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 \(\textrm{N}( μ,σ^2/n )\) .