Spectral representation and estimation for locally stationary wavelet processes
Rainer von Sachs, Guy P. Nason, Gerald Kroisandt · CRM proceedings & lecture notes · 1999
This article develops a wavelet decomposition of a stochastic process which parallels a time--localized Cram'er (Fourier) spectral representation. We provide a time--scale instead of a time--frequency decomposition and, hence, instead of thinking as scale in terms of "inverse frequency" we start from genuine time--scale building blocks or "atoms". Using this class of locally stationary wavelet (LSW) processes, a doubly--indexed array of processes fX t;T g t=1;:::;T ; T 1, we develop a theory for the estimation of the "evolutionary wavelet spectrum". Our asymptotics are based on rescaling in time-- location which allows us to perform rigorous estimation theory starting from a single stretch of observations of fX t;T g. This evolutionary wavelet spectrum measures the local power in the variance--covariance decomposition of the process fX t;T g at a certain scale and a (rescaled) time location. It is possible to estimate the evolutionary wavelet spectrum by means of a wavelet periodogram...