## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 62G05: Estimation

**Keywords**

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

#### Citation

Wright, F. T. A strong Law for Variables Indexed by a Partially Ordered Set with Applications to Isotone Regression. Ann. Probab. 7 (1979), no. 1, 109--127. doi:10.1214/aop/1176995152. https://projecteuclid.org/euclid.aop/1176995152