Convergence of \(U\)-statistics and Mises functionals in mean metric. (Russian, English) Zbl 1050.60021

Sharakhmetov, Sh.

Teor. Jmovirn. Mat. Stat. 68, 152-157 (2003); translation in Theory Probab. Math. Stat. 68, 167-172 (2004).

Let \(x_1,\ldots,x_{n}\) be i.i.d. random variables with values in the measurable space \((\Xi,A)\). Let us define \(U\)-statistic and the Mises functional
\[ U_{n}= {2\over n(n-1)} \sum_{1\leq i<j\leq n}\Phi(x_{i},x_{j}), \qquad V_{n}={1\over n^2} \sum_{i,j=1}^{n}\Phi(x_{i},x_{j}), \]
where \(\Phi: \Xi^2\to R\) is a symmetric function, and let \(g_{i}= E[\Phi(x_{i},x_{j})/x_{i}]\), \(y_{ij}=\Phi(x_{i},x_{j})-g_{i}-g_{j}\), \(E\Phi(x_{i},x_{j})=0\), \(\sigma^2=E[g_{i}]\), \(N\) is standard normal random variable, \(\chi_{s}(X,Y)= s\int| x|^{s-1}| P(X<x)-P(Y<x)|\,dx\). The author proves the following result: If \(0<\sigma<\infty\), \(E| y_{12}|^{4/3}<\infty\), then \(\chi_1({\sqrt n\over2\sigma}U_{n},N)\to0\), \(\chi_1({\sqrt n\over2\sigma}V_{n},N)\to0\), as \(n\to\infty\). Some upper bounds for \(\chi_1({\sqrt n\over2\sigma}U_{n},N)\) and \(\chi_1({\sqrt n\over2\sigma}V_{n},N)\) are obtained.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60F05

Central limit and other weak theorems

Keywords:

convergence in mean metric; \(U\)-statistics; Mises functionals