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

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.

### MSC:

 60F05 Central limit and other weak theorems