  We develop an improved stochastic formalism for the Bethe-Salpeter equation,
based on an exact separation of the effective-interaction $W$ to two parts,
$W=(W-v_W)+v_W$ where the latter is formally any translationally-invariant
interaction $v_W(r-r')$. When optimizing the fit of $v_W$ exchange kernel to
$W$, by using a stochastic sampling of $W$, the difference $W-v_W$ becomes
quite small. Then, in the main BSE routine, this small difference is
stochastically sampled. The number of stochastic samples needed for an accurate
spectrum is then largely independent of system size. While the method is
formally cubic in scaling, the scaling prefactor is small due to the constant
number of stochastic orbitals needed for sampling $W$.
