Abstract
Elementary approaches to classic strong laws of large numbers use a monotonicity argument or a Tauberian argument of summability theory. Together with results on variance of sums of dependent random variables they allow to establish various strong laws of large numbers in case of dependence, especially under mixing conditions. Strong consistency of nonparametric regression estimates of local averaging type (kernel and nearest neighbor estimates), pointwise as well as in L 2, can be considered as a generalization of strong laws of large numbers. Both approaches can be used to establish strong universal consistency in the case of independence and, mostly by sharpened integrability assumptions, consistency under ρ-mixing or α-mixing. In a similar way Rosenblatt-Parzen kernel density estimates are treated.
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Walk, H. (2010). Strong Laws of Large Numbers and Nonparametric Estimation. In: Devroye, L., Karasözen, B., Kohler, M., Korn, R. (eds) Recent Developments in Applied Probability and Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2598-5_8
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