# Nonlinear Spectral Analysis: A Local Gaussian Approach

The spectral distribution $f\left(\omega \right)$ of a stationary time series ${\left\{{Y}_{t}\right\}}_{t\in Z}^{}$ can be used to investigate whether or not periodic structures are present in ${\left\{{Y}_{t}\right\}}_{t\in Z}^{}$, but $f\left(\omega \right)$ has some limitations due to its dependence on the autocovariances $\gamma \left(h\right)$. For example, $f\left(\omega \right)$ can not distinguish white iid noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f\left(\omega \right)$ can be an inadequate tool when ${\left\{{Y}_{t}\right\}}_{t\in Z}^{}$ contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the *local Gaussian autocorrelations*, and these *local measures of dependence* can be used to construct the *local Gaussian spectral density* presented in this paper. A key feature of the new local spectral density is that it coincides with $f\left(\omega \right)$ for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if $f\left(\omega \right)$ is flat, then peaks and troughs of the new local spectral density can indicate nonlinear traits, which potentially might discover *local periodic phenomena* that remain undetected in an ordinary spectral analysis.