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

Olga Yanushkevichiene

Banach Center Publications (2010)

  • Volume: 90, Issue: 1, page 275-284
  • ISSN: 0137-6934

Abstract

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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 ∑ 1 ≤ i L e t q i ( i ≥ 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 < ∞ .

How to cite

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Olga Yanushkevichiene. "Asymptotic rate of convergence in the degenerate U-statistics of second order." Banach Center Publications 90.1 (2010): 275-284. <http://eudml.org/doc/281795>.

@article{OlgaYanushkevichiene2010,
abstract = {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 $T(X₁,...,Xₙ) = n^\{-1\} ∑_\{1≤i Let q_i (i ≥ 1) be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x,y), and q₁ be the largest in absolute value one. We prove that \}$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 $β^\{\prime \}: = E|h(X,X₁)|³ + E|h(X,X₁)|^\{18/5\} + E|g(X)|³ + E|g(X)|^\{18/5\} + 1 < ∞$.},
author = {Olga Yanushkevichiene},
journal = {Banach Center Publications},
keywords = {Kolmogorov distance; eigenvalues},
language = {eng},
number = {1},
pages = {275-284},
title = {Asymptotic rate of convergence in the degenerate U-statistics of second order},
url = {http://eudml.org/doc/281795},
volume = {90},
year = {2010},
}

TY - JOUR
AU - Olga Yanushkevichiene
TI - Asymptotic rate of convergence in the degenerate U-statistics of second order
JO - Banach Center Publications
PY - 2010
VL - 90
IS - 1
SP - 275
EP - 284
AB - 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 $T(X₁,...,Xₙ) = n^{-1} ∑_{1≤i Let q_i (i ≥ 1) be eigenvalues of the Hilbert-Schmidt operator associated with the kernel h(x,y), and q₁ be the largest in absolute value one. We prove that }$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 $β^{\prime }: = E|h(X,X₁)|³ + E|h(X,X₁)|^{18/5} + E|g(X)|³ + E|g(X)|^{18/5} + 1 < ∞$.
LA - eng
KW - Kolmogorov distance; eigenvalues
UR - http://eudml.org/doc/281795
ER -

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