We prove a refined Cramér-type moderate deviation result by taking into account of the skewness in normal approximation for sums of local statistics of independent random variables. We apply the main result to $k$-runs, U-statistics and subgraph counts in the Erdős–Rényi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order tail probability expansions via Stein’s method.
"A refined Cramér-type moderate deviation for sums of local statistics." Bernoulli 26 (3) 2319 - 2352, August 2020. https://doi.org/10.3150/20-BEJ1195