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# 7. Spectral Analysis

1. Ngai Hang Chan

Published Online: 28 JAN 2011

DOI: 10.1002/9781118032466.ch7

## Time Series: Applications to Finance with R and S-Plus, Second Edition

#### How to Cite

Chan, N. H. (2010) Spectral Analysis, in Time Series: Applications to Finance with R and S-Plus, Second Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118032466.ch7

#### Author Information

1. The Chinese University of Hong Kong, Department of Statistics, Shatin, Hong Kong

#### Publication History

1. Published Online: 28 JAN 2011
2. Published Print: 13 SEP 2010

#### Book Series:

1. Wiley Series in Probability and Statistics

#### ISBN Information

Print ISBN: 9780470583623

Online ISBN: 9781118032466

## SEARCH

### Keywords:

• auto-covariance function;
• Fourier transform;
• periodogram;
• spectral analysis;
• spectral representation theorems;
• stationary process;
• time series

### Summary

This chapter discusses the idea of spectral analysis in time series. It also states three spectral representation theorems which are used primarily in deriving theoretical properties. The first theorem states that the covariance function of a stationary process is related to the spectral distribution function through a Fourier transform. This is sometimes known as the spectral representation theorem of the auto-covariance function. The second theorem relates the process {Yt} itself to another stationary process {Z(λ)} defined over the spectral domain. It is sometimes known as the spectral representation theorem of the process. The third theorem relates the spectral density function to the autocovariance function. The chapter also explains the smoothed periodogram illustrating the spectrum of the accidental death data set. Finally, it highlights some limitations of spectral analysis.

#### Controlled Vocabulary Terms

autocovariance function; Fourier transform; periodogram; spectral density function; spectral window; stationary process; time series