  We present new asymptotic formulas for the distribution of OPE coefficients
in conformal field theories. These formulas involve products of four or more
coefficients and include light-light-heavy as well as heavy-heavy-heavy
contributions. They are derived from crossing symmetry of the six and higher
point functions on the plane and should be interpreted as non-Gaussianities in
the statistical distribution of the OPE coefficients. We begin with a formula
for arbitrary operator exchanges (not necessarily primary) valid in any
dimension. This is the first asymptotic formula constraining heavy-heavy-heavy
OPE coefficients in $d>2$. For two-dimensional CFTs, we present refined
asymptotic formulas stemming from exchanges of quasi-primaries as well as
Virasoro primaries.
