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 SST is quantified by carefully analyzing relevant multivariate complex Gaussian random variables. Specifically, we provide the quotient distribution of dependent and improper complex Gaussian random variables. Then, a central limit theorem result for SST is established. Furthermore, we provide a block bootstrap scheme based on the established SST theory to test if a given time series contains oscillatory components.