Limiting distribution transition for degenerate second-order incomplete U-statistics

Determine how the limiting distribution of second-order (k=2) degenerate incomplete U-statistics constructed using minimum-variance designs transitions between a normal distribution and an infinite weighted sum of centered chi-square random variables, as a function of the growth rate of the design size |D| relative to the sample size n.

Background

For complete second-order degenerate U-statistics, the limiting distribution is known to be an infinite weighted sum of centered chi-square random variables. For incomplete U-statistics with k=2, prior work (e.g., Weber 1981) shows that both normal and weighted chi-square limits can occur, and that the choice of design strongly influences the asymptotic behavior.

Despite these findings, the precise characterization of the phase transition between normal and weighted chi-square limits—specifically in relation to how fast the size of a minimum-variance design grows with n—has not been established. Resolving this will provide clear criteria for when Gaussian approximations are valid and when resampling is required in practical inference.