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

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Yanushkevichiene O.

Banach Center Publications

2010

tom 90

nr 1

275-284

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
$Δ_{n} = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ^{'1/6})/(√(|q₁|) n^{1/12})$,
where $G_i$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β': = E|h(X,X₁)|³ + E|h(X,X₁)|^{18/5} + E|g(X)|³ + E|g(X)|^{18/5} + 1 < ∞$.

Ograniczanie wyników

1 Banach Center Publications

1 Yanushkevichiene O.

1 2010

Wyniki wyszukiwania

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