Convergence in distribution

Summary

  • z1,z2,…z1,z2,… is a sequence of random variables where znzn has cumulative distribution function Fn(z)Fn(z) for n=1,2,…n=1,2,… and zz is a random variable with cumulative distribution function F(z)F(z) .
  • We say that the sequence z1,z2,…z1,z2,… converges in distribution to a random variable zz if

Fn(z)→F(z)Fn(z)→F(z)

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

Central limit theorem

  • (Lindeberg–Lévy CLT). x1,x2,...,xn is an IID random sample, E(xi)=μ and Var(xi)=σ2 . Then

√n(ˉxn−μ)→N(0,σ2)

  • or

√n(ˉxn−μ)σ→N(0,1)

  • For n large, ˉxn will be approximately N(μ,σ2/n) .