  We introduce the wavelet scattering spectra which provide non-Gaussian models
of time-series having stationary increments. A complex wavelet transform
computes signal variations at each scale. Dependencies across scales are
captured by the joint correlation across time and scales of wavelet
coefficients and their modulus. This correlation matrix is nearly diagonalized
by a second wavelet transform, which defines the scattering spectra. We show
that this vector of moments characterizes a wide range of non-Gaussian
properties of multi-scale processes. We prove that self-similar processes have
scattering spectra which are scale invariant. This property can be tested
statistically on a single realization and defines a class of wide-sense
self-similar processes. We build maximum entropy models conditioned by
scattering spectra coefficients, and generate new time-series with a
microcanonical sampling algorithm. Applications are shown for highly
non-Gaussian financial and turbulence time-series.
