for $\beta\in(0,1)$ and $m(n)= \sum_{j=1}^n\ell(j)j^{-1}$ for $\beta=1$.

In some of our statements, the observable $v$ is not mentioned. Here, we are speaking of convergence of operators on the Banach space $\mathcal{B}$. So for example, Theorem \[thm-conv\] states that $\sup_{v\in\mathcal{B},\;\|v\|=1}\|m(n)n^{1-\beta}T_nv-d_\beta\int_Y v\, d\mu\|\to0$ as $n\to\infty$. Since $\mathcal{B}$ is embedded in $L^\infty(Y)$, this immediately implies almost sure convergence (at a uniform rate) on $Y$. Redefining sequences on a set of measure zero, we obtain uniform convergence on $Y$. For brevity, we will speak of uniform convergence throughout this paper without further comment.

\[thm-conv\] If $\beta\in(\frac12,1]$, then $\lim_{n\to\infty}m(n) n^{1-\beta}T_n =d_\beta P$.

The next result gives upper bounds on the decay rate of $T_n$ for $\beta\le\frac12$, and an improved upper bound for $\beta\ge\frac12$ when the observable is of mean zero.

\[thm-bound\]

-   If $\beta=\frac12$, then $T_n \ll \ell(n) n^{-\frac12}\int_{1/n}^\pi \ell(1/\theta)^{-2}\theta^{-1}\,
    d\theta$.

-   If $\beta\in(0,\frac12)$, then $T_n \ll \ell(n) n^{-\beta}$.

-   If $\beta\in(0,1)$, and $v\in\mathcal{B}$ satisfies $\int_Y v\,d\mu=0$, then $T_nv \ll \ell(n) n^{-\beta}$.

As indicated in [@GarsiaLamperti62], the estimate in Theorem \[thm-bound\](b) is essentially optimal. However, we recover certain aspects of Theorem \[thm-conv\] even for $\beta\le\frac12$. Recall that $E\subset{{\mathbb N}}$ has [*density zero*]{} if $\lim_{n\to\infty} \frac1n\sum_{j=1}^n 1_E(j)=0$.

\[thm-liminf\] Let $\beta\in(0,\frac12]$ and $v\in\mathcal{B}$.

-   For all $y\in 