DICHOTOMY RESULTS FOR EVENTUALLY ALWAYS HITTING TIME STATISTICS AND ALMOST SURE GROWTH OF EXTREMES
Suppose (f,X, μ) is a measure preserving dynamical system and φ: X → R a measurable function. Consider the maximum process Mn := max{X1, ,Xn}, where Xi = φ ○ fi-1 is a time series of observations on the system. Suppose that (un) is a non-decreasing sequence of real numbers, such that μ(X1 > un) → 0. For certain dynamical systems, we obtain a zero- one measure dichotomy for μ(Mn ≤ un i.o.) dependin