Abstract
Using martingale methods, we obtain some Fuk-Nagaev type inequalities for suprema of unbounded empirical processes associated with independent and identically distributed random variables. We then derive weak and strong moment inequalities. Next, we apply our results to suprema of empirical processes which satisfy a power-type tail condition.
Highlights
For each k = 1, . . . , n, let Xk := (Xk, t)t ∈ T be a collection of centered real-valued random variables such that X1, . . . , Xn are independent and identically distributed according to a law P
The proofs of our results rely on martingales methods, comparison inequalities, and recent concentration inequalities for real-valued martingales proved by Rio [Rio17a]
Let Y1, . . . , Yn be a finite sequence of nonnegative iid random variables and let X1, . . . , Xn be a finite sequence of iid random variables with values in some measurable space (X, F) such that the two sequences are independent
Summary
We recall that the bounded case (that is, for all t ∈ T , Xk, t 1 a.s. or |Xk, t| 1 a.s) is handled by Talagrand’s inequality It is the merit of Talagrand [Tal96] to obtain the first a functional Bennett inequality for suprema of empirical processes, but with unspecified constants. The same authors in [L13, Theorem 8] provide an exponential tail bound for Z under a uniform Bernstein condition To do this, they introduce the Bernstein–Orlicz norm that they combine with a variation of the classical chaining. The proofs of our results rely on martingales methods, comparison inequalities, and recent concentration inequalities for real-valued martingales proved by Rio [Rio17a] Another important point is that we do not seek to directly control the upper tail quantile as usual, but instead we control the Conditional Value-At-Risk of Z − E[Z]. In order to explain our results, let us give some definitions and notation
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