  Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let
$f^{n}$ be its $n$-th iterate. Let $\{a_{n}\}$ be a sequence of complex
numbers. We prove that the series $\sum a_{n}f^{n}(\xi)$ converges at almost
every point $\xi$ of the unit circle if and only if $\sum |a_n|^2 < \infty$.
The main step in the proof is to show that under this assumption, the function
$F= \sum a_n f^n$ has bounded mean oscillation. We also prove that $F$ is
bounded on the unit disc if and only if $\sum |a_n| < \infty$. Finally we
describe the sequences of coefficients $\{a_n \}$ such that $F$ belongs to
other classical function spaces, as the disc algebra and the Dirichlet class.
