  Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as
logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an
observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums.
Given a probability measure $\mu$ on $M$, we consider $v_n$ as a discrete time
random process on the probability space $(M, \mu)$.
  In smooth ergodic theory there are various natural choices of $\mu$, such as
the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They
give rise to different random processes.
  We investigate relation between such processes. We show that in a large class
of measures, it is possible to couple (redefine on a new probability space)
every two processes so that they are almost surely close to each other, with
explicit estimates of "closeness".
  The purpose of this work is to close a gap in the proof of the almost sure
invariance principle for nonuniformly hyperbolic transformations by Melbourne
and Nicol.
