Central limit theorem

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https://en.wikipedia.org/wiki/Central_limit_theorem 

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory.

If ${\textstyle X_{1},X_{2},\dots ,X_{n},\dots }$ are random samples drawn from a population with overall mean ${\textstyle \mu }$ and finite variance ${\textstyle \sigma ^{2}}$, and if ${\textstyle {\bar {X}}_{n}}$ is the sample mean of the first ${\textstyle n}$ samples, then the limiting form of the distribution, ${\textstyle Z=\lim _{n\to \infty }{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma _{\bar {X}}}}\right)}}$, with ${\displaystyle \sigma _{\bar {X}}=\sigma /{\sqrt {n}}}$, is a standard normal distribution.

For example, suppose that a sample is obtained containing many observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the probability distribution of the average will closely approximate a normal distribution. A simple example of this is that if one flips a coin many times, the probability of getting a given number of heads will approach a normal distribution, with the mean equal to half the total number of flips. At the limit of an infinite number of flips, it will equal a normal distribution.

The central limit theorem has several variants. In its common form, the random variables must be independent and identically distributed (i.i.d.). In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, if they comply with certain conditions.

The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.

Independent sequences

A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations; see Illustration of the central limit theorem for further details.

 

Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.

Classical CLT

Let ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ be a sequence of random samples — that is, a sequence of i.i.d. random variables drawn from a distribution of expected value given by ${\textstyle \mu }$ and finite variance given by ${\textstyle \sigma ^{2}}$. Suppose we are interested in the sample average

$${\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}}$$

of the first ${\textstyle n}$ samples.

By the law of large numbers, the sample averages converge almost surely (and therefore also converge in probability) to the expected value ${\textstyle \mu }$ as ${\textstyle n\to \infty }$.

The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number ${\textstyle \mu }$ during this convergence. More precisely, it states that as ${\textstyle n}$ gets larger, the distribution of the difference between the sample average ${\textstyle {\bar {X}}_{n}}$ and its limit ${\textstyle \mu }$, when multiplied by the factor ${\textstyle {\sqrt {n}}}$ (that is ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$) approximates the normal distribution with mean 0 and variance ${\textstyle \sigma ^{2}}$. For large enough n, the distribution of ${\textstyle {\bar {X}}_{n}}$ is close to the normal distribution with mean ${\textstyle \mu }$ and variance ${\textstyle \sigma ^{2}/n}$.

The usefulness of the theorem is that the distribution of ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ approaches normality regardless of the shape of the distribution of the individual ${\textstyle X_{i}}$. Formally, the theorem can be stated as follows:

Lindeberg–Lévy CLT — Suppose ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ is a sequence of i.i.d. random variables with ${\textstyle \mathbb {E} [X_{i}]=\mu }$ and ${\textstyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty }$. Then as ${\textstyle n}$ approaches infinity, the random variables ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ converge in distribution to a normal ${\textstyle {\mathcal {N}}(0,\sigma ^{2})}$:

$${\displaystyle {\sqrt {n}}\left({\bar {X}}_{n}-\mu \right)\ \xrightarrow {d} \ {\mathcal {N}}\left(0,\sigma ^{2}\right).}$$

In the case ${\textstyle \sigma >0}$, convergence in distribution means that the cumulative distribution functions of ${\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}$ converge pointwise to the cdf of the ${\textstyle {\mathcal {N}}(0,\sigma ^{2})}$ distribution: for every real number ${\textstyle z}$,

$${\displaystyle \lim _{n\to \infty }\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]=\lim _{n\to \infty }\mathbb {P} \left[{\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq {\frac {z}{\sigma }}\right]=\Phi \left({\frac {z}{\sigma }}\right),}$$

where ${\textstyle \Phi (z)}$ is the standard normal cdf evaluated at ${\textstyle z}$. The convergence is uniform in ${\textstyle z}$ in the sense that

$${\displaystyle \lim _{n\to \infty }\;\sup _{z\in \mathbb {R} }\;\left|\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]-\Phi \left({\frac {z}{\sigma }}\right)\right|=0~,}$$

where ${\textstyle \sup }$ denotes the least upper bound (or supremum) of the set.

Lyapunov CLT

The theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the central limit theorem the random variables ${\textstyle X_{i}}$ have to be independent, but not necessarily identically distributed. The theorem also requires that random variables ${\textstyle \left|X_{i}\right|}$ have moments of some order ${\textstyle (2+\delta )}$, and that the rate of growth of these moments is limited by the Lyapunov condition given below.

Lyapunov CLT — Suppose ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ is a sequence of independent random variables, each with finite expected value ${\textstyle \mu _{i}}$ and variance ${\textstyle \sigma _{i}^{2}}$. Define

$${\displaystyle s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}.}$$

If for some ${\textstyle \delta >0}$, Lyapunov’s condition

$${\displaystyle \lim _{n\to \infty }\;{\frac {1}{s_{n}^{2+\delta }}}\,\sum _{i=1}^{n}\mathbb {E} \left[\left|X_{i}-\mu _{i}\right|^{2+\delta }\right]=0}$$

is satisfied, then a sum of ${\textstyle {\frac {X_{i}-\mu _{i}}{s_{n}}}}$ converges in distribution to a standard normal random variable, as ${\textstyle n}$ goes to infinity:

$${\displaystyle {\frac {1}{s_{n}}}\,\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)\ \xrightarrow {d} \ {\mathcal {N}}(0,1).}$$

In practice it is usually easiest to check Lyapunov's condition for ${\textstyle \delta =1}$.

If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.

Lindeberg CLT

Main article: Lindeberg's condition

In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920).

Suppose that for every ${\textstyle \varepsilon >0}$

$${\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\mathbb {E} \left[(X_{i}-\mu _{i})^{2}\cdot \mathbf {1} _{\left\{X_{i}:\left|X_{i}-\mu _{i}\right|>\varepsilon s_{n}\right\}}\right]=0}$$

where ${\textstyle \mathbf {1} _{\{\ldots \}}}$ is the indicator function. Then the distribution of the standardized sums

$${\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}$$

converges towards the standard normal distribution ${\textstyle {\mathcal {N}}(0,1)}$.

