Spectral Analysis

Abstract

A univariate discrete time-series x_t is said to be second-order stationary if its mean, variance and autocovariances µ_r = cov(x_t, x_t−r) are all time invariant. If x_t has no strictly cyclical or deterministic components and is stationary, there are two mathematical relationships with important interpretations, the Cramer representation

$${x_t} = \int_{ - \pi }^\pi {{{\text{e}}^{it\omega }}{\text{d}}z\left( \omega \right),}$$

where

$$\begin{array}{*{20}{c}} {E\left[ {{\text{d}}z\left( \omega \right)\overline {{\text{d}}z\left( \lambda \right)} } \right]} \\ { = f\left( \omega \right)dw,} \end{array} = 0,\left. {{}_{\omega = \lambda }^{\omega \ne \lambda }} \right\}$$

and the spectral representation of the autocovariances

$${\mu _r} = \int_{ - \pi }^\pi {{{\text{e}}^{i\tau \omega }}} f\left( \omega \right){\text{d}}w$$

.