What does "$\pi$-almost" mean? I'm reading "Complex Stochastic Systems" edited by Barndorff-Nielsen et al. I found that they use the expression $\pi$-almost consistently throughout Chapter 1. I don't understand what $\pi$-almost means or how to read in plain English the sentences using this expression. Can somebody tell me how to interpret the sentences in the images?



It means that the event asserted in the limit theorems occurs almost surely with respect to the probability measure $\pi$ --- i.e., the asserted event satisfies $\pi(\text{Event}) = 1$.
A useful intuitive way to read assertions of this kind is to read it as, "the limiting behaviour shown in the theorems holds, except possibly on a set of values of $x^{(0)}$ occurring with zero probability under the measure $\pi$".