Multidimensional CLT

Proofs that use characteristic functions can be extended to cases where each individual ${\textstyle \mathbf {X} _{i}}$ is a random vector in ${\textstyle \mathbb {R} ^{k}}$, with mean vector ${\textstyle {\boldsymbol {\mu }}=\mathbb {E} [\mathbf {X} _{i}]}$ and covariance matrix ${\textstyle \mathbf {\Sigma } }$ (among the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done component-wise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.

Let

$${\displaystyle \mathbf {X} _{i}={\begin{bmatrix}X_{i(1)}\\\vdots \\X_{i(k)}\end{bmatrix}}}$$

be the k-vector. The bold in ${\textstyle \mathbf {X} _{i}}$ means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be

$${\displaystyle {\begin{bmatrix}X_{1(1)}\\\vdots \\X_{1(k)}\end{bmatrix}}+{\begin{bmatrix}X_{2(1)}\\\vdots \\X_{2(k)}\end{bmatrix}}+\cdots +{\begin{bmatrix}X_{n(1)}\\\vdots \\X_{n(k)}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}\left[X_{i(1)}\right]\\\vdots \\\sum _{i=1}^{n}\left[X_{i(k)}\right]\end{bmatrix}}=\sum _{i=1}^{n}\mathbf {X} _{i}}$$

and the average is

$${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}\mathbf {X} _{i}={\frac {1}{n}}{\begin{bmatrix}\sum _{i=1}^{n}X_{i(1)}\\\vdots \\\sum _{i=1}^{n}X_{i(k)}\end{bmatrix}}={\begin{bmatrix}{\bar {X}}_{i(1)}\\\vdots \\{\bar {X}}_{i(k)}\end{bmatrix}}=\mathbf {{\bar {X}}_{n}} }$$

and therefore

$${\displaystyle {\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}\left[\mathbf {X} _{i}-\mathbb {E} \left(X_{i}\right)\right]={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}(\mathbf {X} _{i}-{\boldsymbol {\mu }})={\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)~.}$$

The multivariate central limit theorem states that

$${\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\,\xrightarrow {D} \ {\mathcal {N}}_{k}(0,{\boldsymbol {\Sigma }})}$$

where the covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$ is equal to

$${\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\operatorname {Var} \left(X_{1(1)}\right)}&\operatorname {Cov} \left(X_{1(1)},X_{1(2)}\right)&\operatorname {Cov} \left(X_{1(1)},X_{1(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1(1)},X_{1(k)}\right)\\\operatorname {Cov} \left(X_{1(2)},X_{1(1)}\right)&\operatorname {Var} \left(X_{1(2)}\right)&\operatorname {Cov} \left(X_{1(2)},X_{1(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1(2)},X_{1(k)}\right)\\\operatorname {Cov} \left(X_{1(3)},X_{1(1)}\right)&\operatorname {Cov} \left(X_{1(3)},X_{1(2)}\right)&\operatorname {Var} \left(X_{1(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1(3)},X_{1(k)}\right)\\\vdots &\vdots &\vdots &\ddots &\vdots \\\operatorname {Cov} \left(X_{1(k)},X_{1(1)}\right)&\operatorname {Cov} \left(X_{1(k)},X_{1(2)}\right)&\operatorname {Cov} \left(X_{1(k)},X_{1(3)}\right)&\cdots &\operatorname {Var} \left(X_{1(k)}\right)\\\end{bmatrix}}~.}$$

The rate of convergence is given by the following Berry–Esseen type result:

Theorem — Let ${\displaystyle X_{1},\dots ,X_{n},\dots }$ be independent ${\displaystyle \mathbb {R} ^{d}}$-valued random vectors, each having mean zero. Write ${\displaystyle S=\sum _{i=1}^{n}X_{i}}$ and assume ${\displaystyle \Sigma =\operatorname {Cov} [S]}$ is invertible. Let ${\displaystyle Z\sim {\mathcal {N}}(0,\Sigma )}$ be a ${\displaystyle d}$-dimensional Gaussian with the same mean and same covariance matrix as ${\displaystyle S}$. Then for all convex sets ${\displaystyle U\subseteq \mathbb {R} ^{d}}$,

$${\displaystyle \left|\mathbb {P} [S\in U]-\mathbb {P} [Z\in U]\right|\leq C\,d^{1/4}\gamma ~,}$$

where ${\displaystyle C}$ is a universal constant, ${\displaystyle \gamma =\sum _{i=1}^{n}\mathbb {E} \left[\left\|\Sigma ^{-1/2}X_{i}\right\|_{2}^{3}\right]}$, and ${\displaystyle \|\cdot \|_{2}}$ denotes the Euclidean norm on ${\displaystyle \mathbb {R} ^{d}}$.

It is unknown whether the factor ${\textstyle d^{1/4}}$ is necessary.

Generalized theorem

Main article: Stable distribution § A generalized central limit theorem

The central limit theorem states that the sum of a number of independent and identically distributed random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with a power-law tail (Paretian tail) distributions decreasing as ${\textstyle {|x|}^{-\alpha -1}}$ where ${\textstyle 0<\alpha <2}$ (and therefore having infinite variance) will tend to a stable distribution ${\textstyle f(x;\alpha ,0,c,0)}$ as the number of summands grows. If ${\textstyle \alpha >2}$ then the sum converges to a stable distribution with stability parameter equal to 2, i.e. a Gaussian distribution.

Dependent processes

CLT under weak dependence

A useful generalization of a sequence of independent, identically distributed random variables is a mixing random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially strong mixing (also called α-mixing) defined by ${\textstyle \alpha (n)\to 0}$ where ${\textstyle \alpha (n)}$ is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is:

Theorem — Suppose that ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ is stationary and ${\displaystyle \alpha }$-mixing with ${\textstyle \alpha _{n}=O\left(n^{-5}\right)}$ and that ${\textstyle \mathbb {E} [X_{n}]=0}$ and ${\textstyle \mathbb {E} [{X_{n}}^{12}]<\infty }$. Denote ${\textstyle S_{n}=X_{1}+\cdots +X_{n}}$, then the limit

$${\displaystyle \sigma ^{2}=\lim _{n\rightarrow \infty }{\frac {\mathbb {E} \left(S_{n}^{2}\right)}{n}}}$$

exists, and if ${\textstyle \sigma \neq 0}$ then ${\textstyle {\frac {S_{n}}{\sigma {\sqrt {n}}}}}$ converges in distribution to ${\textstyle {\mathcal {N}}(0,1)}$.

