For a LARCH ("linear ARCH") sequence $ (y_n, \sigma_n) $ exhibiting long range dependence, we determine the limiting distribution of sums $\sum f(y_n)$, $ \sum f(\sigma_n) $ for smooth functions $ f $ satisfying $E(y_0 f' (y_0)) \neq 0 $, $ E (\sigma_0 f' (\sigma_0)) \neq 0 $. We also give an approximation formula for the above sums, providing the first term of the asymptotic expansions of $ \sum f (y_n),\break \sum f (\sigma_n)$.