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

Banach Center Publications (2010)

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

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topOlga 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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