Peridynamics

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it 1972, to obtain a bound between the distribution of a sum of $m$-dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform) metric and hence to prove not only a central limit theorem, but also bounds on the rates of convergence for the given metric.

History

At the end of the 1960s, unsatisfied with the by-then known proofs of a specific central limit theorem, Charles Stein developed a new way of proving the theorem for his statistics lecture. The seminal paper was presented in 1970 at the sixth Berkeley Symposium and published in the corresponding proceedings.

Later, his Ph.D. student Louis Chen Hsiao Yun modified the method so as to obtain approximation results for the Poisson distribution, therefore the method is often referred to as Stein-Chen method. Whereas moderate attention was given to the new method in the 70s, it has undergone major development in the 80s, where many important contributions were made and on which today's view of the method are largely based. Probably the most important contributions are the monograph by Stein (1986), where he presents his view of the method and the concept of auxiliary randomisation, in particular using exchangeable pairs, and the articles by Barbour (1988) and Götze (1991), who introduced the so-called generator interpretation, which made it possible to easily adapt the method to many other probability distributions. An important contribution was also an article by Bolthausen (1984) on a long-standing open problem around the so-called combinatorial central limit theorem, which surely helped the method to become widely known. Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

In the 1990s the method was adapted to a variety of distributions, such as Gaussian processes by Barbour (1990), the binomial distribution by Ehm (1991), Poisson processes by Barbour and Brown (1992), the Gamma distribution by Luk (1994), and many others.

The basic approach

Probability metrics

Stein's method is a way to bound the distance of two probability distributions in a specific probability metric. To be tractable with the method, the metric must be given in the form

$(1.1)d(P,Q)={\sup }_{h\in \mathcal{\mathscr{H}}}|\int hdP-\int hdQ|={\sup }_{h\in \mathcal{\mathscr{H}}}|Eh(W)-Eh(Y)|$

Here, $P$ and $Q$ are probability measures on a measurable space $\mathcal{𝒳}$, $W$ and $Y$ are random variables with distribution $P$ and $Q$ respectively, $E$ is the usual expectation operator and $\mathcal{\mathscr{H}}$ is a set of functions from $\mathcal{𝒳}$ to the real numbers. This set has to be large enough, so that the above definition indeed yields a metric. Important examples are the total variation metric, where we let $\mathcal{\mathscr{H}}$ consist of all the indicator functions of measurable sets, the Kolmogorov (uniform) metric for probability measures on the real numbers, where we consider all the half-line indicator functions, and the Lipschitz (first order Wasserstein; Kantorovich) metric, where the underlying space is itself a metric space and we take the set $\mathcal{\mathscr{H}}$ to be all Lipschitz-continuous functions with Lipschitz-constant 1. However, note that not every metric can be represented in the form (1.1).

In what follows we think of $P$ as a complicated distribution (e.g. a sum of dependent random variables), which we want to approximate by a much simpler and tractable distribution $Q$ (e.g. the standard normal distribution to obtain a central limit theorem).

The Stein operator

We assume now that the distribution $Q$ is a fixed distribution; in what follows we shall in particular consider the case where $Q$ is the standard normal distribution, which serves as a classical example of the application of Stein's method.

First of all, we need an operator $\mathcal{𝒜}$ which acts on functions $f$ from $\mathcal{𝒳}$ to the real numbers, and which 'characterizes' the distribution $Q$ in the sense that the following equivalence holds:

$(2.1)E(\mathcal{𝒜}f)(Y)=0\text{ for all }f⟺Y\text{ has distribution }Q.$

We call such an operator the Stein operator. For the standard normal distribution, Stein's lemma exactly yields such an operator:

$(2.2)E(f^{\prime }(Y)-Yf(Y))=0\text{ for all }f\in C_b^1⟺Y\text{ has standard normal distribution.}$

thus we can take

$(2.3)(\mathcal{𝒜}f)(x)=f^{\prime }(x)-xf(x)$

We note that there are in general infinitely many such operators and it still remains an open question, which one to choose. However, it seems that for many distributions there is a particular good one, like (2.3) for the normal distribution.

There are different ways to find Stein operators. But by far the most important one is via generators. This approach was, as already mentioned, introduced by Barbour and Götze. Assume that $Z=(Z_t{)}_{t\ge 0}$ is a (homogeneous) continuous time Markov process taking values in $\mathcal{𝒳}$. If $Z$ has the stationary distribution $Q$ it is easy to see that, if $\mathcal{𝒜}$ is the generator of $Z$, we have $E(\mathcal{𝒜}f)(Y)=0$ for a large set of functions $f$. Thus, generators are natural candidates for Stein operators and this approach will also help us for later computations.

Setting up the Stein equation

Observe now that saying that $P$ is close to $Q$ with respect to $d$ is equivalent to saying that the difference of expectations in (1.1) is close to 0, and indeed if $P=Q$ it is equal to 0. We hope now that the operator $\mathcal{𝒜}$ exhibits the same behavior: clearly if $P=Q$ we have $E(\mathcal{𝒜}f)(W)=0$ and hopefully if $P\approx Q$ we have $E(\mathcal{𝒜}f)(W)\approx 0$.