In fact,

$${\displaystyle \sigma ^{2}=\mathbb {E} \left(X_{1}^{2}\right)+2\sum _{k=1}^{\infty }\mathbb {E} \left(X_{1}X_{1+k}\right),}$$

where the series converges absolutely.

The assumption ${\textstyle \sigma \neq 0}$ cannot be omitted, since the asymptotic normality fails for ${\textstyle X_{n}=Y_{n}-Y_{n-1}}$ where ${\textstyle Y_{n}}$ are another stationary sequence.

There is a stronger version of the theorem: the assumption ${\textstyle \mathbb {E} \left[{X_{n}}^{12}\right]<\infty }$ is replaced with ${\textstyle \mathbb {E} \left[{\left|X_{n}\right|}^{2+\delta }\right]<\infty }$, and the assumption ${\textstyle \alpha _{n}=O\left(n^{-5}\right)}$ is replaced with

$${\displaystyle \sum _{n}\alpha _{n}^{\frac {\delta }{2(2+\delta )}}<\infty .}$$

Existence of such ${\textstyle \delta >0}$ ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2007).

Martingale difference CLT

Main article: Martingale central limit theorem

Theorem — Let a martingale ${\textstyle M_{n}}$ satisfy

• ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\mathbb {E} \left[\left(M_{k}-M_{k-1}\right)^{2}|M_{1},\dots ,M_{k-1}\right]\to 1}$ in probability as n → ∞,
• for every ε > 0, ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\mathbb {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mathbf {1} \left[|M_{k}-M_{k-1}|>\varepsilon {\sqrt {n}}\right]\right]}\to 0}$ as n → ∞,

then ${\textstyle {\frac {M_{n}}{\sqrt {n}}}}$ converges in distribution to ${\textstyle N(0,1)}$ as ${\textstyle n\to \infty }$.

Remarks

Proof of classical CLT

The central limit theorem has a proof using characteristic functions. It is similar to the proof of the (weak) law of large numbers.

Assume ${\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ are independent and identically distributed random variables, each with mean ${\textstyle \mu }$ and finite variance ${\textstyle \sigma ^{2}}$. The sum ${\textstyle X_{1}+\cdots +X_{n}}$ has mean ${\textstyle n\mu }$ and variance ${\textstyle n\sigma ^{2}}$. Consider the random variable

$${\displaystyle Z_{n}={\frac {X_{1}+\cdots +X_{n}-n\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {X_{i}-\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {1}{\sqrt {n}}}Y_{i},}$$

where in the last step we defined the new random variables ${\textstyle Y_{i}={\frac {X_{i}-\mu }{\sigma }}}$, each with zero mean and unit variance (${\textstyle \operatorname {var} (Y)=1}$). The characteristic function of ${\textstyle Z_{n}}$ is given by

$${\displaystyle \varphi _{Z_{n}}\!(t)=\varphi _{\sum _{i=1}^{n}{{\frac {1}{\sqrt {n}}}Y_{i}}}\!(t)\ =\ \varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\varphi _{Y_{2}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\cdots \varphi _{Y_{n}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\ =\ \left[\varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\right]^{n},}$$

where in the last step we used the fact that all of the ${\textstyle Y_{i}}$ are identically distributed. The characteristic function of ${\textstyle Y_{1}}$ is, by Taylor's theorem,

$${\displaystyle \varphi _{Y_{1}}\!\left({\frac {t}{\sqrt {n}}}\right)=1-{\frac {t^{2}}{2n}}+o\!\left({\frac {t^{2}}{n}}\right),\quad \left({\frac {t}{\sqrt {n}}}\right)\to 0}$$

where ${\textstyle o(t^{2}/n)}$ is "little o notation" for some function of ${\textstyle t}$ that goes to zero more rapidly than ${\textstyle t^{2}/n}$. By the limit of the exponential function (${\textstyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}$), the characteristic function of ${\displaystyle Z_{n}}$ equals

$${\displaystyle \varphi _{Z_{n}}(t)=\left(1-{\frac {t^{2}}{2n}}+o\left({\frac {t^{2}}{n}}\right)\right)^{n}\rightarrow e^{-{\frac {1}{2}}t^{2}},\quad n\to \infty .}$$

All of the higher order terms vanish in the limit ${\textstyle n\to \infty }$. The right hand side equals the characteristic function of a standard normal distribution ${\textstyle N(0,1)}$, which implies through Lévy's continuity theorem that the distribution of ${\textstyle Z_{n}}$ will approach ${\textstyle N(0,1)}$ as ${\textstyle n\to \infty }$. Therefore, the sample average

$${\displaystyle {\bar {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}$$

is such that

$${\displaystyle {\frac {\sqrt {n}}{\sigma }}({\bar {X}}_{n}-\mu )}$$

converges to the normal distribution ${\textstyle N(0,1)}$, from which the central limit theorem follows.

Convergence to the limit

The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.

The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment ${\textstyle \operatorname {E} \left[(X_{1}-\mu )^{3}\right]}$ exists and is finite, then the speed of convergence is at least on the order of ${\textstyle 1/{\sqrt {n}}}$ (see Berry–Esseen theorem). Stein's method can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.

The convergence to the normal distribution is monotonic, in the sense that the entropy of ${\textstyle Z_{n}}$ increases monotonically to that of the normal distribution.

The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realizations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity, this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.

Relation to the law of large numbers

The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of S_n as n approaches infinity?" In mathematical analysis, asymptotic series are one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of ${\textstyle f(n)}$:

$${\displaystyle f(n)=a_{1}\varphi _{1}(n)+a_{2}\varphi _{2}(n)+O{\big (}\varphi _{3}(n){\big )}\qquad (n\to \infty ).}$$

Dividing both parts by φ₁(n) and taking the limit will produce a₁, the coefficient of the highest-order term in the expansion, which represents the rate at which f(n) changes in its leading term.

$${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{\varphi _{1}(n)}}=a_{1}.}$$

Informally, one can say: "f(n) grows approximately as a₁φ₁(n)". Taking the difference between f(n) and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about f(n):

$${\displaystyle \lim _{n\to \infty }{\frac {f(n)-a_{1}\varphi _{1}(n)}{\varphi _{2}(n)}}=a_{2}.}$$

Here one can say that the difference between the function and its approximation grows approximately as a₂φ₂(n). The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.

