Abstract: This talk contains three results. First, I will introduce two exponential inequalities for nondegenerate and degenerate U-statistics under different weak dependence conditions. The first is based on a deterministic decomposition of double-indexed partial sum processes, and the second is based on decomposition using random Fourier features and a probabilistic method. Second, I will introduce a new exponential inequality for large autocovariance matrices constructed from high dimensional structural time series. It extends the inequality for product measures of Rudelson, and the proof method is based on the Cantor-set blocking argument put forward by Merlevede et al. (2011) and Banna et al. (2016) in case of geometrically strongly mixing scalar-valued or absolutely regular matrix-valued sequences.