Monetary policy reaction function

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In the theory of probability, the Glivenko–Cantelli theorem, named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows.[1] The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets.[2] The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.

Assume that are independent and identically-distributed random variables in with common cumulative distribution function . The empirical distribution function for is defined by

where is the indicator function of the set . For every (fixed) , is a sequence of random variables which converge to almost surely by the strong law of large numbers, that is, converges to pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of to .

Theorem

almost surely.[3]

This theorem originates with Valery Glivenko,[4] and Francesco Cantelli,[5] in 1933.

Remarks

Empirical measures

One can generalize the empirical distribution function by replacing the set by an arbitrary set C from a class of sets to obtain an empirical measure indexed by sets

Further generalization is the map induced by on measurable real-valued functions f, which is given by

Then it becomes an important property of these classes that the strong law of large numbers holds uniformly on or .

Glivenko–Cantelli class

Consider a set with a sigma algebra of Borel subsets A and a probability measure P. For a class of subsets,

and a class of functions

define random variables

where is the empirical measure, is the corresponding map, and

, assuming that it exists.

Definitions

  • A class is called a Glivenko–Cantelli class (or GC class) with respect to a probability measure P if any of the following equivalent statements is true.
1. almost surely as .
2. in probability as .
3. , as (convergence in mean).
The Glivenko–Cantelli classes of functions are defined similarly.
  • A class is called a universal Glivenko–Cantelli class if it is a GC class with respect to any probability measure P on (S,A).
  • A class is called uniformly Glivenko–Cantelli if the convergence occurs uniformly over all probability measures P on (S,A):

Theorem (Vapnik and Chervonenkis, 1968)[7]

A class of sets is uniformly GC if and only if it is a Vapnik–Chervonenkis class.

Examples

, that is is uniformly Glivenko–Cantelli class.

See also

Notes

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References

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Further reading

  • Dudley, R. M. (1999). Uniform Central Limit Theorems, Cambridge University Press. ISBN 0-521-46102-2.
  • Shorack, G.R., Wellner J.A. (1986) Empirical Processes with Applications to Statistics, Wiley. ISBN 0-471-86725-X.
  • van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes, Springer. ISBN 0-387-94640-3.
  • Aad W. van der Vaart, Jon A. Wellner (1996) Glivenko-Cantelli Theorems, Springer.
  • Aad W. van der Vaart, Jon A. Wellner (2000) Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes, Springer
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  4. Glivenko, V. (1933). Sulla determinazione empirica della legge di probabilita. Giorn. Ist. Ital. Attuari 4, 92-99.
  5. Cantelli, F. P. (1933). Sulla determinazione empirica delle leggi di probabilita. Giorn. Ist. Ital. Attuari 4, 221-424.
  6. 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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    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