  We start with an i.i.d. sequence and consider the product of two
polynomial-forms moving averages based on that sequence. The coefficients of
the polynomial forms are asymptotically slowly decaying homogeneous functions
so that these processes have long memory. The product of these two polynomial
forms is a stationary nonlinear process. Our goal is to obtain limit theorems
for the normalized sums of this product process in three cases: exclusion of
the diagonal terms of the polynomial form, inclusion, or the mixed case (one
polynomial form excludes the diagonals while the other one includes them). In
any one of these cases, if the product has long memory, then the limits are
given by Wiener chaos. But the limits in each of the cases are quite different.
If the diagonals are excluded, then the limit is expressed as in the product
formula of two Wiener-It\^{o} integrals. When the diagonals are included, the
limit stochastic integrals are typically due to a single factor of the product,
namely the one with the strongest memory. In the mixed case, the limit
stochastic integral is due to the polynomial form without the diagonals
irrespective of the strength of the memory.
