Exponential Inequalities, with Constants, for U-statistics of Order Two

Abstract

Abstract. A martingale proof of a sharp exponential inequality (with constants) is given for U-statistics of order two as well as for double integrals of Poisson processes.

Adaptive estimation of a quadratic functional of a density by model selection

“…We deduce from Theorem 3.4 in Houdré and Reynaud [13] that there exists some absolute constant C > 0 such that for all t > 0,…”

“…A crucial point in the proof of our results is an exponential inequality for U -statistics of order 2 due to Houdré and Reynaud [13].…”

“…3) In order to prove Proposition 2, we use an exponential inequality with explicit constants for U -statistics of order 2 due to Houdré and Reynaud [13]. It is worth mentioning the paper by Giné, Latala and Zinn [11] where an exponential inequality for general U -statistics is given, and the paper by Bretagnolle [5] where an exponential inequality for U -statistics of order 2 is also established.…”

“…To do this, we use an exponential inequality for U -statistics of order 2 due to Houdré and Reynaud [13] in order to control the term U n (H J ). In order to control the term P n (h J ) − f − f J 2 2 , we use Bernstein's inequality.…”

The LIL for U-Statistics in Hilbert Spaces

“…Necessary and sufficient conditions were found for the strong law of large numbers [17], the central limit theorem [19,10] and the law of the iterated logarithm [11,2]. Also some sharp exponential inequalities for canonical U -statistics have been found [8,1,14]. Analysis of the aforementioned results shows an interesting phenomenon.…”