## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 3 (1980), 630-635.

### Maxima of Partial Sums and a Monotone Regression Estimator

#### Abstract

Let $\{t_k\}$ be a sequence of points in $d$-dimensional Euclidean space. Let $\{X_k\}$ be a sequence of random variables with zero mean, i.i.d. or nearly so. If $\mathscr{A}$ is a class of subsets of $R^d$, let $$M_n(\omega) = \sup_{A\in\mathscr{A}}\Sigma_{\{k\leqslant n: t_k \in A\}}X_k(\omega).$$ $M_n$ is related to a commonly used estimator in monotone regression. Under various conditions on $\mathscr{A}$ and the points $\{t_k\}$, we study the a.s. convergence to zero of $M_n/n$ as $n \rightarrow \infty$.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 3 (1980), 630-635.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994734

**Digital Object Identifier**

doi:10.1214/aop/1176994734

**Mathematical Reviews number (MathSciNet)**

MR573300

**Zentralblatt MATH identifier**

0434.60031

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 62G05: Estimation

**Keywords**

Independent random variables stationary ergodic sequences maxima of partial sums monotone regression subadditive processes

#### Citation

Smythe, R. T. Maxima of Partial Sums and a Monotone Regression Estimator. Ann. Probab. 8 (1980), no. 3, 630--635. doi:10.1214/aop/1176994734. https://projecteuclid.org/euclid.aop/1176994734