Normal Approximation of Studentized U-Statistics Under Exact Fourth-Moment Conditions and Random Sample Sizes

Studentized U-statistics constitute an important class of nonlinear statistics whose asymptotic analysis is considerably more challenging than that of standardized U-statistics because the normalization involves a random variance estimator. Although extensive research has been devoted to Berry–Esseen approximations for U-statistics and self-normalized statistics, comparatively little attention has been given to the combined effects of studentization and random sample sizes. In this paper, we investigate the normal approximation of studentized U-statistics under a finite fourth-moment condition on the kernel. Using a direct probabilistic approach, we establish a Berry–Esseen-type bound of order $$O(n^{-1/4})$$ while requiring only fourth-order integrability, thereby relaxing the classical condition $$E|h(X_1,X_2)|^{9/2}<\infty $$ . The analysis is then extended to randomly indexed studentized U-statistics, yielding a normal approximation theorem that explicitly separates the effects of studentization and random-index variability. An expanded Monte Carlo study examines the approximation under normal, Laplace, moderately and highly skewed gamma, standardized Student $$t_5$$ , and contaminated normal distributions, together with several random sample-size mechanisms. The results indicate favorable robustness under symmetric heavy tails and moderate contamination, while pronounced skewness produces larger finite-sample deviations. The proposed methodology provides a unified framework for studying studentization and random indexing within U-statistic theory and extends normal approximation techniques to an important class of randomly indexed stochastic systems.