Asymptotic rate of convergence in the degenerate U-statistics of second order

Olga Yanushkevichiene

Banach Center Publications, Tome 89 (2010), p. 275-284 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T (X ₁, . . ., X ₙ) = n^{- 1} \sum_{1 \leq i L e t q_{i} (i \geq 1) b e e i g e n v a l u e s o f t h e H i l b e r t - S c h m i d t o p e r a t o r a s s o c i a t e d w i t h t h e k e r n e l h (x, y), a n d q ₁ b e t h e l a r g e s t i n a b s o l u t e v a l u e o n e . W e p r o v e t h a t}$ Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12) $,$ where $G_{i}$ , 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β^{'} : = E | h (X, X ₁) | ³ + {E | h (X, X ₁) |}^{18 / 5} + E | g (X) | ³ + {E | g (X) |}^{18 / 5} + 1 < \infty$ .

Publié le : 2010-01-01

Zbl 1229.62056

EUDML-ID : urn:eudml:doc:281795

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     author = {Olga Yanushkevichiene},
     title = {Asymptotic rate of convergence in the degenerate U-statistics of second order},
     journal = {Banach Center Publications},
     volume = {89},
     year = {2010},
     pages = {275-284},
     zbl = {1229.62056},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc90-0-17}
}

Olga Yanushkevichiene. Asymptotic rate of convergence in the degenerate U-statistics of second order. Banach Center Publications, Tome 89 (2010) pp. 275-284. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc90-0-17/