Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Abstract

Conditioned, in the sense of Renyi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L p -norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Renyi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.