Series criteria for growth rates of partial maxima of iterated ergodic map values (Q5960458)

Let \( f \) be a measurable real-valued function and let the mapping \( T \) be measure-preserving and ergodic. The author discusses the a.s. convergence \[ M_n := \max_{k \leq n} f \circ T^k \uparrow \text{ess sup } f. \] Here the maximal rate of convergence is characterized by the statement: \[ P(M_n > x_n , \text{ i.o.}) = 0 \text{ or } 1 \iff \sum_n P(f > x_n) \quad \text{converges or diverges}, \] where \( \{ x_n \} \) is a non-decreasing sequence of real numbers. Denote \( A_n := \{ M_n \leq x_n \} \). It is shown that: \[ P(M_n \leq x_n , \text{ i.o.}) = 1 \iff \sum_j P(A_j \mid A^c_1 \cap \dots \cap A^c_{j-1}) \quad \text{diverges}, \] which characterizes the minimal rate of convergence.