Convergence in distribution

Summary

• $z_1,z_2,…$ is a sequence of random variables where $z_n$ has cumulative distribution function $F_n\left( z \right)$ for $n=1,2,…$ and $z$ is a random variable with cumulative distribution function $F\left( z \right)$ .
• We say that the sequence $z_1,z_2,…$ converges in distribution to a random variable $z$ if

$$F_n\left( z \right)→F(z)$$

• as $n→∞$ for all $z∈R$ at which $F$ is continuous.

Central limit theorem

• (Lindeberg–Lévy CLT). $x_1,x_2,...,x_n$ is an IID random sample, $E\left( x_i \right)=μ$ and $Var\left( x_i \right)=σ^2$ . Then

$$\sqrt{n}\left( {\bar{x}}_n-μ \right) \to \textrm{N}( 0,σ^2 )$$

• or

$$\frac{\sqrt{n}\left( {\bar{x}}_n-μ \right)}{σ} \to \textrm{N}( 0,1 )$$

• For $n$ large, ${\bar{x}}_n$ will be approximately $\textrm{N}( μ,σ^2/n )$ .