  Time-averaged autocorrelation functions of a dichotomous random process
switching between 1 and 0 and governed by wide power law sojourn time
distribution are studied. Such a process, called a L\'evy walk, describes
dynamical behaviors of many physical systems, fluorescence intermittency of
semiconductor nanocrystals under continuous laser illumination being one
example. When the mean sojourn time diverges the process is non-ergodic. In
that case, the time average autocorrelation function is not equal to the
ensemble averaged autocorrelation function, instead it remains random even in
the limit of long measurement time. Several approximations for the distribution
of this random autocorrelation function are obtained for different parameter
ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the
power spectrum of the process is briefly discussed, and a nonstationary
Wiener-Khintchine theorem, relating the correlation functions and the power
spectrum is presented. The considered situation is in full contrast to the
usual assumptions of ergodicity and stationarity.
