A note on logarithmic tail asymptotics and mixing

Abstract

Let Y 1, Y 2,… be a stationary, ergodic sequence of non-negative random variables with heavy tails. Under mixing conditions, we derive logarithmic tail asymptotics for the distributions of the arithmetic mean. If not all moments of Y 1 are finite, these logarithmic asymptotics amount to a weaker form of the Baum–Katz law. Roughly, the sum of i.i.d. heavy-tailed non-negative random variables has the same behaviour as the largest term in the sum, and this phenomenon persists for weakly dependent random variables. Under mixing conditions, the rate of convergence in the law of large numbers is, as in the i.i.d. case, determined by the tail of the distribution of Y 1. There are many results which make these statements more precise. The paper describes a particularly simple way to carry over logarithmic tail asymptotics from the i.i.d. to the mixing case.