# 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

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