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| == As Venezuelans look ahead Michael Kors Handbags Sydney == | | The '''Wiener–Khinchin theorem''' (also known as the '''Wiener–Khintchine theorem''' and sometimes as the '''Wiener–Khinchin–Einstein theorem''' or the '''Khinchin–Kolmogorov theorem''') states that the [[autocorrelation]] function of a [[wide-sense-stationary random process]] has a spectral decomposition given by the [[power spectrum]] of that process.<ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | pages = 94–95}}</ref><ref>{{cite book | title = Time Series | author = Norbert Wiener | publisher = M.I.T. Press, Cambridge, Massachusetts | year = 1964 | page = 42}}</ref><ref>Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, ''The New Palgrave: A Dictionary of Economics. Time Series and Statistics'', Macmillan, London, 1990, p. 271.</ref><ref>{{cite book | title = Echo Signal Processing | author = Dennis Ward Ricker | publisher = Springer | year = 2003 | isbn = 1-4020-7395-X | url = http://books.google.com/books?id=NF2Tmty9nugC&pg=PA23&dq=%22power+spectral+density%22+%22energy+spectral+density%22&lr=&as_brr=3&ei=HZMvSPSWFZyStwPWsfyBAw&sig=1ZZcHwxXkErvNXtAHv21ijTXoP8#PPA23,M1 }}</ref><ref>{{cite book | title = Digital and Analog Communications Systems | author = Leon W. Couch II | edition = sixth | publisher = Prentice Hall, New Jersey | year = 2001 | isbn=0-13-522583-3| pages = 406–409}}</ref><ref>{{cite book | title = Wireless Technologies: Circuits, Systems, and Devices | author = Krzysztof Iniewski | publisher = CRC Press | year = 2007 | isbn = 0-8493-7996-2 | url = http://books.google.com/books?id=JJXrpazX9FkC&pg=PA390&dq=Wiener-Khinchin-Einstein&ei=1SxlSPGhB4jgsQPr5b3lDw&sig=ACfU3U2Phnk-zwJi57XrvNmdfosyg55FVA }}</ref><ref>{{cite book | title = Statistical Optics | author = Joseph W. Goodman | publisher = Wiley-Interscience | year = 1985 | isbn=0-471-01502-4}}</ref> |
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| | ==History== |
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| | [[Norbert Wiener]] first published this [[theorem]] in 1930;<ref>{{cite journal|last=Wiener|first=Norbert|title=Generalized Harmonic Analysis|journal=Acta Mathematica|year=1930|volume=55|pages=117–258}}</ref> [[Aleksandr Khinchin]] independently<ref>{{cite book|last=Nahin|first=Paul J.|title=Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills|year=2011|publisher=Princeton University Press|isbn=9780691150376|pages=225|url=http://books.google.com/books?id=GvSg5HQ7WPcC&pg=PA225}}</ref> discovered the result and published it in 1934.<ref>{{cite journal|last=Khintchine|first=A.|title=Korrelationstheorie der stationären stochastischen Prozesse |journal=[[Mathematische Annalen]]|year=1934 |volume=109 |issue=1 |pages=604–615 |doi=10.1007/BF01449156 }}</ref> [[Albert Einstein]] had probably anticipated the idea in a brief two-page memo in 1914.<ref>{{cite book|title=The Legacy of Norbert Wiener: A Centennial Symposium (Proceedings of Symposia in Pure Mathematics)|page=95|first1=David|last1=Jerison|first2=Isadore Manuel|last2=Singer|first3=Daniel W.|last3=Stroock|publisher=American Mathematical Society|year=1997|isbn=0-8218-0415-4}}</ref> |
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| == addresses Tiffany Jewelry Australia == | | ==The case of a continuous time process== |
| | For continuous time, the Wiener—Khinchin theorem <ref>Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, ''The New Palgrave: A Dictionary of Economics. Time Series and Statistics'', Macmillan, London, 1990, p. 271.</ref><ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | pages = 94–95}}</ref> says that if <math> x </math> is a wide-sense stationary process such that its [[autocorrelation function]] (sometimes called [[autocovariance]]) defined in terms of statistical [[expected value]] E, |
| | <math>r_{xx}(\tau) = \operatorname{E}\big[\, x(t)x^*(t-\tau) \, \big] \ </math> |
| | exists and is finite at every lag <math> \tau </math>, then there exists a monotone function <math> F(f) </math> in the frequency domain <math> -\infty < f < \infty </math> such that |
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| | r_{xx} (\tau) = \int_{-\infty}^\infty e^{2\pi i\tau f} dF(f) |
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| | where the integral is a [[Stieltjes integral]]. This is a kind of spectral decomposition of the auto-correlation function. F is called the power spectral distribution function, and is a statistical distribution function. It is sometimes called the integrated spectrum. |
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| | (The asterisk denotes complex conjugate, and of course it can be omitted if the random process is real-valued.) |
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| | But if <math> F(f) </math> is absolutely continuous, for example if the process is purely indeterministic, then one can define the power [[spectral density]] of <math>x(t)\,</math> by taking the derivative of <math> F </math>, putting <math> S_{xx}(f) = F'(f) </math> almost everywhere,<ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | page = 96}}</ref> and the theorem simplifies to |
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| | r_{xx} (\tau) = \int_{-\infty}^\infty S_{xx}(f) e^{2\pi i\tau f} df. |
| | </math> |
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| | If now one assumes that r and S satisfy the necessary conditions for Fourier inversion to be valid, the Wiener—Khinchin theorem takes the simple form of saying that r and S are a Fourier transform pair, and |
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| | ==The case of a discrete time process== |
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| | For the discrete-time case, the power spectral density of the function with discrete values <math>x[n]\,</math> is |
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| | :<math>r_{xx}[k] = \operatorname{E}\big[ \, x[n] x^*[n-k] \, \big] \ </math> |
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| | ==Application== |
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| | The theorem is useful for analyzing [[LTI system theory|linear time-invariant systems]], LTI systems, when the inputs and outputs are not square integrable, so their Fourier transforms do not exist. A corollary is that the Fourier transform of the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of the autocorrelation function of the input of the system times the squared magnitude of the Fourier transform of the system impulse response.<ref> |
| <li>[http://www.harrietoneilllaw.com/node/559959 http://www.harrietoneilllaw.com/node/559959]</li>
| | {{cite book |
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| | | title = Random signals and noise: a mathematical introduction |
| <li>[http://www.haina263.com/home.php?mod=space&uid=73260&do=blog&quickforward=1&id=801800 http://www.haina263.com/home.php?mod=space&uid=73260&do=blog&quickforward=1&id=801800]</li>
| | | author = Shlomo Engelberg |
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| | | publisher = CRC Press |
| <li>[http://www.ltc1688.com/forum.php?mod=viewthread&tid=1477345&fromuid=110316 http://www.ltc1688.com/forum.php?mod=viewthread&tid=1477345&fromuid=110316]</li>
| | | year = 2007 |
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| | | isbn = 978-0-8493-7554-5 |
| <li>[http://www.zjzgol.com/news/html/?71122.html http://www.zjzgol.com/news/html/?71122.html]</li>
| | | page = 130 |
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| | | url = http://books.google.com/books?id=Zl51JGnoww4C&pg=PA130 |
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| | }}</ref> |
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| | This works even when the Fourier transforms of the input and output signals do not exist because these signals are not square integrable, so the system inputs and outputs cannot be directly related by the Fourier transform of the impulse response. |
| </ul>
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| == Where you sell Mbt Stockists Australia ==
| | Since the Fourier transform of the autocorrelation function of a signal is the power spectrum of the signal, this corollary is equivalent to saying that the power spectrum of the output is equal to the power spectrum of the input times the power [[transfer function]]. |
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| | In many textbooks and in much of the technical literature it is tacitly assumed that Fourier inversion of the [[autocorrelation]] function and the power spectral density is valid, and the Wiener—Khinchin theorem is stated, very simply, as if it said that the Fourier transform of the autocorrelation function was equal to the power [[spectral density]], ignoring all questions of convergence.<ref>{{cite book | title = The Analysis of Time Series—An Introduction | author = C. Chatfield | edition = fourth | publisher = Chapman and Hall, London | year = 1989 | isbn=0-412-31820-2 | page = 98}}</ref> (Einstein is an example.) |
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| | But the theorem (as stated here), was applied by [[Norbert Wiener]] and [[Aleksandr Khinchin]] to the sample functions (signals) of [[wide-sense-stationary random process]]es, signals whose Fourier transforms do not exist. |
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| | The whole point of Wiener's contribution was to make sense of the spectral decomposition of the autocorrelation function of a sample function of a [[wide-sense-stationary random process]] even when the integrals for the Fourier transform and Fourier inversion do not make sense. |
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| | Some authors refer to R as the autocovariance function. They then proceed to normalise it, by dividing by R(0), to obtain what they refer to as the autocorrelation function. |
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| | ==References== |
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| | {{Reflist}} |
| <li>[http://www.observatoiredesreligions.fr/spip.php?article11 http://www.observatoiredesreligions.fr/spip.php?article11]</li>
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| <li>[http://www.observatoiredesreligions.fr/spip.php?article11 http://www.observatoiredesreligions.fr/spip.php?article11]</li>
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| <li>[http://general.assembly.codesria.org/spip.php?article87&lang=pt/ http://general.assembly.codesria.org/spip.php?article87&lang=pt/]</li>
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| == Dunoyer P Nike Air Force Australia == | | ==Further reading== |
| | *{{cite book |last=Brockwell |first=Peter A. |last2=Davis |first2=Richard J. |title=Introduction to Times Series and Forecasting |edition=Second |publisher=Springer-Verlag |location=New York |year=2002 |isbn=038721657X }} |
| | *{{cite book |last=Chatfield |first=C. |title=The Analysis of Time Series—An Introduction |edition=Fourth |publisher=Chapman and Hall |location=London |year=1989 |isbn=0412318202 }} |
