## The Annals of Probability

### A strong Law for Variables Indexed by a Partially Ordered Set with Applications to Isotone Regression

F. T. Wright

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 7, Number 1 (1979), 109-127.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995152

Digital Object Identifier
doi:10.1214/aop/1176995152

Mathematical Reviews number (MathSciNet)
MR515817

Zentralblatt MATH identifier
0392.60033

JSTOR