On Consistency in Monotonic Regression

For each t in some subset T of N-dimensional Euclidean space let Fₜ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of t. Let t₁, t₂,⋯ be a sequence of points in T and let Y₁, Y₂,⋯ be an independent sequence of random variables such that the distribution function of Yₖ is Ftₖ. Estimators m̂ₙ(t; Y₁,⋯, Yₙ) of $m(t)$ which are monotone in each coordinate of t and which minimize ∑ⁿᵢ₌₁ ̂ₙ(tᵢ; Y₁,⋯, Yₙ) - Yᵢ² are already known. Brunk has investigated their consistency when $N = 1$. In this paper additional consistency results are obtained when $N = 1$ and some results are obtained in the case $N = 2$. In addition, we prove several lemmas about the law of large numbers which we believe to be of independent interest.