Long Memory In Fractional Gaussian Random Fields

Long memory is an important concept in current research areas. Fractional Brownian sheet (fBs) plays an important role in modeling anisotropic Gaussian random fields with long-range dependence and self-similarity. In this thesis, we study the long-memory property based on frac-tional Brownian sheet using multidimensional wavelet analysis, regression analysis and probabil-ity limit theory, include the estimation of Hurst parameters of fractional Brownian sheet and the simulation of fractional Brownian sheet. This work is important for the improvement of diffusion-tensor imaging (DTI) of nuclear magnetic resonance (NMR) in brain and the study of stochastic differential equations driven by fBs.In Chapter 1 and 2, we introduce the backgrounds of this problem, state the main contents of this thesis and the basic theory of our work.In Chapter 3, we present two wavelet-based estimators in the case of one-dimension and carry out some numerical computations in the case of fractional Brownian motion (fBm), include comparison of methods, selection of parameters and analysis of empirical bias. The obtained results add to the existing studies on the estimation of Hurst parameter in one-dimension case, can be a guide for extending the wavelet-based estimator to multidimensional case.In Chapter 4, we estimate the Hurst parameters of fractional Brownian sheet using multidi-mensional wavelet transform. First we obtain the wavelet-based estimator of Hurst parameters of fractional Brownian sheet after assuming the dependence between wavelet coefficients, and validate this estimator using simulated fractional Brownian sheet. The estimator performs well in both the standard deviation and the root mean square error. Second we remove the assumption of dependence, reconstruct the wavelet-based estimator in the strict mathematical sense, and prove the asymptotic normality of this estimator. Furthermore, this estimator is realized using two-step method, and its accuracy is validated using numerical simulation.In Chapter 5, we use wavelet analysis to simulate fractional Brownian sheet. First we propose the wavelet representation of multi-parameter Gaussian white noise in the sense of generalized functions. Based on the integral definition of fractional Brownian sheet, the wavelet representa-tion of fBs is obtained through partial fractional integrals of the wavelet representation of multi-parameter Gaussian white noise. The almost sure convergence of this representation is proved. Based on the wavelet representation of fBs, the simulation of fBs is realized, and it is validated by evaluating the distance between the correlation coefficient of the simulated fBs and the true one.We summarize current works and some problems for further research in chapter 6.