To make this statement rigorous we could find a function $f$, such that, for a given function $h$,

$(3.1)E(\mathcal{𝒜}f)(W)=Eh(W)-Eh(Y),$

so that the behavior of the right hand side is reproduced by the operator $\mathcal{𝒜}$ and $f$. However, this equation is too general. We solve instead the more specific equation

$(3.2)(\mathcal{𝒜}f)(x)=h(x)-Eh(Y),\text{for all }x,$

which is called Stein equation. Replacing $x$ by $W$ and taking expectation with respect to $W$, we are back to (3.1), which is what we effectively want. Now all the effort is worth only if the left hand side of (3.1) is easier to bound than the right hand side. This is, surprisingly, often the case.

If $Q$ is the standard normal distribution and we use (2.3), the corresponding Stein equation is

$(3.3)f^{\prime }(x)-xf(x)=h(x)-Eh(Z),\text{for all }x,$

which is just an ordinary differential equation.

Solving the Stein equation

Now, in general, we cannot say much about how the equation (3.2) is to be solved. However, there are important cases, where we can.

Analytic methods. We see from (3.3) that equation (3.2) can in particular be a differential equation (if $Q$ is concentrated on the integers, it will often turn out to be a difference equation). As there are many methods available to treat such equations, we can use them to solve the equation. For example, (3.3) can be easily solved explicitly:

$(4.1)f(x)=e^{x^2/2}\underset{-\mathrm{\infty }}{\overset{x}{\int }}[h(s)-Eh(Y)]e^{-s^2/2}ds.$

Generator method. If $\mathcal{𝒜}$ is the generator of a Markov process $(Z_t{)}_{t\ge 0}$as explained before, we can give a general solution to (3.2):

$(4.2)f(x)=-\underset{0}{\overset{\mathrm{\infty }}{\int }}[E^{x}h(Z_t)-Eh(Y)]dt.$

where $E^x$ denotes expectation with respect to the process $Z$ being started in $x$. However, one still has to prove that the solution (4.2) exists for all desired functions $h\in \mathcal{\mathscr{H}}$.

Properties of the solution to the Stein equation

After showing the existence of a solution to (3.2) we can now try to analyze its properties. Usually, one tries to give bounds on $f$ and its derivatives (which has to be carefully defined if $\mathcal{𝒳}$ is a more complicated space) or differences in terms of $h$ and its derivatives or differences, that is, inequalities of the form

$(5.1)\left|\right|D^{k}f\left|\right|\le C_{k,l}\left|\right|D^{l}h\left|\right|,$

for some specific $k,l=0,1,2,\dots$ (typically $k\ge l$ or $k\ge l-1$, respectively, depending on the form of the Stein operator) and where often $\left|\right|\cdot \left|\right|$ is taken to be the supremum norm. Here, $D^k$ denotes the differential operator, but in discrete settings it usually refers to a difference operator. The constants $C_{k,l}$ may contain the parameters of the distribution $Q$. If there are any, they are often referred to as Stein factors or magic factors.

In the case of (4.1) we can prove for the supremum norm that

$(5.2)\left|\right|f|{|}_{\mathrm{\infty }}\le \min \{\sqrt{\pi /2}\left|\right|h|{|}_{\mathrm{\infty }},2|\left|h^{\prime }\right|{|}_{\mathrm{\infty }}\},|\left|f^{\prime }\right|{|}_{\mathrm{\infty }}\le \min \{2|\left|h\right|{|}_{\mathrm{\infty }},4\left|\right|h^{\prime }|{|}_{\mathrm{\infty }}\},\left|\right|f^{″}|{|}_{\mathrm{\infty }}\le 2|\left|h^{\prime }\right|{|}_{\mathrm{\infty }},$

where the last bound is of course only applicable if $h$ is differentiable (or at least Lipschitz-continuous, which, for example, is not the case if we regard the total variation metric or the Kolmogorov metric!). As the standard normal distribution has no extra parameters, in this specific case, the constants are free of additional parameters.

Note that, up to this point, we did not make use of the random variable $W$. So, the steps up to here in general have to be calculated only once for a specific combination of distribution $Q$, metric $d$ and Stein operator $\mathcal{𝒜}$. However, if we have bounds in the general form (5.1), we usually are able to treat many probability metrics together. Furthermore as there is often a particular 'good' Stein operator for a distribution (e.g., no other operator than (2.3) has been used for the standard normal distribution up to now), one can often just start with the next step below, if bounds of the form (5.1) are already available (which is the case for many distributions).

An abstract approximation theorem

We are now in a position to bound the left hand side of (3.1). As this step heavily depends on the form of the Stein operator, we directly regard the case of the standard normal distribution.

Now, at this point we could directly plug in our random variable $W$ which we want to approximate and try to find upper bounds. However, it is often fruitful to formulate a more general theorem using only abstract properties of $W$. Let us consider here the case of local dependence.

