标题： 线性过程的长记忆参数的小波估计量的渐近正态性 显示英文标题

标题： Asymptotic normality of wavelet estimators of the memory parameter for linear processes

摘要： 我们考虑线性过程，不一定是高斯过程，具有长记忆、短记忆或负记忆。 记忆参数通过从过程的样本$X_1,...,X_n$中使用小波进行半参数估计。 我们处理对数回归小波估计量和小波Whittle估计量。 我们证明这些估计量在样本大小$n\to\infty$时渐近正态，并得到了极限方差的显式表达式。 这些结果来源于线性过程的经验小波图的渐近正态性的普遍结果，该结果经过适当中心化和标准化。 小波图是一个由观测样本的二次形式组成的数组，从该样本的小波系数计算得出。 与基于傅里叶系数计算的二次形式（如周期图）不同，小波图涉及的协方差不会随着样本大小$n\to\infty$的增加而消失。 显示英文摘要

摘要： We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size $n\to\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$.