Maximal inequalities for averages of i.i.d. and 2-exchangeable random variables

Abstract

Let {X,Xn:n⩾1} be a sequence of 2-exchangeable random variables, i.e., any two random variables in the sequence have the same joint probability distribution function as any other two. Any sequence of random variables in which the random variables are either i.i.d., or pairwise independent, identically distributed, or exchangeable is also 2-exchangeable. Let Sn=∑i=1nXi. We will obtain upper and lower bounds for the distribution function of max1⩽i⩽n|Si/i|. For i.i.d. real valued random variables our result translates into,∑i=1nP{|X|>10λi}1+∑i=1nP{|X|>10λi}⩽Pmax1⩽i⩽nSii>5λ⩽10∑i=1nP{|X|>λi},for every positive integer n and λ>0.