Weak law from a characteristic-function expansion
= Weak law from a characteristic-function expansion
If <independent and identically distributed random variables> have <characteristic function> $\varphi(t)=1+iat+o(|t|)$ at zero, then their <sample mean> has <convergence in distribution> to $a$: its <characteristic function> is $\varphi(t/n)^n\to e^{iat}$. This is also <convergence in probability> by <convergence in distribution to a constant implies convergence in probability>.