Periodogram

A function $ I _ {N} ( \lambda ) $, $ - \infty < \lambda < \infty $, with $ N $ a positive integer, defined on a sample $ X( 1) \dots X( N) $ of a stationary stochastic process $ X( t) $, $ t = 0, \pm 1 \dots $ as follows:

$$ I _ {N} ( \lambda ) = \frac{1}{2 \pi N } | d _ {N} ^ {( X)} ( \lambda ) | ^ {2} , $$

where

$$ d _ {N} ^ {( X)} ( \lambda ) = \sum _ { t= 1} ^ { N } \mathop{\rm exp} \{ - it \lambda \} X( t). $$

A periodogram is a function that is periodic in $ \lambda $ with period $ 2 \pi $. The differentiable spectral density $ f( \lambda ) $ of the stationary process $ X( t) $ with mean $ c = {\mathsf E} X( t) $ can be estimated by means of the periodogram for $ \lambda \neq 0 $ $ ( \mathop{\rm mod} 2 \pi ) $:

$$ {\mathsf E} I _ {N} ( \lambda ) = f( \lambda ) + \frac{1}{2 \pi N } \frac{\sin ^ {2} N \lambda /2 }{\sin ^ {2} \lambda /2 } c ^ {2} + O( N ^ {- 1} ). $$

At the same time, the periodogram is not a consistent estimator for $ f( \lambda ) $ (cf. [1]). Consistent estimators of the spectral density (cf. Spectral density, estimator of the) can be obtained by some further constructions that employ the asymptotic lack of correlation for the periodograms for different frequencies $ \lambda _ {1} \neq \lambda _ {2} $, with the result that averaging $ I _ {N} ( x) $ with respect to frequencies close to $ \lambda $ may lead to an asymptotically-consistent estimator. In the case of an $ n $-dimensional stochastic process $ X( t) = \{ X _ {k} ( t) \} _ {k= 1} ^ {n} $, the matrix periodogram $ I _ {N} ( \lambda ) $ is determined by its elements

$$ I _ {N} ^ {( i, j) } ( \lambda ) = \ \frac{1}{2 \pi N } d _ {N} ^ {X _ {i} } ( \lambda ) {d _ {N} ^ {X _ {j} } ( \lambda ) } bar . $$

Along with $ I _ {N} ( \lambda ) $, which is also called a second-order periodogram, one sometimes also considers the periodogram of order $ m $:

$$ I _ {N} ^ {( k _ {1} \dots k _ {m} ) } ( \lambda _ {1} \dots \lambda _ {m} ) = \frac{1}{( 2 \pi ) ^ {m- 1 }N } \prod _ { j= 1 }^ { m } d _ {N} ^ {X _ {k _ {j} } } ( \lambda _ {j} ), $$

which is used in constructing $ m $-th order estimators of the spectral density (see Spectral semi-invariant).

References

[1]	D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1974)
[2]	E.J. Hannan, "Multiple time series" , Wiley (1970)