Open Access
June, 1980 Maxima of Partial Sums and a Monotone Regression Estimator
R. T. Smythe
Ann. Probab. 8(3): 630-635 (June, 1980). DOI: 10.1214/aop/1176994734


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 $n \rightarrow \infty$.


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R. T. Smythe. "Maxima of Partial Sums and a Monotone Regression Estimator." Ann. Probab. 8 (3) 630 - 635, June, 1980.


Published: June, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0434.60031
MathSciNet: MR573300
Digital Object Identifier: 10.1214/aop/1176994734

Primary: 60F15
Secondary: 62G05

Keywords: Independent random variables , maxima of partial sums , monotone regression , stationary ergodic sequences , Subadditive processes

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 3 • June, 1980
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