On the Maximal Inequalities for Partial Sums of Strong Mixing Random Variables with Applications

Guo-dong Xing, Shan-chao Yang

Abstract

Maximal inequalities for partial sums of strong mixing random variables are established. To show the applications of the inequalities obtained, we discuss the strong consistency of Gasser-M¨uller estimator of fixed design regression estimate and obtain the almost sure convergence rate $n^{-1/2}log log n)^{1/\xi}log^{3/2}n$ with any $0<\xi<2$, which closes to the optimal achievable convergence rate for independent random variables under an iterated logarithm.