To this end, assume that $W=\sum _{i=1}^n{X}_i$ is a sum of random variables such that the $EW=0$ and variance $VarW=1$. Assume that, for every $i=1,\dots ,n$, there is a set $A_i\subset \{1,2,\dots ,n\}$, such that $X_i$ is independent of all the random variables $X_j$ with $j\in ̸A_i$. We call this set the 'neighborhood' of $X_i$. Likewise let $B_i\subset \{1,2,\dots ,n\}$ be a set such that all $X_j$ with $j\in A_i$ are independent of all $X_k$, $k\in ̸B_i$. We can think of $B_i$ as the neighbors in the neighborhood of $X_i$, a second-order neighborhood, so to speak. For a set $A\subset \{1,2,\dots ,n\}$ define now the sum $X_A:=\sum _{j\in A}{X}_j$.

Using basically only Taylor expansion, it is possible to prove that

$(6.1)|E(f^{\prime }(W)-Wf(W))|\le \left|\right|f^{″}|{|}_{\mathrm{\infty }}\sum _{i=1}^n\left(\frac{1}{2}E\right|X_i{X}_{A_i}^2|+E|X_i{X}_{A_i}{X}_{B_i\setminus A_i}|+E|X_i{X}_{A_i}\left|E\right|X_{B_i}\left|\right)$

Note that, if we follow this line of argument, we can bound (1.1) only for functions where $\left|\right|h^{\prime }|{|}_{\mathrm{\infty }}$ is bounded because of the third inequality of (5.2) (and in fact, if $h$ has discontinuities, so will $f^{″}$). To obtain a bound similar to (6.1) which contains only the expressions $\left|\right|f|{|}_{\mathrm{\infty }}$ and $\left|\right|f^{\prime }|{|}_{\mathrm{\infty }}$, the argument is much more involved and the result is not as simple as (6.1); however, it can be done.

Theorem A. If $W$ is as described above, we have for the Lipschitz metric $d_W$ that

$(6.2)d_W(\mathcal{ℒ}(W),N(0,1))\le 2\sum _{i=1}^n\left(\frac{1}{2}E\right|X_i{X}_{A_i}^2|+E|X_i{X}_{A_i}{X}_{B_i\setminus A_i}|+E|X_i{X}_{A_i}\left|E\right|X_{B_i}\left|\right).$

Proof. Recall that the Lipschitz metric is of the form (1.1) where the functions $h$ are Lipschitz-continuous with Lipschitz-constant 1, thus $\left|\right|h^{\prime }\left|\right|\le 1$. Combining this with (6.1) and the last bound in (5.2) proves the theorem.

Thus, roughly speaking, we have proved that, to calculate the Lipschitz-distance between a $W$ with local dependence structure and a standard normal distribution, we only need to know the third moments of $X_i$ and the size of the neighborhoods $A_i$ and $B_i$.

Application of the theorem

We can treat the case of sums of independent and identically distributed random variables with Theorem A. So assume now that $E{X}_i=0$, $Var{X}_i=1$ and $W=n^{-1/2}\sum X_i$. We can take $A_i=B_i=\{i\}$ and we obtain from Theorem A that

$(7.1)d_W(\mathcal{ℒ}(W),N(0,1))\le \frac{5E|X_1{|}^3}{n^{1/2}}.$

Connections to other methods

• Lindeberg's method. Lindeberg (1922) introduced in a seminal article a method, where the difference in (1.1) is directly bounded. This method usually also heavily relies on Taylor expansion and thus shows some similarities with Stein's method.

• Tikhomirov's method. Clearly the approach via (1.1) and (3.1) does not involve characteristic functions. However, Tikhomirov (1980) presented a proof of a central limit theorem based on characteristic functions and a differential operator similar to (2.3). The basic observation is that the characteristic function $\psi (t)$ of the standard normal distribution satisfies the differential equation ${\psi }^{\prime }(t)+t\psi (t)=0$ for all $t$. Thus, if the characteristic function ${\psi }_W(t)$ of $W$ is such that $\psi {\text{'}}_W(t)+t{\psi }_W(t)\approx 0$ we expect that ${\psi }_W(t)\approx \psi (t)$ and hence that $W$ is close to the normal distribution. Tikhomirov states in his paper that he was inspired by Stein's seminal paper.

Notes

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References

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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
  
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  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
• 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
• 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
• 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
• 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
• 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
• 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

Literature

The following text is advanced, and gives a comprehensive overview of the normal case

• 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

Another advanced book, but having some introductory character, is

• 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

A standard reference is the book by Stein,

• 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

which contains a lot of interesting material, but may be a little hard to understand at first reading.

Despite its age, there are few standard introductory books about Stein's method available. The following recent textbook has a chapter (Chapter 2) devoted to introducing Stein's method:

• 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

Although the book

• 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

is by large parts about Poisson approximation, it contains nevertheless a lot of information about the generator approach, in particular in the context of Poisson process approximation.

The following textbook has a chapter (Chapter 10) devoted to introducing Stein's method of Poisson approximation:

• 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