Slow convergence of Birkhoff ergodic averages

Let $\{T^z\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, ${\sum }_i^{\mathrm{\infty }}{a}_i<\mathrm{\infty }$, $a_i>0$. For any sequence $M_k\to +\mathrm{\infty }$ there exist $N_k>M_k$ and a function $f\in L_1(X,m)$ such that $$m(x:\mid A(x,N_k,f)-\int fdm\mid >a_k)\to 1,$$ where $$A(x,N,f)=\frac{1}{N^n}\sum _{z\in Q_N}f(T^{z}x),$$ $$Q_N=\{(z_1,\dots ,z_n)\}:1\le z_1,\dots ,z_n\le N\}\mathrm{.}$$