TITLE: "Typical" convergence rate for the von Neumann mean ergodic theorem
QUESTION [5 upvotes]: The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$,
where ${\cal H}$ is a Hilbert space, the following holds:
$$
\lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \Pi_U v, 
$$
where $\Pi_U$ is the orthogonal projection onto the invariant subspace of $U$.
Following answers by Coudy and Asaf, and comment by Terry Tao:
Suppose one wishes to describe the "typical" behavior of the convergence rate, say,
how close does the sum vector $\frac{1}{N}\sum_{n=1}^N U^n v$ get to the invariant subspace of $U$ in terms of $l_2$-norm.
In general the convergence rate depends heavily on the unitary $U$, and in some cases it is not even computable (see Tao's comment on metastability).
Hence there is no "typical" behavior to speak of.
Would it be then reasonable to conjecture a "density"-type argument as follows:
For every $U$, there exists a "nearby" $U'$, $\|U - U'\| = o(1)$, such that $U'$ 
convergence rate is at least, say inverse-polynomial in $dim({\cal H})$ ?

REPLY [5 votes]: It depends on $U$ of course. If $U$ is the identity, the convergence is pretty fast.
It also depends on $f$. If $f$ is a coboundary ($f=g-Ug$), then the convergence is of the order $1/n$.
When $U$ comes from a dynamical system, there are a number of results showing that there is usually no rate of convergence in the ergodic theorem. These theorems are  stated as counterexamples to the a.e. convergence.
Let me just mention  one result from the book of Petersen, "Ergodic Theory" (3.2 B).
Theorem (Kakutani-Petersen)
Let $T$ an ergodic measure transformation of a non-atomic probability space and $(b_i)$ a positive sequence with divergent sum. Then there is a bounded function $f$ with zero integral such that 
$$
sup_N \left|\sum_{n=1}^N b_n {1\over n} S_nf(x) \right| = \infty \quad for\ a.e. \ x.
$$
denoting as usual $Uf = f\circ T$ and $S_Nf = {1\over N} \sum_{k=1}^NU^kf$.
This implies that $\sum b_n \| {1\over n} S_nf\|_2$ is divergent and shows that
for any positive sequence $(a_n)$ going to zero, there can't be a bound 
$\| {1\over n} S_nf\|_2 \leq a_n$ (choose $b_n$ such that $\sum b_n = \infty$ and $\sum b_n a_n < \infty$).
In contrast, assume that the sequence $(f\circ T^n)$ is independent, then there is a convergence rate given by the iterated law of the logarithm or the Kolmogorov three-series theorem.