There's no estimate that works in general. Krengel, "On the speed of convergence in the ergodic theorem", shows that for any ergodic transformation of $[0,1]$ and any sequence $(a_n)$ converging to $0$, no matter how slowly, there is a set $A$ such that $$\lim_{N\rightarrow\infty}\frac{1}{a_N}|A_N\chi_A(x)-\mu(A)|=\infty$$ for almost every $x$, where $A_N=\frac{1}{N}\sum_{i<N}f(T^ix)$. (The same function $\chi_A$ has similarly slow convergence in the $L^p$ norm for all $p\in[1,\infty)$ as well.)

As Vaughn Climenhaga has already pointed out, many additional assumptions have consequences for the rate of convergence. If you can't work with one of those assumptions, there's a big literature on the oscillations of the ergodic averages, both as "oscillations" and as "upcrossings", and those results suffice for many purposes.

There's no estimate that works in general. Krengel, "On the speed of convergence in the ergodic theorem", shows that for any ergodic transformation of $[0,1]$ and any sequence $(a_n)$ converging to $0$, no matter how slowly, there is a set $A$ such that $$\limsup_{N\rightarrow\infty}\frac{1}{a_N}|A_N\chi_A(x)-\mu(A)|=\infty$$ for almost every $x$, where $A_N=\frac{1}{N}\sum_{i<N}f(T^ix)$. (The same function $\chi_A$ has similarly slow convergence in the $L^p$ norm for all $p\in[1,\infty)$ as well.)

As Vaughn Climenhaga has already pointed out, many additional assumptions have consequences for the rate of convergence. If you can't work with one of those assumptions, there's a big literature on the oscillations of the ergodic averages, both as "oscillations" and as "upcrossings", and those results suffice for many purposes.

There's no estimate that works in general. Krengel, "On the speed of convergence in the ergodic theorem", shows that for any ergodic transformation of $[0,1]$ and any sequence $(a_n)$ converging to $0$, no matter how slowly, there is a set $A$ such that $$\lim_{N\rightarrow\infty}\frac{1}{a_N}|A_N\chi_A(x)-\mu(A)|=\infty$$ for almost every $x$, where $A_N=\frac{1}{N}\sum_{i<N}f(x)$. (The same function $\chi_A$ has similarly slow convergence in the $L^p$ norm for all $p\in[1,\infty)$ as well.)

As Vaughn Climenhaga has already pointed out, many additional assumptions have consequences for the rate of convergence. If you can't work with one of those assumptions, there's a big literature on the oscillations of the ergodic averages, both as "oscillations" and as "upcrossings", and those results suffice for many purposes.

There's no estimate that works in general. Krengel, "On the speed of convergence in the ergodic theorem", shows that for any ergodic transformation of $[0,1]$ and any sequence $(a_n)$ converging to $0$, no matter how slowly, there is a set $A$ such that $$\lim_{N\rightarrow\infty}\frac{1}{a_N}|A_N\chi_A(x)-\mu(A)|=\infty$$ for almost every $x$, where $A_N=\frac{1}{N}\sum_{i<N}f(T^ix)$. (The same function $\chi_A$ has similarly slow convergence in the $L^p$ norm for all $p\in[1,\infty)$ as well.)

As Vaughn Climenhaga has already pointed out, many additional assumptions have consequences for the rate of convergence. If you can't work with one of those assumptions, there's a big literature on the oscillations of the ergodic averages, both as "oscillations" and as "upcrossings", and those results suffice for many purposes.