Generalized Strong Laws of Large Numbers for Intermediately Trimmed Sums for Non-negative Stationary Processes

Abstract

We consider intermediately trimmed sums for non-negative identically distributed random variables. Here we distinguish three cases, namely independent random variables, observables of an underlying dynamical system with a spectral gap, and à -mixing random variables. We show that in all three cases it is possible to find a proper trimming function for every distribution function such that an intermediate trimmed strong law holds. For the case that the distribution function has regularly varying tails and the random variables are independent we give sharp conditions on the trimming function for an intermediate trimmed strong law. The same trimming rate holds for observables of a dynamical system with a spectral gap. For the case of mixing random variables we show some convergence results with stronger conditions on the trimming rate dependent on the mixing coefficient.