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 $n \rightarrow \infty$.
Publié le : 1980-06-14
Classification:
Independent random variables,
stationary ergodic sequences,
maxima of partial sums,
monotone regression,
subadditive processes,
60F15,
62G05
@article{1176994734,
author = {Smythe, R. T.},
title = {Maxima of Partial Sums and a Monotone Regression Estimator},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 630-635},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994734}
}
Smythe, R. T. Maxima of Partial Sums and a Monotone Regression Estimator. Ann. Probab., Tome 8 (1980) no. 6, pp. 630-635. http://gdmltest.u-ga.fr/item/1176994734/