Inference of synchrosqueezing transform -- toward a unified statistical analysis of nonlinear-type time-frequency analysis

We provide a statistical analysis of a tool in nonlinear-type time-frequency analysis, the synchrosqueezing transform (SST), for both the null and non-null cases. The intricate nonlinear interaction of different quantities in the SST is quantified by carefully analyzing relevant multivariate complex Gaussian random variables. Several new results for such random variables are provided, and a central limit theorem result for the SST is established. The analysis sheds lights on bridging time-frequency analysis to time series analysis and diffusion geometry.