Informally, something along these lines happens when the sum, S_n, of independent identically distributed random variables, X₁, ..., X_n, is studied in classical probability theory. If each X_i has finite mean μ, then by the law of large numbers, S_n/n → μ. If in addition each X_i has finite variance σ², then by the central limit theorem,

$${\displaystyle {\frac {S_{n}-n\mu }{\sqrt {n}}}\to \xi ,}$$

where ξ is distributed as N(0,σ²). This provides values of the first two constants in the informal expansion

$${\displaystyle S_{n}\approx \mu n+\xi {\sqrt {n}}.}$$

In the case where the X_i do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:

$${\displaystyle {\frac {S_{n}-a_{n}}{b_{n}}}\rightarrow \Xi ,}$$

or informally

$${\displaystyle S_{n}\approx a_{n}+\Xi b_{n}.}$$

Distributions Ξ which can arise in this way are called stable. Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined. The scaling factor b_n may be proportional to n^c, for any c ≥ 1/2; it may also be multiplied by a slowly varying function of n.

The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function √n log log n, intermediate in size between n of the law of large numbers and √n of the central limit theorem, provides a non-trivial limiting behavior.

Alternative statements of the theorem

Density functions

The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov for a particular local limit theorem for sums of independent and identically distributed random variables.

Characteristic functions

Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.

An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.

Calculating the variance

Let S_n be the sum of n random variables. Many central limit theorems provide conditions such that S_n/√Var(S_n) converges in distribution to N(0,1) (the normal distribution with mean 0, variance 1) as n → ∞. In some cases, it is possible to find a constant σ² and function f(n) such that S_n/(σ√n⋅f(n)) converges in distribution to N(0,1) as n→ ∞.

Lemma — Suppose ${\displaystyle X_{1},X_{2},\dots }$ is a sequence of real-valued and strictly stationary random variables with ${\displaystyle \mathbb {E} (X_{i})=0}$ for all ${\displaystyle i}$, ${\displaystyle g:[0,1]\to \mathbb {R} }$, and ${\displaystyle S_{n}=\sum _{i=1}^{n}g\left({\tfrac {i}{n}}\right)X_{i}}$. Construct

$${\displaystyle \sigma ^{2}=\mathbb {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\mathbb {E} (X_{1}X_{1+i})}$$

1. If ${\displaystyle \sum _{i=1}^{\infty }\mathbb {E} (X_{1}X_{1+i})}$ is absolutely convergent, ${\displaystyle \left|\int _{0}^{1}g(x)g'(x)\,dx\right|<\infty }$, and ${\displaystyle 0<\int _{0}^{1}(g(x))^{2}dx<\infty }$ then ${\displaystyle \mathrm {Var} (S_{n})/(n\gamma _{n})\to \sigma ^{2}}$ as ${\displaystyle n\to \infty }$ where ${\displaystyle \gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}\left(g\left({\tfrac {i}{n}}\right)\right)^{2}}$.
2. If in addition ${\displaystyle \sigma >0}$ and ${\displaystyle S_{n}/{\sqrt {\mathrm {Var} (S_{n})}}}$ converges in distribution to ${\displaystyle {\mathcal {N}}(0,1)}$ as ${\displaystyle n\to \infty }$ then ${\displaystyle S_{n}/(\sigma {\sqrt {n\gamma _{n}}})}$ also converges in distribution to ${\displaystyle {\mathcal {N}}(0,1)}$ as ${\displaystyle n\to \infty }$.

Extensions

Products of positive random variables

The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different random factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called Gibrat's law.

Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.

Beyond the classical framework

Asymptotic normality, that is, convergence to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.

Convex body

Theorem — There exists a sequence ε_n ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X₁, ..., X_n have a log-concave joint density f such that f(x₁, ..., x_n) = f(|x₁|, ..., |x_n|) for all x₁, ..., x_n, and E(X²
_k) = 1 for all k = 1, ..., n. Then the distribution of

$${\displaystyle {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}}$$

is ε_n-close to N(0,1) in the total variation distance.

These two ε_n-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Another example: f(x₁, ..., x_n) = const · exp(−(|x₁|^α + ⋯ + |x_n|^α)^β) where α > 1 and αβ > 1. If β = 1 then f(x₁, ..., x_n) factorizes into const · exp (−|x₁|^α) … exp(−|x_n|^α), which means X₁, ..., X_n are independent. In general, however, they are dependent.

The condition f(x₁, ..., x_n) = f(|x₁|, ..., |x_n|) ensures that X₁, ..., X_n are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the way, pairwise independence cannot replace independence in the classical central limit theorem.

Here is a Berry–Esseen type result.

Theorem — Let X₁, ..., X_n satisfy the assumptions of the previous theorem, then 

$${\displaystyle \left|\mathbb {P} \left(a\leq {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq {\frac {C}{n}}}$$

for all a < b; here C is a universal (absolute) constant. Moreover, for every c₁, ..., c_n ∈ R such that c²
₁ + ⋯ + c²
_n = 1,

$${\displaystyle \left|\mathbb {P} \left(a\leq c_{1}X_{1}+\cdots +c_{n}X_{n}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq C\left(c_{1}^{4}+\dots +c_{n}^{4}\right).}$$

The distribution of X₁ + ⋯ + X_n/√n need not be approximately normal (in fact, it can be uniform). However, the distribution of c₁X₁ + ⋯ + c_nX_n is close to N(0,1) (in the total variation distance) for most vectors (c₁, ..., c_n) according to the uniform distribution on the sphere c²
₁ + ⋯ + c²
_n = 1.

Lacunary trigonometric series

Theorem (Salem–Zygmund) — Let U be a random variable distributed uniformly on (0,2π), and X_k = r_k cos(n_kU + a_k), where

• n_k satisfy the lacunarity condition: there exists q > 1 such that n_{k + 1} ≥ qn_k for all k,
• r_k are such that
  $${\displaystyle r_{1}^{2}+r_{2}^{2}+\cdots =\infty \quad {\text{ and }}\quad {\frac {r_{k}^{2}}{r_{1}^{2}+\cdots +r_{k}^{2}}}\to 0,}$$

  
• 0 ≤ a_k < 2π.

