  While we are usually focused on forecasting future values of time series, it
is often valuable to additionally predict their entire probability
distributions, e.g. to evaluate risk, Monte Carlo simulations. On example of
time series of $\approx$ 30000 Dow Jones Industrial Averages, there will be
presented application of hierarchical correlation reconstruction for this
purpose: MSE estimating polynomial as joint density for (current value,
context), where context is for example a few previous values. Then substituting
the currently observed context and normalizing density to 1, we get predicted
probability distribution for the current value. In contrast to standard machine
learning approaches like neural networks, optimal polynomial coefficients here
have inexpensive direct formula, have controllable accuracy, are unique and
independently calculated, each has a specific cumulant-like interpretation, and
such approximation can asymptotically approach complete description of any real
joint distribution - providing universal tool to quantitatively describe and
exploit statistical dependencies in time series, systematically enhancing
ARMA/ARCH-like approaches, also based on different distributions than Gaussian
which turns out improper for daily log returns. There is also discussed
application for non-stationary time series like calculating linear time trend,
or adapting coefficients to local statistical behavior.
