  We present a novel static analysis technique to derive higher moments for
program variables for a large class of probabilistic loops with potentially
uncountable state spaces. Our approach is fully automatic, meaning it does not
rely on externally provided invariants or templates. We employ algebraic
techniques based on linear recurrences and introduce program transformations to
simplify probabilistic programs while preserving their statistical properties.
We develop power reduction techniques to further simplify the polynomial
arithmetic of probabilistic programs and define the theory of moment-computable
probabilistic loops for which higher moments can precisely be computed. Our
work has applications towards recovering probability distributions of random
variables and computing tail probabilities. The empirical evaluation of our
results demonstrates the applicability of our work on many challenging
examples.
