Geodesic
Cram\'er type large deviations for some $U$-statistics
Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 4, pp. 858-868

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We prove Cramér type large deviations for some $U$-statistics of degree two with kernel $h(x,y)$ being of bounded variation on bounded rectangles. The proof consists of two basic steps. First some explicit bounds (similar to Helmers' bounds for $L$-statistics) for the $U$-statistics are obtained. Then Linnik's result and some results exploiting strong approximations are applied.
Keywords: $U$-statistics, large deviations, strong approximations. 
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     author = {T. Inglot and T. Ledwina},
     title = {Cram\'er type large deviations for some $U$-statistics},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {858--868},
     publisher = {mathdoc},
     volume = {38},
     number = {4},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_1993_38_4_a8/}
}
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T. Inglot; T. Ledwina. Cram\'er type large deviations for some $U$-statistics. Teoriâ veroâtnostej i ee primeneniâ, Tome 38 (1993) no. 4, pp. 858-868. http://geodesic.mathdoc.fr/item/TVP_1993_38_4_a8/