# Dependency-dependent Bounds for Sums of Dependent Random Variables

@article{Lampert2018DependencydependentBF, title={Dependency-dependent Bounds for Sums of Dependent Random Variables}, author={Christoph H. Lampert and Liva Ralaivola and Alexander Zimin}, journal={arXiv: Probability}, year={2018} }

We consider the problem of bounding large deviations for non-i.i.d. random variables that are allowed to have arbitrary dependencies. Previous works typically assumed a specific dependence structure, namely the existence of independent components. Bounds that depend on the degree of dependence between the observations have only been studied in the theory of mixing processes, where variables are time-ordered. Here, we introduce a new way of measuring dependences within an unordered set of…

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## References

SHOWING 1-10 OF 31 REFERENCES

Large deviations for sums of partly dependent random variables

- Mathematics, Computer Science
- 2004

A method by Hoeffding is used and extended to obtain strong large deviation bounds for sums of dependent random variables with suitable dependency structure and applications are given to U-statistics, random strings and random graphs.

Concentration Inequalities for Dependent Random Variables via the Martingale Method

- Mathematics
- 2008

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms…

Probability Inequalities for sums of Bounded Random Variables

- Mathematics
- 1963

Abstract Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S…

Concentration Inequalities - A Nonasymptotic Theory of Independence

- MathematicsConcentration Inequalities
- 2013

Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.

Weighted dependency graphs

- Mathematics
- 2016

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and…

Chernoff-Hoeffding bounds for applications with limited independence

- Computer Science, MathematicsSODA '93
- 1993

The limited independence result implies that a reduced amount and weaker sources of randomness are sufficient for randomized algorithms whose analyses use the CH bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routing.

Auto-correlation Dependent Bounds for Relational Data

- Computer Science
- 2013

This paper derives distribution free bounds for the relational setting where the class of data generation models it considers are inspired from the type joint distributions that are represented by relational classication models developed by the SRL community.

An Exponential Inequality for U-Statistics Under Mixing Conditions

- Computer Science, Mathematics
- 2016

A novel exponential inequality for U-statistics under the time series setting is proved via exploiting the temporal correlatedness structure and applications to high-dimensional time series inference are discussed.

A Bernstein-type Inequality for Some Mixing Processes and Dynamical Systems with an Application to Learning

- Mathematics, Computer Science
- 2015

We establish a Bernstein-type inequality for a class of stochastic processes that include the classical geometrically $\phi$-mixing processes, Rio's generalization of these processes, as well as many…

Constructive Proofs of Concentration Bounds

- MathematicsAPPROX-RANDOM
- 2010

We give a combinatorial proof of the Chernoff-Hoeffding concentration bound [9,16], which says that the sum of independent {0, 1}- valued random variables is highly concentrated around the expected…