process X with increment exponent $$0<\alpha <1$$ . We prove that, under mild assumptions on the covariance of X, the law of the weak $$\nu $$ -symmetric Riemann sums converge in the Skorohod topology when $$\alpha =(2\ell +1)^{-1}$$ , where $$\ell $$ denotes the largest positive integer satisfying $$\int _{0}^{1}x^{2j}\nu (\mathrm{d}x)=(2j+1)^{-1}$$ for all $$j=0,\dots , \ell -1$$ . In the case $$\alpha >(2\ell +1)^{-1}$$ , we prove that the convergence holds in probability.

Central limit theorem for functionals of a generalized self-similar Gaussian process

We consider a class of self-similar, continuous Gaussian processes that do not necessarily have stationary increments. We prove a version of the Breuer–Major theorem for this class, that is, subject to conditions on the covariance function, a generic functional of the process increments converges in law to a Gaussian random variable. The proof is based on the Fourth Moment Theorem. We give examples of five non-stationary processes that satisfy these conditions.

Chung’s law of the iterated logarithm for subfractional Brownian motion

Let XH = {XH(t), t ∈ ℝ+} be a subfractional Brownian motion in ℝd. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that XH has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of XH, we establish Chung’s law of the iterated logarithm for XH.

From infinite urn schemes to decompositions of self-similar Gaussian processes

We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $\mathbb B(t^\alpha), \alpha\in(0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$. The decomposition in the latter case is a special case of the decompositions of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as correlated random walks, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.

Parisian ruin of self-similar Gaussian risk processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

Parisian ruin of self-similar Gaussian risk processes

In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.