On the distribution function of the majorant of ergodic means

Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let ${\displaystyle f\cdot ={\supset }_N\left\{\frac{1}{N}\right\}{\sum }_{m=0}^{N-1}f\circ T^m}$. In this paper we mainly investigate the question of whether (i) ${\displaystyle {ʃ}_a^{\infty }|\mu (f\cdot >t)-\left\{\frac{1}{t}\right\}{ʃ}_{(f\cdot >t)}fd\mu |dt<\infty }$ and whether (ii) ${\displaystyle {ʃ}_a^{\infty }|\mu (f\cdot >t)-\left\{\frac{1}{t}\right\}{ʃ}_{(f>t)}fd\mu |dt<\infty }$ for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables ${\displaystyle f\circ T^m}$ are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries.