Exponential inequalities for self-normalized martingales with applications

By Bernard Bercu, A. Touati. 2007

138 citations recorded by the source corpus.

We propose several exponential inequalities for self-normalized martingales similar to those established by De la Pe\~{n}a. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.

Subfields: Artificial Intelligence, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26eb046d1181b5debb812ba32b3b43e1d10dafe6

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In Artificial Intelligence

In Management Science and Operations Research

In Mathematical Physics

Citations in the corpus

Hoeffding’s inequality for supermartingales (2011, methodology): “Further, inequalities for the probabilities P (Xn > x) have been obtained by Prohorov [26], Hoeffding [17], Azuma [1], Steiger [28,29], Freedman [13], Nagaev [21], Haeusler [16], McDiarmid [20], Pinelis [23], Talagrand [30], De La Peña [10], Lesigne and Volný [18], Dzhaparidze and van Zanten [11], N”
Conditional Quantile Estimation for Generalized Autoregressive Conditional Heteroscedasticity Models (2009, unknown): “For the first term, by exponential inequality for martingale difference sequences (see, e.g., Bercu and Touati 2008), we have N(n)max k Pr (∣∣∣∣∑ t λ ξt ∣∣∣∣> √nε;S2n + ω2n ≤ M) ≤ 2N(n) exp ( −n 2 2M ) .”
Exponential inequalities for martingales with applications (2013, methodology): “(45) When (φk) and (εk) are sub-Gaussian, exponential inequalities on the convergence of θn − θ have been established by Bercu and Touati [4].”
Regularized estimation in sparse high-dimensional time series models (2013, background): “The standard martingale inequalities involve some measure of predictable quadratic variation (Bercu and Touati, 2008), which, in our context, corresponds to T − 1 t = d − 1 ( X tj ) 2 .”