A law of the iterated logarithm for an estimate of frequency
A form of the law of the iterated logarithm is proved for the estimate of the frequency, ω 0, of a sinusoidal oscillation when observed subject to stationary noise. The estimate, ω , is the location of the maximum of the periodogram from T observations. The form of the law is unusual since it is { T...
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| Published in | Stochastic processes and their applications Vol. 22; no. 1; pp. 103 - 109 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
01.05.1986
Elsevier Science Elsevier |
| Series | Stochastic Processes and their Applications |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0304-4149 1879-209X |
| DOI | 10.1016/0304-4149(86)90117-1 |
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| Abstract | A form of the law of the iterated logarithm is proved for the estimate of the frequency,
ω
0, of a sinusoidal oscillation when observed subject to stationary noise. The estimate,
ω
, is the location of the maximum of the periodogram from
T observations. The form of the law is unusual since it is
{
T
3
log log T
}
1/2 (
ω−ω
0
whose limit superior is a.s. finite. |
|---|---|
| AbstractList | A form of the law of the iterated logarithm is proved for the estimate of the frequency,[omega]0, of a sinusoidal oscillation when observed subject to stationary noise. The estimate,, is the location of the maximum of the periodogram from T observations. The form of the law is unusual since it is whose limit superior is a.s. finite. A form of the law of the iterated logarithm is proved for the estimate of the frequency, ω 0, of a sinusoidal oscillation when observed subject to stationary noise. The estimate, ω , is the location of the maximum of the periodogram from T observations. The form of the law is unusual since it is { T 3 log log T } 1/2 ( ω−ω 0 whose limit superior is a.s. finite. |
| Author | Mackisack, M. Hannan, E.J. |
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| Keywords | periodogram weak mixing integrated logarithm oscillatory frequency Stochastic convergence Law of iterated logarithm |
| Language | English |
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| References | Gordin (BIB2) 1969; 10 Hannan (BIB4) 1973; 14 Rozanov (BIB6) 1967 Hannan (BIB3) 1973; 10 Zygmund (BIB9) 1968 Walters (BIB8) 1975 Hong-zhi, Zhao-guo, Harman (BIB1) 1983; 13 Stout (BIB7) 1974 Hannan (BIB5) 1978; 10 Gordin (10.1016/0304-4149(86)90117-1_BIB2) 1969; 10 Hannan (10.1016/0304-4149(86)90117-1_BIB5) 1978; 10 Hannan (10.1016/0304-4149(86)90117-1_BIB4) 1973; 14 Rozanov (10.1016/0304-4149(86)90117-1_BIB6) 1967 Walters (10.1016/0304-4149(86)90117-1_BIB8) 1975 Stout (10.1016/0304-4149(86)90117-1_BIB7) 1974 Hong-zhi (10.1016/0304-4149(86)90117-1_BIB1) 1983; 13 Hannan (10.1016/0304-4149(86)90117-1_BIB3) 1973; 10 Zygmund (10.1016/0304-4149(86)90117-1_BIB9) 1968 |
| References_xml | – year: 1967 ident: BIB6 article-title: Stationary Random Processes – year: 1975 ident: BIB8 article-title: Ergodic Theory—Introductory Lectures – volume: 10 start-page: 1174 year: 1969 end-page: 1176 ident: BIB2 article-title: The central limit theorem for stationary processes publication-title: Soviet Math. Dokl. – year: 1974 ident: BIB7 article-title: Almost Sure Convergence – volume: 10 start-page: 510 year: 1973 end-page: 519 ident: BIB3 article-title: The estimation of frequency publication-title: J. Appl. Prob. – volume: 13 start-page: 383 year: 1983 end-page: 400 ident: BIB1 article-title: The maximum of the periodogram publication-title: J. Multivar. Analysis – volume: 14 start-page: 157 year: 1973 end-page: 170 ident: BIB4 article-title: Central limit theorems for time series regression publication-title: Z. Wahrsch. Verw. Geb. – volume: 10 start-page: 740 year: 1978 end-page: 743 ident: BIB5 article-title: Rates of convergence for time series regression publication-title: Adv. Appl. Prob. – year: 1968 ident: BIB9 article-title: Trigometric Series – year: 1974 ident: 10.1016/0304-4149(86)90117-1_BIB7 – year: 1975 ident: 10.1016/0304-4149(86)90117-1_BIB8 – year: 1968 ident: 10.1016/0304-4149(86)90117-1_BIB9 – volume: 13 start-page: 383 year: 1983 ident: 10.1016/0304-4149(86)90117-1_BIB1 article-title: The maximum of the periodogram publication-title: J. Multivar. Analysis doi: 10.1016/0047-259X(83)90017-9 – volume: 14 start-page: 157 year: 1973 ident: 10.1016/0304-4149(86)90117-1_BIB4 article-title: Central limit theorems for time series regression publication-title: Z. Wahrsch. Verw. Geb. doi: 10.1007/BF00533484 – volume: 10 start-page: 1174 year: 1969 ident: 10.1016/0304-4149(86)90117-1_BIB2 article-title: The central limit theorem for stationary processes publication-title: Soviet Math. Dokl. – year: 1967 ident: 10.1016/0304-4149(86)90117-1_BIB6 – volume: 10 start-page: 510 year: 1973 ident: 10.1016/0304-4149(86)90117-1_BIB3 article-title: The estimation of frequency publication-title: J. Appl. Prob. doi: 10.2307/3212772 – volume: 10 start-page: 740 year: 1978 ident: 10.1016/0304-4149(86)90117-1_BIB5 article-title: Rates of convergence for time series regression publication-title: Adv. Appl. Prob. doi: 10.2307/1426656 |
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| Snippet | A form of the law of the iterated logarithm is proved for the estimate of the frequency,
ω
0, of a sinusoidal oscillation when observed subject to stationary... A form of the law of the iterated logarithm is proved for the estimate of the frequency,[omega]0, of a sinusoidal oscillation when observed subject to... |
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| StartPage | 103 |
| SubjectTerms | Exact sciences and technology integrated logarithm integrated logarithm oscillatory frequency periodogram weak mixing Limit theorems Mathematics oscillatory frequency periodogram Probability and statistics Probability theory and stochastic processes Sciences and techniques of general use weak mixing |
| Title | A law of the iterated logarithm for an estimate of frequency |
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