On a normal approximation of $U$-statistics

We consider $U$-statistics of order 2 constructed upon independent identically distributed random variables $X_1,\ldots,X_n$ with values in a measurable space $(\mathfrak{X,B})$. For $U$-statistics with a nondegenerate kernel and canonical functions $g\colon \mathfrak{X}\mapsto\mathbf{R}$ and $h\colon \mathfrak{X}^2\mapsto\mathbf{R}$, we investigate a problem on the estimation of the rate of convergence in the central limit theorem. The result obtained implies that the estimate of order $n^{-1/2}$ depends only on the third moment $\mathbf{E}|g(X_1)|^3$ and the weak moment $\sup_{x > 0}(x^{5/3} \mathbf{P}\{|h(X_1,\,X_2)| > x\})$ of order ${\frac{5}{3}}$.