| | *{{cite book |last=Fuller |first=Wayne |title=Introduction to Statistical Time Series |series=Wiley Series in Probability and Statistics |edition=Second |publisher=Wiley |location=New York |year=1996 |isbn=0471552399 }} |
| | *{{cite paper |last=Wiener |first=Norbert |title=Extrapolation, Interpolation, and Smoothing of Stationary Time Series |publisher=Technology Press and Johns Hopkins Univ. Press |location=Cambridge, Massachusetts |year=1949 }} (a classified document written for the Dept. of War in 1943). |
| | *{{cite book |last=Yaglom |first=A. M. |title=An Introduction to the Theory of Stationary Random Functions |publisher=Prentice-Hall |location=Englewood Cliffs, New Jersey |year=1962 }} |
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The Wiener–Khinchin theorem (also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem) states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process.[1][2][3][4][5][6][7]
History
Norbert Wiener first published this theorem in 1930;[8] Aleksandr Khinchin independently[9] discovered the result and published it in 1934.[10] Albert Einstein had probably anticipated the idea in a brief two-page memo in 1914.[11]
The case of a continuous time process
For continuous time, the Wiener—Khinchin theorem [12][13] says that if is a wide-sense stationary process such that its autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value E,
exists and is finite at every lag , then there exists a monotone function in the frequency domain such that
where the integral is a Stieltjes integral. This is a kind of spectral decomposition of the auto-correlation function. F is called the power spectral distribution function, and is a statistical distribution function. It is sometimes called the integrated spectrum.
(The asterisk denotes complex conjugate, and of course it can be omitted if the random process is real-valued.)
Note that the Fourier transform of does not exist in general, because stationary random functions are not generally either square integrable or absolutely integrable. Nor is assumed to be absolutely integrable, so it need not have a Fourier transform, either.
But if is absolutely continuous, for example if the process is purely indeterministic, then one can define the power spectral density of by taking the derivative of , putting almost everywhere,[14] and the theorem simplifies to
If now one assumes that r and S satisfy the necessary conditions for Fourier inversion to be valid, the Wiener—Khinchin theorem takes the simple form of saying that r and S are a Fourier transform pair, and
The case of a discrete time process
For the discrete-time case, the power spectral density of the function with discrete values is
- ,
where
is the discrete autocorrelation function of , provided this is absolutely integrable. Being a sampled and discrete-time sequence, the spectral density is periodic in the frequency domain.
Application
The theorem is useful for analyzing linear time-invariant systems, LTI systems, when the inputs and outputs are not square integrable, so their Fourier transforms do not exist. A corollary is that the Fourier transform of the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of the autocorrelation function of the input of the system times the squared magnitude of the Fourier transform of the system impulse response.[15]
This works even when the Fourier transforms of the input and output signals do not exist because these signals are not square integrable, so the system inputs and outputs cannot be directly related by the Fourier transform of the impulse response.
Since the Fourier transform of the autocorrelation function of a signal is the power spectrum of the signal, this corollary is equivalent to saying that the power spectrum of the output is equal to the power spectrum of the input times the power transfer function.
This corollary is used in the parametric method for power spectrum estimation.
Discrepancies in terminology
In many textbooks and in much of the technical literature it is tacitly assumed that Fourier inversion of the autocorrelation function and the power spectral density is valid, and the Wiener—Khinchin theorem is stated, very simply, as if it said that the Fourier transform of the autocorrelation function was equal to the power spectral density, ignoring all questions of convergence.[16] (Einstein is an example.)
But the theorem (as stated here), was applied by Norbert Wiener and Aleksandr Khinchin to the sample functions (signals) of wide-sense-stationary random processes, signals whose Fourier transforms do not exist.
The whole point of Wiener's contribution was to make sense of the spectral decomposition of the autocorrelation function of a sample function of a wide-sense-stationary random process even when the integrals for the Fourier transform and Fourier inversion do not make sense.
Some authors refer to R as the autocovariance function. They then proceed to normalise it, by dividing by R(0), to obtain what they refer to as the autocorrelation function.
References
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Further reading
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- Template:Cite paper (a classified document written for the Dept. of War in 1943).
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, The New Palgrave: A Dictionary of Economics. Time Series and Statistics, Macmillan, London, 1990, p. 271.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ Hannan, E.J., "Stationary Time Series", in: John Eatwell, Murray Milgate, and Peter Newman, editors, The New Palgrave: A Dictionary of Economics. Time Series and Statistics, Macmillan, London, 1990, p. 271.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534