Laws of the iterated logarithm are derived for row sums of triangular arrays of independent random variables, in the context of nonparametric regression estimators. These laws provide exact strong convergence rates for kernel type nonparametric regression estimators. They apply to the important case where design points are conditioned upon, and permit the design to be multivariate. We impose minimal conditions on the error distribution; in fact, only finite variance is needed.
Peter Hall. "On Iterated Logarithm Laws for Linear Arrays and Nonparametric Regression Estimators." Ann. Probab. 19 (2) 740 - 757, April, 1991. https://doi.org/10.1214/aop/1176990449