  In numerous applications data are observed at random times and an estimated
graph of the spectral density may be relevant for characterizing and explaining
phenomena. By using a wavelet analysis, one derives a nonparametric estimator
of the spectral density of a Gaussian process with stationary increments (or a
stationary Gaussian process) from the observation of one path at random
discrete times. For every positive frequency, this estimator is proved to
satisfy a central limit theorem with a convergence rate depending on the
roughness of the process and the moment of random durations between successive
observations. In the case of stationary Gaussian processes, one can compare
this estimator with estimators based on the empirical periodogram. Both
estimators reach the same optimal rate of convergence, but the estimator based
on wavelet analysis converges for a different class of random times. Simulation
examples and application to biological data are also provided.
