Local periodogram smoothing ($\hat{f}(\omega )={\sum }_j{w}_{Tj}(\omega )I_T({\omega }_j)$)

Average neighboring periodogram ordinates with nonnegative weights summing to one. Let the number $L_T$ of averaged ordinates tend to infinity while their frequency span, of order $L_T/T$, tends to zero. For a smooth spectral density and suitable short-memory or fourth-order dependence conditions, the bias tends to zero and variance is of order $1/L_T$. Smooth second-order spectral density alone does not ensure this: a time-constant random scale multiplying independent white noise has a smooth unconditional spectrum but a nonergodic random limiting local power.