# 13.1 Convergence and the central limit theorem

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The central limit theorem (CLT) asserts that the sum of a large class of independent random variables, each with reasonable distributions,is approximately normally distributed. Various versions of this theorem have been studied intensively. On the other hand, certain common forms serve as the basis of an extraordinary amount of applied work. In the statistics of large samples, the sample average is approximately normal—whether or not the population distribution is normal. In much of the theory of errors of measurement, the observed error is the sum of a large number of independent random quantities which contribute additively to the result. Similarly, in the theory of noise, the noise signal is the sum of a large number of random components, independently produced. In such situations, the assumption of a normal population distribution is frequently quite appropriate

## The central limit theorem

The central limit theorem (CLT) asserts that if random variable X is the sum of a large class of independent random variables, each with reasonable distributions, then X is approximately normally distributed. This celebrated theorem has been the object of extensive theoretical research directed toward the discoveryof the most general conditions under which it is valid. On the other hand, this theorem serves as the basis of an extraordinary amount of applied work.In the statistics of large samples, the sample average is a constant times the sum of the random variables in the sampling process . Thus, for large samples,the sample average is approximately normal—whether or not the population distribution is normal. In much of the theory of errors of measurement,the observed error is the sum of a large number of independent random quantities which contribute additively to the result. Similarly, in the theory of noise, thenoise signal is the sum of a large number of random components, independently produced. In such situations, the assumption of a normal population distribution is frequentlyquite appropriate.

We consider a form of the CLT under hypotheses which are reasonable assumptions in many practical situations. We sketch a proof of this version of the CLT,known as the Lindeberg-Lévy theorem, which utilizes the limit theorem on characteristic functions, above, along with certain elementary facts from analysis. Itillustrates the kind of argument used in more sophisticated proofs required for more general cases.

Consider an independent sequence $\left\{{X}_{n}:1\le n\right\}$ of random variables. Form the sequence of partial sums

${S}_{n}=\sum _{i=1}^{n}{X}_{i}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\ge 1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{with}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}E\left[{S}_{n}\right]=\sum _{i=1}^{n}E\left[{X}_{i}\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[{S}_{n}\right]=\sum _{i=1}^{n}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[{X}_{i}\right]$

Let ${S}_{n}^{*}$ be the standardized sum and let F n be the distribution function for ${S}_{n}^{*}$ . The CLT asserts that under appropriate conditions, ${F}_{n}\left(t\right)\to \Phi \left(t\right)$ as $n\to \infty$ for all t . We sketch a proof of the theorem under the condition the X i form an iid class.

Central Limit Theorem (Lindeberg-Lévy form)

If $\left\{{X}_{n}:1\le n\right\}$ is iid, with

$E\left[{X}_{i}\right]=\mu ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[{X}_{i}\right]={\sigma }^{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{S}_{n}^{*}=\frac{{S}_{n}-n\mu }{\sigma \sqrt{n}}$

then

${F}_{n}\left(t\right)\to \Phi \left(t\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t$

IDEAS OF A PROOF

There is no loss of generality in assuming $\mu =0$ . Let φ be the common characteristic function for the X i , and for each n let φ n be the characteristic function for ${S}_{n}^{*}$ . We have

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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or in general
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tahir
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anybody can imagine what will be happen after 100 years from now in nano tech world
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
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silver nanoparticles could handle the job?
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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how did you get the value of 2000N.What calculations are needed to arrive at it
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive