The danger of confusing long-range dependence with non-stationarity has been pointed out by many authors. Finding an answer to this difficult question is of importance to model time-series showing trend-like behavior, such as river runoff in hydrology, historical temperatures in the study of climates changes, or packet counts in network traffic engineering.The main goal of this paper is to develop a test procedure to detect the presence of non-stationarity for a class of processes whose K-th order difference is stationary.Contrary to most of the proposed methods, the test procedure has the same distribution for short-range and long-range dependence covariance stationary processes, which means that this test is able to detect the presence of non-stationarity for processes showing long-range dependence or which are unit root.The proposed test is formulated in the wavelet domain, where a change in the generalized spectral density results in a change in the variance of wavelet coefficients at one or several scales. Such tests have been already proposed in Whitcher et al.(2001), but these authors do not have taken into account the dependence of the wavelet coefficients within scales and between scales. Therefore, the asymptotic distribution of the test they have proposed was erroneous; as a consequence, the level of the test under the null hypothesis of stationarity was wrong.In this contribution, we introduce two test procedures, both using an estimator of the variance of the scalogram at one or several scales. The asymptotic distribution of the test under the null is rigorously justified. The pointwise consistency of the test in the presence of a single jump in the general spectral density is also be presented.A limited Monte-Carlo experiment is performed to illustrate our findings.H 0 , when the asymptotic level is set to 0.05. White noise n 512 1024 2048 4096 8192 J = 3 KSM 0.02 0.01 0.03 0.02 0.02 J = 3 CV M 0.05 0.045 0.033 0.02 0.02 J = 4 KSM 0.047 0.04 0.04 0.02 0.02 J = 4 CV M 0.041 0.02 0.016 0.016 0.01 J = 5 KSM 0.09 0.031 0.02 0.025 0.02 J = 5 CV M 0.086 0.024 0.012 0.012 0.02 Table 1.3. Empirical level of KSM − CVM for a white noise. 1 Testing for homogeneity of variance in the wavelet domain. 29 MA(1)[θ = 0.9] n 512 1024 2048 4096 8192 J = 3 KSM 0.028 0.012 0.012 0.012 0.02 J = 3 CV M 0.029 0.02 0.016 0.016 0.01 J = 4 KSM 0.055 0.032 0.05 0.025 0.02 J = 4 CV M 0.05 0.05 0.03 0.02 0.02 J = 5 KSM 0.17 0.068 0.02 0.02 0.02 J = 5 CV M 0.13 0.052 0.026 0.021 0.02 Table 1.4. Empirical level of KSM − CVM for a M A(q) process. AR(1)[φ = 0.9] n 512 1024 2048 4096 8192 J = 3 KSM 0.083 0.073 0.072 0.051 0.04 J = 3 CV M 0.05 0.05 0.043 0.032 0.03 J = 4 KSM 0.26 0.134 0.1 0.082 0.073 J = 4 CV M 0.14 0.092 0.062 0.04 0.038 J = 5 KSM 0.547 0.314 0.254 0.22 0.11 J = 5 CV M 0.378 0.221 0.162 0.14 0.093

In this paper, we study robust estimators of the memory parameter d of a (possibly) non stationary Gaussian time series with generalized spectral density f . This generalized spectral density is characterized by the memory parameter d and by a function f * which specifies the short-range dependence structure of the process. Our setting is semi-parametric since both f * and d are unknown and d is the only parameter of interest. The memory parameter d is estimated by regressing the logarithm of the estimated variance of the wavelet coefficients at different scales. The two estimators of d that we consider are based on robust estimators of the variance of the wavelet coefficients, namely the square of the scale estimator proposed by [27] and the median of the square of the wavelet coefficients. We establish a Central Limit Theorem for these robust estimators as well as for the estimator of d based on the classical estimator of the variance proposed by [19]. Some Monte-Carlo experiments are presented to illustrate our claims and compare the performance of the different estimators.The properties of the three estimators are also compared on the Nile River data and the Internet traffic packet counts data. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the memory parameter d of Gaussian time series.

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Consider the discrete wavelet transform (DWT) of a time series = with weakly stationary th differences. Such time series are encountered in many situations, including unit root or long memory processes. If the wavelet has at least vanishing moments, the DWT is jointly stationary across scales provided that small scales coefficients are reshaped in appropriate blocks to cope with the downsampling embedded in the DWT. Our goal is to compute the covariance matrix or the joint spectral density of the DWT, given the autocovariance of the th differences. We assume that the DWT corresponds to a multiresolution analysis (MRA), which allows us to compute wavelet coefficients iteratively through a succession of finite impulse response (FIR) filters and downsampling. This iterative scheme, however, is not suitable for our purpose in the case where the process itself is not stationary. Hence, we first derive an iterative algorithm with the same DWT output but with input the th differences of the time series. An iterative low complexity scheme is then deduced to compute the exact covariance matrix and spectral density of the DWT. This new algorithm is an opportunity to investigate how using exact DWT covariance computations improves previously proposed statistical methods that rely on approximated computations. Numerical experiments are used for comparisons. Two cases are examined: 1) a local semi-parametric likelihood estimation of long memory processes in the wavelet domain and 2) the computation of a test statistic for detecting change points in the wavelet domain for long memory processes. A real data set analysis is also presented. Namely, economic structural changes are investigated by looking for change points in the daily S&P 500 absolute log returns.

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