  Consider a nonuniformly hyperbolic map $ T $ modelled by a Young tower with
tails of the form $ O(n^{-\beta}) $, $ \beta>2 $. We prove optimal moment
bounds for Birkhoff sums $ \sum_{i=0}^{n-1}v\circ T^i $ and iterated sums $
\sum_{0\le i<j<n}v\circ T^i\, w\circ T^j $, where $ v,w:M\to \Bbb{R}$ are
(dynamically) H\"older observables. Previously iterated moment bounds were only
known for $ \beta>5$. Our method of proof is as follows; (i) prove that $ T $
satisfies an abstract functional correlation bound, (ii) use a weak dependence
argument to show that the functional correlation bound implies moment
estimates.
  Such iterated moment bounds arise when using rough path theory to prove
deterministic homogenisation results. Indeed, by a recent result of Chevyrev,
Friz, Korepanov, Melbourne & Zhang we have convergence an It\^o diffusion for
fast-slow systems of the form \[
x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \quad
y_{k+1}=T y_k \] in the optimal range $ \beta>2. $
