The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Ciprian A. Tudor - One of the best experts on this subject based on the ideXlab platform.
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limit behavior of the Rosenblatt ornstein uhlenbeck process with respect to the hurst index
arXiv: Probability, 2019Co-Authors: Meryem Slaoui, Ciprian A. TudorAbstract:We study the convergence in distribution, as $H\to \frac{1}{2}$ and as $H\to 1$, of the integral $\int_{\mathbb{R}} f(u) dZ^{H}(u) $, where $Z ^{H}$ is a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right) $ and $f$ is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.
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parameter estimation for the Rosenblatt ornstein uhlenbeck process with periodic mean
arXiv: Probability, 2019Co-Authors: Radomyra Shevchenko, Ciprian A. TudorAbstract:We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.
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Limit behavior of the Rosenblatt Ornstein-Uhlenbeck process with respect to the Hurst index
Theory of Probability and Mathematical Statistics, 2018Co-Authors: Meryem Slaoui, Ciprian A. TudorAbstract:We study the convergence in distribution, as H → 1 2 and as H → 1, of the integral R f (u)dZ H (u), where Z H is a Rosenblatt process with self-similarity index H ∈ 1 2 , 1 and f is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.
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Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process
Journal of Multivariate Analysis, 2014Co-Authors: Jean-marc Bardet, Ciprian A. TudorAbstract:The purpose of this paper is to estimate the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations.
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on the distribution of the Rosenblatt process
Statistics & Probability Letters, 2013Co-Authors: Makoto Maejima, Ciprian A. TudorAbstract:We prove that the multivariate Rosenblatt distribution belongs to the Thorin class which is a subset of the class of selfdecomposable distributions. Using this fact we derive new properties of the Rosenblatt distribution.
Vladas Pipiras - One of the best experts on this subject based on the ideXlab platform.
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Wavelet-based synthesis of the Rosenblatt process
Signal Processing, 2006Co-Authors: Patrice Abry, Vladas PipirasAbstract:Based on a wavelet-type expansion of the Rosenblatt process, we introduce and examine two different practical ways to simulate the Rosenblatt process. The synthesis procedures proposed here are obtained by either truncating the series of the approximation term or using the approximation coefficients in the wavelet-type expansion of the Rosenblatt process. Both benefit from the low computational cost usually associated with the discrete wavelet transform. We show that the number of zero moments of a related orthogonal multiresolution analysis plays an important role. We study in detail the wavelet-based simulation in terms of uniform convergence. We also discuss at length the importance of the choices of the initial and final resolutions, the specific case of the simulation on the integer grid as well as the usefulness of the wavelet-based simulation. Matlab routines implementing these synthesis procedures as well as their analysis are available upon request.
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wavelet type expansion of the Rosenblatt process
Journal of Fourier Analysis and Applications, 2004Co-Authors: Vladas PipirasAbstract:The Rosenblatt process is an important example of self-similar stationary increments stochastic processes whose finite-dimensional distributions are non-Gaussian with all their moments finite. We show that the Rosenblatt process admits a wavelet-type expansion which is almost surely convergent uniformly on compact intervals and which can be thought as decorrelating the high frequencies. Our wavelet expansion of the Rosenblatt process is different from standard wavelet decompositions used in the wavelet literature. It nevertheless yields natural approximations to the Rosenblatt process, possesses a multiresolution-like structure and can be used for simulation of the Rosenblatt process in practice based on the usual Mallat-type pyramidal algorithm.
Murad S Taqqu - One of the best experts on this subject based on the ideXlab platform.
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behavior of the generalized Rosenblatt process at extreme critical exponent values
Annals of Probability, 2017Co-Authors: Murad S TaqquAbstract:The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0,1]C[0,1]. These limits cannot be strengthened to convergence in L2(Ω)L2(Ω).
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behavior of the generalized Rosenblatt process at extreme critical exponent values
arXiv: Probability, 2016Co-Authors: Murad S TaqquAbstract:The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in $C[0,1]$. These limits cannot be strengthened to convergence in $L^2(\Omega)$.
