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February, 1979 A strong Law for Variables Indexed by a Partially Ordered Set with Applications to Isotone Regression
F. T. Wright
Ann. Probab. 7(1): 109-127 (February, 1979). DOI: 10.1214/aop/1176995152


In studying the asymptotic properties of certain isotone regression estimators, one is led to consider the maximum of sums of independent random variables indexed by a partially ordered set. An index set which is a sequence of $\beta$ dimensional vectors, $\{t_k\}^\infty_{k = 1}$, and the usual partial order on $R_\beta$, the $\beta$ dimensional reals, are considered here. The random variables are assumed to satisfy a condition equivalent to a finite first moment in the identically distributed case and are assumed to be centered at their means. For $A \subset R_\beta$, let $S_n(A)$ denote the sum of those random variables with indices $t_k \in A$ and $k \leqslant n$. It is shown that if the sequence $\{t_k\}$ satisfies a certain condition, then the maximum, over all upper layers $U$ in $R_\beta$, of $S_n(U)/n$ converges almost surely to zero. As a corollary to this result one obtains the strong consistency of this isotone regression estimator. If the sequence $\{t_k\}$ is a realization of a sequence of independent, identically distributed, $\beta$ dimensional random vectors and if the probability induced by such a vector is discrete, absolutely continuous or a mixture of the two, then the condition on the sequence $\{t_k\}$ is satisfied almost surely. Some nondiscrete, singular induced probabilities of interest in these regression problems are considered also.


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F. T. Wright. "A strong Law for Variables Indexed by a Partially Ordered Set with Applications to Isotone Regression." Ann. Probab. 7 (1) 109 - 127, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60033
MathSciNet: MR515817
Digital Object Identifier: 10.1214/aop/1176995152

Primary: 60F15
Secondary: 62G05

Keywords: isotone regression and strong consistency , partilly ordered sets , Strong law of large numbers

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 1 • February, 1979
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