Maxima of partial sums indexed by geometrical structures
type: Item
instance of: scholarly article
title: Maxima of partial sums indexed by geometrical structures
zbMATH Open document ID: 1014.60024
author: Tiefeng Jiang
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \((X_i)_{i\in \mathbb{N}^d}\) be a family of i.i.d. random variables with \(E(X_i)<0\), \(P(X_i> 0)>0\), and \(E(e^{tX_i}) <\infty\) for \(t>0\). For a rectangle or cube \(\Delta\subset \Delta_n:=\{1, \dots,n\}^d\) consider \(S_\Delta:= \sum_{i\in \Delta} X_i\) and \(W_n:=\max_{\Delta \subset\Delta_n} S_\Delta\). The author derives several limit theorems for the random variables \(W_n\) where for the dimensions \(d=1\), \(d=2\) and \(d\geq 3\) sometimes a different limit behavior appears. The results generalize the case \(d=1\), which is already treated in the classical books of W. Feller and F. Spitzer; they are motivated by possible applications in protein folding. Proofs are based on large deviation techniques, the Chen-Stein method, and inequalities for empirical processes.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 1906102
zbMATH Keywords: maxima over partial sums
zbMATH Keywords: large deviations
zbMATH Keywords: protein folding
zbMATH Keywords: Chen-Stein method
reviewed by: Michael Voit
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1039548374
OpenAlex ID: W2086617294
cites work: Two moments suffice for Poisson approximations: The Chen-Stein method
cites work: On Deviations of the Sample Mean
cites work: Strong large deviation and local limit theorems
cites work: Q4040465
cites work: Limit laws of Erdős-Rényi-Shepp type
cites work: Critical phenomena for sequence matching with scoring
cites work: Limit distribution of maximal non-aligned two-sequence segmental score
cites work: Q4391441
cites work: Central limit theorems for empirical measures
cites work: Q5624436
cites work: Q3856819
cites work: Extreme Values in the GI/G/1 Queue
cites work: A comparison of scores of two protein structures with foldings
cites work: Convergence of stochastic processes
cites work: Maxima and exceedances of stationary Markov chains
cites work: Approximate tail probabilities for the maxima of some random fields
cites work: Using the generalized likelihood ratio statistic for sequential detection of a change-point
cites work: Tail probabilities for the null distribution of scanning statistics
cites work: Q5731217
cites work: On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
DOI: 10.1214/AOP/1039548374
description: scientific article; zbMATH DE number 1906102