Large Deviation inequalities for sums of positive correlated variables with clustering

TL;DR

This paper develops large deviation inequalities for sums of positively correlated Bernoulli variables, especially in clustered scenarios, using spectral methods to extend classical results to dependent variables.

Contribution

It introduces novel upper deviation inequalities for positively correlated Bernoulli variables, a case scarcely addressed in existing literature, using spectral decomposition techniques.

Findings

Established upper deviation bounds for positively correlated Bernoulli variables

Demonstrated the applicability of spectral methods in dependent variable inequalities

Provided illustrative examples of the inequalities in clustered processes

Abstract

Large deviation inequalities for ergodic sums is an important subject since the seminal contribution of Bernstein for independent random variables with finite variances, followed by the Chernoff method and the Hoefding result for independent bounded variables. Very few results appears in the literature for the non independent case. Here we consider the, barely treated in the literature, case of positively correlated Bernoulli variables. This case represents the appearance in clusters of a certain fixed phenomena in the overlying stochastic process. Under a very mild condition we prove several upper deviation inequalities. The results follow by a spectral decomposition of an appropriated recursive operator. We illustrate with examples.