Then

$${\displaystyle {\frac {X_{1}+\cdots +X_{k}}{\sqrt {r_{1}^{2}+\cdots +r_{k}^{2}}}}}$$

converges in distribution to N(0, 1/2).

Gaussian polytopes

Theorem — Let A₁, ..., A_n be independent random points on the plane R² each having the two-dimensional standard normal distribution. Let K_n be the convex hull of these points, and X_n the area of K_n Then

$${\displaystyle {\frac {X_{n}-\mathbb {E} (X_{n})}{\sqrt {\operatorname {Var} (X_{n})}}}}$$

converges in distribution to N(0,1) as n tends to infinity.

The same also holds in all dimensions greater than 2.

The polytope K_n is called a Gaussian random polytope.

A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.

Linear functions of orthogonal matrices

A linear function of a matrix M is a linear combination of its elements (with given coefficients), M ↦ tr(AM) where A is the matrix of the coefficients; see Trace (linear algebra)#Inner product.

A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group O(n,R); see Rotation matrix#Uniform random rotation matrices.

Theorem — Let M be a random orthogonal n × n matrix distributed uniformly, and A a fixed n × n matrix such that tr(AA*) = n, and let X = tr(AM). Then the distribution of X is close to N(0,1) in the total variation metric up to 2√3/n − 1.

Subsequences

Theorem — Let random variables X₁, X₂, ... ∈ L₂(Ω) be such that X_n → 0 weakly in L₂(Ω) and X
_n → 1 weakly in L₁(Ω). Then there exist integers n₁ < n₂ < ⋯ such that

$${\displaystyle {\frac {X_{n_{1}}+\cdots +X_{n_{k}}}{\sqrt {k}}}}$$

converges in distribution to N(0,1) as k tends to infinity.

Random walk on a crystal lattice

The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.

Applications and examples

Simple example

This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the chi-squared values that quantify the goodness of the fit (the fit is good if the reduced chi-squared value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/√n), which is called the standard deviation of the mean (since it refers to the spread of sample means).

A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.

Comparison of probability density functions, **p(k) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).

 

Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 512. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.

Real applications

Published literature contains a number of useful and interesting examples and applications relating to the central limit theorem. One source states the following examples:

• The probability distribution for total distance covered in a random walk (biased or unbiased) will tend toward a normal distribution.
• Flipping many coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).

From another viewpoint, the central limit theorem explains the common appearance of the "bell curve" in density estimates applied to real world data. In cases like electronic noise, examination grades, and so on, we can often regard a single measured value as the weighted average of many small effects. Using generalisations of the central limit theorem, we can then see that this would often (though not always) produce a final distribution that is approximately normal.

In general, the more a measurement is like the sum of independent variables with equal influence on the result, the more normality it exhibits. This justifies the common use of this distribution to stand in for the effects of unobserved variables in models like the linear model.

Regression

Regression analysis and in particular ordinary least squares specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error term. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.

Other illustrations

Main article: Illustration of the central limit theorem

Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.

History

Dutch mathematician Henk Tijms writes:

The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician Abraham de Moivre who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie analytique des probabilités, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician Aleksandr Lyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.

Sir Francis Galton described the Central Limit Theorem in this way:

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.

The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper. Pólya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word central in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails". The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya in 1920 translates as follows.

The occurrence of the Gaussian probability density 1 = e^−x2 in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by Liapounoff. ...

A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Cauchy's, Bessel's and Poisson's contributions, is provided by Hald. Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by von Mises, Pólya, Lindeberg, Lévy, and Cramér during the 1920s, are given by Hans Fischer. Le Cam describes a period around 1935. Bernstein presents a historical discussion focusing on the work of Pafnuty Chebyshev and his students Andrey Markov and Aleksandr Lyapunov that led to the first proofs of the CLT in a general setting.

A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's College at the University of Cambridge. Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.

Exascale computing

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Exascale_computing 

Exascale computing refers to computing systems capable of calculating at least "10¹⁸ IEEE 754 Double Precision (64-bit) operations (multiplications and/or additions) per second (exaFLOP)"; it is a measure of supercomputer performance.

Exascale computing is a significant achievement in computer engineering: primarily it will allow improved scientific applications and better prediction such as in weather forecasting, climate modeling and personalised medicine. Exascale also reaches the estimated processing power of the human brain at the neural level, a target of the Human Brain Project. There has been a race to be the first country to build an exascale computer, typically ranked in the TOP500 list.

In 2022, the world's first public exascale computer, Frontier, was announced; As of June 2022, it is the world's fastest supercomputer.

Definitions

Further information: FLOPS, TOP500, and computer performance

Floating point operations per second (FLOPS) are one measure of computer performance. FLOPS can be recorded in different measures of precision, however the standard measure (used by the TOP500 supercomputer list) uses 64 bit (double-precision floating-point format) operations per second using the High Performance LINPACK (HPLinpack) benchmark.

Whilst a distributed computing system had broken the 1 exaFLOP barrier before Frontier, the metric typically refers to single computing systems. Supercomputers had also previously broken the 1 exaFLOP barrier using alternative precision measures; again these do not meet the criteria for exascale computing using the standard metric. It has been recognised that HPLinpack may not be a good general measure of supercomputer utility in real world application, however it is the common standard for performance measurement.

Technological challenges

It has been recognized that enabling applications to fully exploit capabilities of exascale computing systems is not straightforward. Developing data-intensive applications over exascale platforms requires the availability of new and effective programming paradigms and runtime systems. The Folding@home project, the first to break this barrier, relied on a network of servers sending pieces of work to hundreds of thousands of clients using a client–server model network architecture.

History

The first petascale (10¹⁵ FLOPS) computer entered operation in 2008. At a supercomputing conference in 2009, Computerworld projected exascale implementation by 2018. In June 2014, the stagnation of the Top500 supercomputer list had observers question the possibility of exascale systems by 2020.

Although exascale computing was not achieved by 2018, in the same year the Summit OLCF-4 supercomputer performed 1.8×10¹⁸ calculations per second using an alternative metric whilst analysing genomic information. The team performing this won the Gordon Bell Prize at the 2018 ACM/IEEE Supercomputing Conference.

