Lower bound for the maximum of a stochastic process
type: Item
instance of: scholarly article
title: Lower bound for the maximum of a stochastic process
zbMATH Open document ID: 0838.60041
DOI: 10.1007/BF02266701
published in: Mathematical Notes
publication date: 1996-02-01T00:00:00Z
review text: The authors extend a well-known result on Rademacher functions to functions bounded in \(L_3\)-norm, and show that if both \(\xi_j\) and \(\varphi_j (x)\) are independent random variables on possibly different spaces, and both of them are bounded in \(L_3\)-norm, then (on the product space) the expectation of the \(L_\infty\)-norm of the sum \(\sum^n_{j = 1} \xi_j \varphi_j (x)\) is bounded from below by a constant times \((n \log n)^{1/2}\). A major step in the proof is to establish a lower bound of \[ E [\max_{1 \leq u \leq n} |\sum^n_{j = 1} \xi_j \varphi_j (x_u) |], \] where \(x_u\) is a properly chosen sequence of points.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60F25
zbMATH DE Number: 830780
zbMATH Keywords: sums of independent random variables
zbMATH Keywords: expected value
zbMATH Keywords: two-sided bounds
zbMATH Keywords: Rademacher functions
author: B. S. Kashin
author: L. A. Tsafriri
reviewed by: Janos Galambos
MaRDI profile type: Publication
cites work: Some properties of trigonometric series whose terms have random signs
cites work: Random Fourier Series with Applications to Harmonic Analysis. (AM-101)
cites work: Q3997990
cites work: Invertibility of ``large'' submatrices with applications to the geometry of Banach spaces and harmonic analysis
description: scientific article; zbMATH DE number 830780