Lévy continuity theorem

If the characteristic functions ${\phi }_n$ of probability distributions ${\mu }_n$ converge pointwise to a function $\phi$ that is continuous at zero, then $\phi$ is a characteristic function and ${\mu }_n$ converges in distribution to its distribution. Conversely, convergence in distribution implies pointwise convergence of the characteristic functions.

For a probability measure on the real line with characteristic function $\phi$, averaging $1-\cos (tx)$ over $[-h,h]$ gives $1-\sin (hx)/(hx)$. This is at least $1/2$ when $|x|\ge 2/h$. Integrating gives the displayed bound. Consequently pointwise convergence of characteristic functions to a function continuous at zero implies tightness of probability measures, by the dominated convergence theorem on $[-h,h]$ and a separate tail bound for the finitely many initial measures.

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