The exaFLOPS barrier was first broken in March 2020 by the distributed computing network Folding@home coronavirus research project.

In June 2020 the Japanese supercomputer Fugaku achieved 1.42 exaFLOPS using the alternative HPL-AI benchmark.

Development

United States

In 2008, two United States of America governmental organisations within the US Department of Energy, the Office of Science and the National Nuclear Security Administration, provided funding to the Institute for Advanced Architectures for the development of an exascale supercomputer; Sandia National Laboratory and the Oak Ridge National Laboratory were also to collaborate on exascale designs. The technology was expected to be applied in various computation-intensive research areas, including basic research, engineering, earth science, biology, materials science, energy issues, and national security.

In January 2012, Intel purchased the InfiniBand product line from QLogic for US$125 million in order to fulfill its promise of developing exascale technology by 2018.

By 2012, the United States had allotted $126 million for exascale computing development.

In February 2013, the Intelligence Advanced Research Projects Activity started the Cryogenic Computer Complexity (C3) program, which envisions a new generation of superconducting supercomputers that operate at exascale speeds based on superconducting logic. In December 2014 it announced a multi-year contract with IBM, Raytheon BBN Technologies and Northrop Grumman to develop the technologies for the C3 program.

On 29 July 2015, Barack Obama signed an executive order creating a National Strategic Computing Initiative calling for the accelerated development of an exascale system and funding research into post-semiconductor computing. The Exascale Computing Project (ECP) hopes to build an exascale computer by 2021.

On 18 March 2019, the United States Department of Energy and Intel announced the first exaFLOPS supercomputer would be operational at Argonne National Laboratory by late 2022. The computer, named Aurora is to be delivered to Argonne by Intel and Cray (now Hewlett Packard Enterprise), and is expected to use Intel Xe GPGPUs alongside a future Xeon Scalable CPU, and cost US$600 Million.

On 7 May 2019, the U.S. Department of Energy announced a contract with Cray (now Hewlett Packard Enterprise) to build the Frontier supercomputer at Oak Ridge National Laboratory. Frontier is anticipated to be fully operational in 2022 and, with a performance of greater than 1.5 exaFLOPS, should then be the world's most powerful computer.

On 4 March 2020, the U.S. Department of Energy announced a contract with Hewlett Packard Enterprise and AMD to build the El Capitan supercomputer at a cost of US$600 million, to be installed at the Lawrence Livermore National Laboratory (LLNL). It is expected to be used primarily (but not exclusively) for nuclear weapons modeling. El Capitan was first announced in August 2019, when the DOE and LLNL revealed the purchase of a Shasta supercomputer from Cray. El Capitan will be operational in early 2023 and have a performance of 2 exaFLOPS. It will use AMD CPUs and GPUs, with 4 Radeon Instinct GPUs per EPYC Zen 4 CPU, to speed up artificial intelligence tasks. El Capitan should consume around 40 MW of electric power.

As of November 2021, the United States has three of the five fastest supercomputers in the world.

Japan

In Japan, in 2013, the RIKEN Advanced Institute for Computational Science began planning an exascale system for 2020, intended to consume less than 30 megawatts. In 2014, Fujitsu was awarded a contract by RIKEN to develop a next-generation supercomputer to succeed the K computer. The successor is called Fugaku, and aims to have a performance of at least 1 exaFLOPS, and be fully operational in 2021. In 2015, Fujitsu announced at the International Supercomputing Conference that this supercomputer would use processors implementing the ARMv8 architecture with extensions it was co-designing with ARM Limited. It was partially put into operation in June 2020 and achieved 1.42 exaFLOPS (fp16 with fp64 precision) in HPL-AI benchmark making it the first ever supercomputer that achieved 1 exaOPS. Named after Mount Fuji, Japan's tallest peak, Fugaku retained the No. 1 ranking on the Top 500 supercomputer calculation speed ranking announced on November 17 2020, reaching a calculation speed of 442 quadrillion calculations per second, or 0.442 exaFLOPS.

China

As of June 2022, China had two of the Top Ten fastest supercomputers in the world. According to the national plan for the next generation of high performance computers and the head of the school of computing at the National University of Defense Technology (NUDT), China was supposed to develop an exascale computer during the 13th Five-Year-Plan period (2016–2020) which would enter service in the latter half of 2020. The government of Tianjin Binhai New Area, NUDT and the National Supercomputing Center in Tianjin are working on the project. After Tianhe-1 and Tianhe-2, the exascale successor is planned to be named Tianhe-3.

European Union

See also Supercomputing in Europe

In 2011, several projects aiming at developing technologies and software for exascale computing were started in the EU. The CRESTA project (Collaborative Research into Exascale Systemware, Tools and Applications), the DEEP project (Dynamical ExaScale Entry Platform), and the project Mont-Blanc. A major European project based on exascale transition is the MaX (Materials at the Exascale) project. The Energy oriented Centre of Excellence (EoCoE) exploits exascale technologies to support carbon-free energy research and applications.

In 2015, the Scalable, Energy-Efficient, Resilient and Transparent Software Adaptation (SERT) project, a major research project between the University of Manchester and the STFC Daresbury Laboratory in Cheshire, was awarded c. £1million from the UK's Engineering and Physical Sciences Research Council. The SERT project was due to start in March 2015. It will be funded by EPSRC under the Software for the Future II programme, and the project will partner with the Numerical Analysis Group (NAG), Cluster Vision and the Science and Technology Facilities Council (STFC).

On 28 September 2018, the European High-Performance Computing Joint Undertaking (EuroHPC JU) was formally established by the EU. The EuroHPC JU aims to build an exascale supercomputer by 2022/2023. The EuroHPC JU will be jointly funded by its public members with a budget of around €1 billion. The EU's financial contribution is €486 million.

Taiwan

In June 2017, Taiwan's National Center for High-Performance Computing initiated the effort towards designing and building the first Taiwanese exascale supercomputer by funding construction of a new intermediary supercomputer based on a full technology transfer from Fujitsu corporation of Japan, which is currently building the fastest and most powerful A.I. based supercomputer in Japan. Additionally, numerous other independent efforts have been made in Taiwan with the focus on the rapid development of exascale supercomputing technology, such as Foxconn Corporation which recently designed and built the largest and fastest supercomputer in all of Taiwan. This new Foxconn supercomputer is designed to serve as a stepping stone in research and development towards the design and building of a state of the art exascale supercomputer.

