We derive sharp probability bounds on the tails of a product of symmetric non-negative
random variables using only information about their first two moments. If the covariance
matrix of the random variables is known exactly, these bounds can be computed numerically
using semidefinite programming. If only an upper bound on the covariance matrix is avail-
able, the probability bounds on the right tails can be evaluated analytically. The bounds
under precise and imprecise covariance information coincide for all left tails as well as for all
right tails corresponding to quantiles that are either sufficiently small or sufficiently large.
We also prove that all left probability bounds reduce to the trivial bound 1 if the number
of random variables in the product exceeds an explicit threshold. Thus, in the worst case,
the weak-sense geometric random walk defined through the running product of the random
variables is absorbed at 0 with certainty as soon as time exceeds the given threshold. The
techniques devised for constructing Chebyshev bounds for products can also be used to de-
rive Chebyshev bounds for sums, maxima and minima of non-negative random variables.