Editing Central limit theorem (section)

===Lyapunov CLT===
In this variant of the central limit theorem the random variables <math display="inline">X_i</math> have to be independent, but not necessarily identically distributed. The theorem also requires that random variables <math display="inline">\left| X_i\right|</math> have [[moment (mathematics)|moment]]s of some order {{nowrap|<math display="inline">(2+\delta)</math>,}} and that the rate of growth of these moments is limited by the Lyapunov condition given below.

{{math theorem | name = Lyapunov CLT{{sfnp|Billingsley|1995|p=362}} | math_statement =
Suppose <math display="inline">\{X_1, \ldots, X_n, \ldots\}</math> is a sequence of independent random variables, each with finite expected value <math display="inline">\mu_i</math> and variance {{nowrap|<math display="inline">\sigma_i^2</math>.}} Define

<math display="block">s_n^2 = \sum_{i=1}^n \sigma_i^2 .</math>

If for some {{nowrap|<math display="inline">\delta > 0</math>,}} ''Lyapunov’s condition''

<math display="block">\lim_{n\to\infty} \; \frac{1}{s_{n}^{2+\delta}} \, \sum_{i=1}^{n} \operatorname E\left[\left|X_{i} - \mu_{i}\right|^{2+\delta}\right] = 0</math>

is satisfied, then a sum of <math display="inline">\frac{X_i - \mu_i}{s_n}</math> converges in distribution to a standard normal random variable, as <math display="inline">n</math> goes to infinity:

<math display="block">\frac{1}{s_n}\,\sum_{i=1}^{n} \left(X_i - \mu_i\right) \mathrel{\overset{d}{\longrightarrow}} \mathcal{N}(0,1) .</math>}}

In practice it is usually easiest to check Lyapunov's condition for {{nowrap|<math display="inline">\delta = 1</math>.}}

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