In this paper Laws of the Iterated Logarithm for maximums of absolute values of partial sums of permuted random variables are derived under conditions that are the same as or similar to conditions used by Kolmogorov, Hartman and Wintner, Petrov and Csaki in deriving Laws of the Iterated Logarithm for sums of random variables or semimartingales. These results are then applied to obtain logarithmic convergence rates for estimators of non-decreasing regression functions and integral regression functions.
"Laws of the Iterated Logarithm for Permuted Random Variables and Regression Applications." Ann. Statist. 1 (5) 872 - 887, September, 1973. https://doi.org/10.1214/aos/1176342508