Wavelet analysis and covariance structure of some classes of non-stationary processes

Processes with stationary n-increments are known to be characterized by the stationarity of their continuous wavelet coefficients. We extend this resu lt to the case of processes with stationary fractional increments and local ly stationary processes. Then we give two applications of these properties. First, we derive the explicit covariance structure of processes with stati onary n-increments. Second, for fractional Brownian motion, the stationarit y of the fractional increments of order greater than the Hurst exponent is recovered.