A Remark on Convergence in Distribution of $U$-Statistics
Gine, Evarist ; Zinn, Joel
Ann. Probab., Tome 22 (1994) no. 4, p. 117-125 / Harvested from Project Euclid
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It is proved that, for $h$ measurable and symmetric in its arguments and $X_i$ i.i.d., if the sequence $\{n^{-m/2} \sum_{i_1,\ldots,i_m\leq n,i_j\neq i_k \text{if} j\neq k} h(X_{i_1},\ldots, X_{i_m})\}^\infty_{n=1}$ is stochastically bounded, then $Eh^2 < \infty$ and $Eh(X_1,x_2,\ldots,x_m) = 0$ a.s.
Publié le : 1994-01-14
Classification:  $U$-statistics,  necessary conditions for convergence in distribution,  decoupling,  60F05,  60E15 
@article{1176988850,
     author = {Gine, Evarist and Zinn, Joel},
     title = {A Remark on Convergence in Distribution of $U$-Statistics},
     journal = {Ann. Probab.},
     volume = {22},
     number = {4},
     year = {1994},
     pages = { 117-125},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176988850}
}

Gine, Evarist; Zinn, Joel. A Remark on Convergence in Distribution of $U$-Statistics. Ann. Probab., Tome 22 (1994) no. 4, pp.  117-125. http://gdmltest.u-ga.fr/item/1176988850/