Editing Central limit theorem (section)

===CLT for the sum of a random number of random variables===

Rather than summing an integer number <math>n</math> of random variables and taking <math>n \to \infty</math>, the sum can be of a random number <math>N</math> of random variables, with conditions on <math>N</math>.

{{math theorem | name = Robbins CLT<ref>{{cite journal |last1=Robbins |first1=Herbert |title=The asymptotic distribution of the sum of a random number of random variables |journal=Bull. Amer. Math. Soc. |date=1948 |volume=54 |issue=12 |pages=1151–1161 |doi=10.1090/S0002-9904-1948-09142-X |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-54/issue-12/The-asymptotic-distribution-of-the-sum-of-a-random-number/bams/1183513324.full|doi-access=free }}</ref><ref>{{cite book |last1=Chen |first1=Louis H.Y. |last2=Goldstein |first2=Larry |last3=Shao |first3=Qi-Man |title=Normal Approximation by Stein's Method |date=2011 |publisher=Springer-Verlag |location=Berlin Heidelberg |pages=270–271}}</ref> | math_statement =
Let <math>\{X_i, i \geq 1\}</math> be independent, identically distributed random variables with <math>E(X_i) = \mu</math> and <math>\text{Var}(X_i) = \sigma^2</math>, and let <math>\{N_n, n \geq 1\}</math> be a sequence of non-negative integer-valued random variables that are independent of <math>\{X_i, i \geq 1\}</math>. Assume for each <math>n = 1, 2, \dots</math> that <math>E(N_n^2) < \infty</math> and

<math display="block">
  \frac{N_n - E(N_n)}{\sqrt{\text{Var}(N_n)}} \xrightarrow{\quad d \quad} \mathcal{N}(0,1)
</math>

where <math>\xrightarrow{\,d\,}</math> denotes convergence in distribution and <math>\mathcal{N}(0,1)</math> is the normal distribution with mean 0, variance 1.
Then

<math display="block">
  \frac{\sum_{i=1}^{N_n} X_i - \mu E(N_n)}{\sqrt{\sigma^2E(N_n) + \mu^2\text{Var}(N_n)}} \xrightarrow{\quad d \quad} \mathcal{N}(0,1)
</math>
}}