New insights on concentration inequalities for self-normalized martingales

Bercu, Bernard; Touati, Taieb

doi:10.1214/19-ecp269

Cited by 6 publications

(7 citation statements)

References 28 publications

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“…Bercu and Touati [3] obtained that if η k is heavy on left, then (2.14) holds. Thus our condition is analogy of (2.10) in continuous time case.…”

Section: The Main Results and Their Proofsmentioning

confidence: 99%

On De la Peña Type Inequalities for Point Processes

Nai-qi¹,

Ulyanov²,

Wang³

2022

Preprint

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There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades.De la Peña [5] establishes a nice exponential inequality for discrete time locally square integrable martingale . In this paper, we obtain de la Peña's inequalities for stochastic integral of multivariate point processes. The proof is primarily based on Doléans-Dade exponential formula and the optional stopping theorem.As application, we obtain an exponential inequality for block counting process in Λ−coalescents.

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“…Bercu and Touati [3] obtained that if η k is heavy on left, then (2.14) holds. Thus our condition is analogy of (2.10) in continuous time case.…”

Section: The Main Results and Their Proofsmentioning

confidence: 99%

On De la Peña Type Inequalities for Point Processes

Nai-qi¹,

Ulyanov²,

Wang³

2022

Preprint

View full text Add to dashboard Cite

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“…Then, ξ is called heavy on left. Bercu and Touati [14] extended Theorem 1 to general case. Let S = (S n ) n≥0 be a locally square integrable on…”

Section: Propositionmentioning

confidence: 97%

“…Bercu and Touati [14] found that if η k is heavy on the left, then (17) holds. Thus, our condition is an analogy of ( 13) in continuous time case.…”

Section: Propositionmentioning

confidence: 99%

On De la Peña Type Inequalities for Point Processes

2022

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There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain de la Peña’s inequalities for a stochastic integral of multivariate point processes. The proof is primarily based on Doléans–Dade exponential formula and the optional stopping theorem. As an application, we obtain an exponential inequality for block counting process in Λ−coalescent.

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“…In the proofs we will make frequent use of the following iterated exponential martingale inequality which builds on the exponential martingale inequality by Bercu & Touati (2008), see also Bercu et al (2015), Bercu & Touati (2018).…”

Section: A1 Iterated Martingale Inequalitymentioning

confidence: 99%

The Analysis of Marked and Weighted Empirical Processes of Estimated Residuals

2019

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An extended and improved theory is presented for marked and weighted empirical processes of residuals of time series regressions. The theory is motivated by 1step Huber-skip estimators, where a set of good observations are selected using an initial estimator and an updated estimator is found by applying least squares to the selected observations. In this case, the weights and marks represent powers of the regressors and the regression errors, respectively. The inclusion of marks is a non-trivial extention to previous theory and requires refined martingale arguments.

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New insights on concentration inequalities for self-normalized martingales

Cited by 6 publications

References 28 publications

On De la Peña Type Inequalities for Point Processes

On De la Peña Type Inequalities for Point Processes

On De la Peña Type Inequalities for Point Processes

The Analysis of Marked and Weighted Empirical Processes of Estimated Residuals

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