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Rosenblatt distribution subordinated to gaussian random fields with long range dependence
arXiv: Statistics Theory, 2015Co-Authors: Nikolai N Leonenko, M D Ruizmedina, Murad S TaqquAbstract:The Karhunen-Lo\`eve expansion and the Fredholm determinant formula are used to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of Gaussian stationary random fields on R^d displaying long-range dependence. This distribution reduces to the usual Rosenblatt distribution when d=1. Several properties of this new distribution are obtained. Specifically, its series representation in terms of independent chi-squared random variables is given, the asymptotic behavior of the eigenvalues, its L\`evy-Khintchine representation, as well as its membership to the Thorin subclass of self-decomposable distributions. The existence and boundedness of its probability density is then a direct consequence.
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structure of the third moment of the generalized Rosenblatt distribution
Statistics & Probability Letters, 2014Co-Authors: Shuyang Bai, Murad S TaqquAbstract:Abstract The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the one in Maejima and Tudor (2012) and Tudor (2013). Evaluating this formula numerically, we are able to confirm that the class of generalized Hermite processes is strictly richer than the class of Hermite processes.
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properties and numerical evaluation of the Rosenblatt distribution
Bernoulli, 2013Co-Authors: Mark S Veillette, Murad S TaqquAbstract:This paper studies various distributional properties of the Rosenblatt distribution. We begin by describing a technique for computing the cumulants. We then study the expansion of the Rosenblatt distribution in terms of shifted chi-squared distributions. We derive the coefficients of this expansion and use these to obtain the Levy–Khintchine formula and derive asymptotic properties of the Levy measure. This allows us to compute the cumulants, moments, coefficients in the chi-square expansion and the density and cumulative distribution functions of the Rosenblatt distribution with a high degree of precision. Tables are provided and software written to implement the methods described here is freely available by request from the authors.
Sangyeol Lee - One of the best experts on this subject based on the ideXlab platform.
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the bickel Rosenblatt test for continuous time stochastic volatility models
Test, 2014Co-Authors: Liang Ching Lin, Sangyeol Lee, Meihui GuoAbstract:In this paper, we consider the Bickel–Rosenblatt test for continuous time stochastic volatility models. The test is constructed based on discretely observed samples by measuring integrated squared deviations between the nonparametric kernel density estimate from the observations and a parametric fit of the density. It is shown that under the null, the proposed test is asymptotically normal. To evaluate the proposed test, a simulation study is performed for illustration.
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the bickel Rosenblatt test for diffusion processes
Statistics & Probability Letters, 2006Co-Authors: Sangyeol LeeAbstract:Abstract In this paper, we consider the Bickel–Rosenblatt test for a class of diffusion processes that covers the Ornstein–Uhlenbeck process. Using the discrete sampling scheme, we calculate residuals and construct the residual based Bickel–Rosenblatt test. We show that the test statistic is asymptotically normal under regularity conditions. The result is applicable to the test for the existence of jumps in diffusion models.
Neil J Bershad - One of the best experts on this subject based on the ideXlab platform.
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convergence models for Rosenblatt s perceptron learning algorithm
IEEE Transactions on Signal Processing, 1995Co-Authors: Suhas Diggavi, J J Shynk, Neil J BershadAbstract:Presents a stochastic analysis of the steady-state and transient convergence properties of a single-layer perceptron for fast learning (large step-size, input-power product). The training data are modeled using a system identification formulation with zero-mean Gaussian inputs. The perceptron weights are adjusted by a learning algorithm equivalent to Rosenblatt's perceptron convergence procedure. It is shown that the convergence points of the algorithm depend on the step size /spl mu/ and the input signal power (variance) /spl sigma//sub x//sup 2/, and that the algorithm is stable essentially for /spl mu/>0. Two coupled nonlinear recursions are derived that accurately model the transient behavior of the algorithm. The authors also examine how these convergence results are affected by noisy perceptron input vectors. Computer simulations are presented to verify the analytical models. >
