Non-Asymptotic Analysis of Classical Spectrum Estimators with $L$-mixing Time-series Data

Abstract: Spectral estimation is a fundamental problem for time series analysis, which is widely applied in economics, speech analysis, seismology, and control systems. The asymptotic convergence theory for classical, non-parametric estimators, is well-understood, but the non-asymptotic theory is still rather limited. Our recent work gave the first non-asymptotic error bounds on the well-known Bartlett and Welch methods, but under restrictive assumptions. In this paper, we derive non-asymptotic error bounds for a class of non-parametric spectral estimators, which includes the classical Bartlett and Welch methods, under the assumption that the data is an $L$-mixing stochastic process. A broad range of processes arising in time-series analysis, such as autoregressive processes and measurements of geometrically ergodic Markov chains, can be shown to be $L$-mixing. In particular, $L$-mixing processes can model a variety of nonlinear phenomena which do not satisfy the assumptions of our prior work. Our new error bounds for $L$-mixing processes match the error bounds in the restrictive settings from prior work up to logarithmic factors.