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

• Autorzy: Olga Yanushkevichiene
• Języki publikacji: EN

Abstrakty

EN

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 <br> $T(X₁,...,Xₙ) = n^{-1} ∑_{1≤i<k≤n} h(X_i,X_k) + n^{-1/2} ∑_{1≤i≤n} g(X_i)$. <br> 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 <br> $Δ_{n} = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ^{'1/6})/(√(|q₁|) n^{1/12})$, <br> where $G_i$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β': = E|h(X,X₁)|³ + E|h(X,X₁)|^{18/5} + E|g(X)|³ + E|g(X)|^{18/5} + 1 < ∞$.

Kategorie tematyczne

• 60F99: None of the above, but in this section
• 46A32: Spaces of linear operators; topological tensor products; approximation properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 90
• Numer: 1
• Strony: 275-284

Daty

wydano

2010

Twórcy

autor

Olga Yanushkevichiene

• Vilnius Pedagogical University, Studentu 39, Vilnius LT-08106, Lithuania
• Institute of Mathematics and Informatics, Akademijos, 4, Vilnius LT-08663, Lithuania

Identyfikatory

DOI

10.4064/bc90-0-17