India

In 2012, the Indian Government proposed to commit US$2.5 billion to supercomputing research during the 12th five-year plan period (2012–2017). The project was to be handled by Indian Institute of Science (IISc), Bangalore. Additionally, it was later revealed that India plans to develop a supercomputer with processing power in the exaFLOPS range. It will be developed by C-DAC within the subsequent five years of approval. These supercomputers will use indigenously developed microprocessors by C-DAC in India.

Carotene

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Carotene 

A 3-dimensional stick diagram of β-carotene

 

Carotene is responsible for the orange colour of carrots and the colours of many other fruits and vegetables and even some animals.

 

Lesser Flamingos in the Ngorongoro Crater, Tanzania. The pink colour of wild flamingos is due to astaxanthin (a carotenoid) they absorb from their diet of brine shrimp. If fed a carotene-free diet they become white.

The term carotene (also carotin, from the Latin carota, "carrot") is used for many related unsaturated hydrocarbon substances having the formula C₄₀H_x, which are synthesized by plants but in general cannot be made by animals (with the exception of some aphids and spider mites which acquired the synthesizing genes from fungi). Carotenes are photosynthetic pigments important for photosynthesis. Carotenes contain no oxygen atoms. They absorb ultraviolet, violet, and blue light and scatter orange or red light, and (in low concentrations) yellow light.

Carotenes are responsible for the orange colour of the carrot, after which this class of chemicals is named, and for the colours of many other fruits, vegetables and fungi (for example, sweet potatoes, chanterelle and orange cantaloupe melon). Carotenes are also responsible for the orange (but not all of the yellow) colours in dry foliage. They also (in lower concentrations) impart the yellow coloration to milk-fat and butter. Omnivorous animal species which are relatively poor converters of coloured dietary carotenoids to colourless retinoids have yellowed-coloured body fat, as a result of the carotenoid retention from the vegetable portion of their diet. The typical yellow-coloured fat of humans and chickens is a result of fat storage of carotenes from their diets.

Carotenes contribute to photosynthesis by transmitting the light energy they absorb to chlorophyll. They also protect plant tissues by helping to absorb the energy from singlet oxygen, an excited form of the oxygen molecule O₂ which is formed during photosynthesis.

β-Carotene is composed of two retinyl groups, and is broken down in the mucosa of the human small intestine by β-carotene 15,15'-monooxygenase to retinal, a form of vitamin A. β-Carotene can be stored in the liver and body fat and converted to retinal as needed, thus making it a form of vitamin A for humans and some other mammals. The carotenes α-carotene and γ-carotene, due to their single retinyl group (β-ionone ring), also have some vitamin A activity (though less than β-carotene), as does the xanthophyll carotenoid β-cryptoxanthin. All other carotenoids, including lycopene, have no beta-ring and thus no vitamin A activity (although they may have antioxidant activity and thus biological activity in other ways).

Animal species differ greatly in their ability to convert retinyl (beta-ionone) containing carotenoids to retinals. Carnivores in general are poor converters of dietary ionone-containing carotenoids. Pure carnivores such as ferrets lack β-carotene 15,15'-monooxygenase and cannot convert any carotenoids to retinals at all (resulting in carotenes not being a form of vitamin A for this species); while cats can convert a trace of β-carotene to retinol, although the amount is totally insufficient for meeting their daily retinol needs.

Molecular structure

Carotenes are polyunsaturated hydrocarbons containing 40 carbon atoms per molecule, variable numbers of hydrogen atoms, and no other elements. Some carotenes are terminated by rings, on one or both ends of the molecule. All are coloured, due to the presence of conjugated double bonds. Carotenes are tetraterpenes, meaning that they are derived from eight 5-carbon isoprene units (or four 10-carbon terpene units).

Carotenes are found in plants in two primary forms designated by characters from the Greek alphabet: alpha-carotene (α-carotene) and beta-carotene (β-carotene). Gamma-, delta-, epsilon-, and zeta-carotene (γ, δ, ε, and ζ-carotene) also exist. Since they are hydrocarbons, and therefore contain no oxygen, carotenes are fat-soluble and insoluble in water (in contrast with other carotenoids, the xanthophylls, which contain oxygen and thus are less chemically hydrophobic).

History

The discovery of carotene from carrot juice is credited to Heinrich Wilhelm Ferdinand Wackenroder, a finding made during a search for antihelminthics, which he published in 1831. He obtained it in small ruby-red flakes soluble in ether, which when dissolved in fats gave 'a beautiful yellow colour'. William Christopher Zeise recognised its hydrocarbon nature in 1847, but his analyses gave him a composition of C₅H₈. It was Léon-Albert Arnaud in 1886 who confirmed its hydrocarbon nature and gave the formula C₂₆H₃₈, which is close to the theoretical composition of C₄₀H₅₆. Adolf Lieben in studies, also published in 1886, of the colouring matter in corpora lutea, first came across carotenoids in animal tissue, but did not recognise the nature of the pigment. Johann Ludwig Wilhelm Thudichum, in 1868–1869, after stereoscopic spectral examination, applied the term 'luteine'(lutein) to this class of yellow crystallizable substances found in animals and plants. Richard Martin Willstätter, who gained the Nobel Prize in Chemistry in 1915, mainly for his work on chlorophyll, assigned the composition of C₄₀H₅₆, distinguishing it from the similar but oxygenated xanthophyll, C₄₀H₅₆O₂. With Heinrich Escher, in 1910, lycopene was isolated from tomatoes and shown to be an isomer of carotene. Later work by Escher also differentiated the 'luteal' pigments in egg yolk from that of the carotenes in cow corpus luteum.

Dietary sources

The following foods contain carotenes in notable amounts:

• carrots
• wolfberries (goji)
• cantaloupe
• mangoes
• red bell pepper
• papaya
• spinach
• kale
• sweet potato
• tomato
• dandelion greens
• broccoli
• collard greens
• winter squash
• pumpkin
• cassava

Absorption from these foods is enhanced if eaten with fats, as carotenes are fat soluble, and if the food is cooked for a few minutes until the plant cell wall splits and the color is released into any liquid. 12 μg of dietary β-carotene supplies the equivalent of 1 μg of retinol, and 24 µg of α-carotene or β-cryptoxanthin provides the equivalent of 1 µg of retinol.

