Skewness correction in tail probability approximations for sums of local statistics

Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erd\"os-R\'enyi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cram\'er-type moderate deviations via Stein's method.