A New Moment Bound and Weak Laws for Mixingale Arrays without Memory or Heterogeneity Restrictions, with Applications to Tail Trimmed Arrays

We present a new weak law of large numbers for dependent heterogeneous triangular arrays fyn;tg with applications to tail trimming. The law follows from a new partial sum moment bound Ej Pn t=1 yn;tj p K Pn t=1 Ejyn;tj p for zero mean Lp-mixingale arrays fyn;t;=tg, p 2 [1; 2], where =t is a -…eld. We do not require uniform integrability, nor restrict mixingale dependence or heterogeneity. The weak law is applied to tail trimmed heavy tailed data, and we characterize self-scaled laws Pn t=1 jyn;tj= Pn t=1 Ejyn;tj p ! 1 for potentially very heavy tailed data (Ejyn;tj ! 1 is possible). Finally, we characterize a minimal rate of convergence for the tail trimmed self-scaled law when distribution tails are regularly varying and possibly non-stationary, including trending tails. 1. Introduction We present a new partial sum moment bound and weak laws of large numbers [WLLN] for zero mean mixingale triangular arrays fyn;tg = fyn;t : 1 t ngn 1. The array fyn;tg may exhibit any rate of memory decay and any degree of heterogeneity. The primary application is to tail trimmed arrays in Sections 3 and 4. Let f=tg be a sequence of non-decreasing -…elds on a probability space ( ;=; P ), denote the Lp-norm jjxjjp := (Ejxj), and let K > 0 be a …nite constant that may change from place to place. We rule out degenerate cases by assuming