Law of iterated logarithm for inner functions

In a recent work [\emph{Adv.

Math.} 401 (2022), Paper No.

108318], a central limit theorem was established for the linear combinations of the iterates of a non-rotational inner function fixing the origin.

In this paper, we prove the law of iterated logarithm (LIL) in the same setup, with a very mild condition on the coefficients.

We also identify the full set of subsequential limit points at the LIL scale.

Using the Aleksandrov--Clark decomposition and measure-preserving properties of the inner functions, one can construct a reverse martingale that is close to the linear combinations of inner functions.

We prove the LIL for the partial sums of reverse martingale differences under a Feller-type assumption, which then transfers to the linear combinations of iterates of the inner functions.