Forms of carotene

α-carotene

 

β-carotene

 

γ-carotene

 

δ-carotene

The two primary isomers of carotene, α-carotene and β-carotene, differ in the position of a double bond (and thus a hydrogen) in the cyclic group at one end (the right end in the diagram at right).

β-Carotene is the more common form and can be found in yellow, orange, and green leafy fruits and vegetables. As a rule of thumb, the greater the intensity of the orange colour of the fruit or vegetable, the more β-carotene it contains.

Carotene protects plant cells against the destructive effects of ultraviolet light so β-Carotene is an antioxidant.

β-Carotene and physiology

β-Carotene and cancer

An article on the American Cancer Society says that The Cancer Research Campaign has called for warning labels on β-carotene supplements to caution smokers that such supplements may increase the risk of lung cancer.

The New England Journal of Medicine published an article in 1994 about a trial which examined the relationship between daily supplementation of β-carotene and vitamin E (α-tocopherol) and the incidence of lung cancer. The study was done using supplements and researchers were aware of the epidemiological correlation between carotenoid-rich fruits and vegetables and lower lung cancer rates. The research concluded that no reduction in lung cancer was found in the participants using these supplements, and furthermore, these supplements may, in fact, have harmful effects.

The Journal of the National Cancer Institute and The New England Journal of Medicine published articles in 1996 about a trial with a goal to determine if vitamin A (in the form of retinyl palmitate) and β-carotene (at about 30 mg/day, which is 10 times the Reference Daily Intake) supplements had any beneficial effects to prevent cancer. The results indicated an increased risk of lung and prostate cancers for the participants who consumed the β-carotene supplement and who had lung irritation from smoking or asbestos exposure, causing the trial to be stopped early.

A review of all randomized controlled trials in the scientific literature by the Cochrane Collaboration published in JAMA in 2007 found that synthetic β-carotene increased mortality by 1-8 % (Relative Risk 1.05, 95% confidence interval 1.01–1.08). However, this meta-analysis included two large studies of smokers, so it is not clear that the results apply to the general population. The review only studied the influence of synthetic antioxidants and the results should not be translated to potential effects of fruits and vegetables.

β-Carotene and photosensitivity

Oral β-carotene is prescribed to people suffering from erythropoietic protoporphyria. It provides them some relief from photosensitivity.

Carotenemia

Main article: Carotenodermia

Carotenemia or hypercarotenemia is excess carotene, but unlike excess vitamin A, carotene is non-toxic. Although hypercarotenemia is not particularly dangerous, it can lead to an oranging of the skin (carotenodermia), but not the conjunctiva of eyes (thus easily distinguishing it visually from jaundice). It is most commonly associated with consumption of an abundance of carrots, but it also can be a medical sign of more dangerous conditions.

Production

Algae farm ponds in Whyalla, South Australia, used to produce β-carotene.

Carotenes are produced in a general manner for other terpenoids and terpenes, i.e. by coupling, cyclization, and oxygenation reactions of isoprene derivatives. Lycopene is the key precursor to carotenoids. It is formed by coupling of geranylgeranyl pyrophosphate and[geranyllinalyl pyrophosphate.

Most of the world's synthetic supply of carotene comes from a manufacturing complex located in Freeport, Texas and owned by DSM. The other major supplier BASF also uses a chemical process to produce β-carotene. Together these suppliers account for about 85% of the β-carotene on the market. In Spain Vitatene produces natural β-carotene from fungus Blakeslea trispora, as does DSM but at much lower amount when compared to its synthetic β-carotene operation. In Australia, organic β-carotene is produced by Aquacarotene Limited from dried marine algae Dunaliella salina grown in harvesting ponds situated in Karratha, Western Australia. BASF Australia is also producing β-carotene from microalgae grown in two sites in Australia that are the world's largest algae farms. In Portugal, the industrial biotechnology company Biotrend is producing natural all-trans-β-carotene from a non-genetically-modified bacteria of the genus Sphingomonas isolated from soil.

Carotenes are also found in palm oil, corn, and in the milk of dairy cows, causing cow's milk to be light yellow, depending on the feed of the cattle, and the amount of fat in the milk (high-fat milks, such as those produced by Guernsey cows, tend to be yellower because their fat content causes them to contain more carotene).

Carotenes are also found in some species of termites, where they apparently have been picked up from the diet of the insects.

Synthesis

There are currently two commonly used methods of total synthesis of β-carotene. The first was developed by BASF and is based on the Wittig reaction with Wittig himself as patent holder:

The second is a Grignard reaction, elaborated by Hoffman-La Roche from the original synthesis of Inhoffen et al. They are both symmetrical; the BASF synthesis is C20 + C20, and the Hoffman-La Roche synthesis is C19 + C2 + C19.

Nomenclature

Carotenes are carotenoids containing no oxygen. Carotenoids containing some oxygen are known as xanthophylls.

The two ends of the β-carotene molecule are structurally identical, and are called β-rings. Specifically, the group of nine carbon atoms at each end form a β-ring.

The α-carotene molecule has a β-ring at one end; the other end is called an ε-ring. There is no such thing as an "α-ring".

These and similar names for the ends of the carotenoid molecules form the basis of a systematic naming scheme, according to which:

• α-carotene is β,ε-carotene;
• β-carotene is β,β-carotene;
• γ-carotene (with one β ring and one uncyclized end that is labelled psi) is β,ψ-carotene;
• δ-carotene (with one ε ring and one uncyclized end) is ε,ψ-carotene;
• ε-carotene is ε,ε-carotene
• lycopene is ψ,ψ-carotene

ζ-Carotene is the biosynthetic precursor of neurosporene, which is the precursor of lycopene, which, in turn, is the precursor of the carotenes α through ε.

Food additive

Carotene is used to colour products such as juice, cakes, desserts, butter and margarine. It is approved for use as a food additive in the EU (listed as additive E160a) Australia and New Zealand (listed as 160a) and the US.