Geodesic
An exponential inequality for $U$-statistics of i.i.d.\ data
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 508-533

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We establish an exponential inequality for degenerated $U$-statistics of order $r$ of independent and identically distributed (i.i.d.) data. This inequality gives a control of the tail of the maxima absolute values of the $U$-statistic by the sum of the two terms: an exponential term and one involving the tail of $h(X_1,\dots,X_r)$. We also give a version for not necessarily degenerated $U$-statistics having a symmetric kernel and furnish an application to the convergence rates in the Marcinkiewicz law of large numbers. Application to the invariance principle in Hölder spaces is also considered.
Keywords: $U$-statistics, exponential inequality. 
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     author = {D. Giraudo},
     title = {An exponential inequality for $U$-statistics of i.i.d.\ data},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {508--533},
     publisher = {mathdoc},
     volume = {66},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a5/}
}
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D. Giraudo. An exponential inequality for $U$-statistics of i.i.d.\ data. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 508-533. http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a5/