[Levy's continuity theorem][1] states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)$,  pointwise convergence $\varphi_n(t)\rightarrow\varphi(t)$ implies convergence in distribution $X_n\xrightarrow{D}X$.

Are any converse results known for this?  

---
**Why?**

I am interested in the moments of the maximum $M_n=\max (X_1,\ldots,X_n)$ where $X_1,\ldots,X_n$ are i.i.d. and have finite support $[x_l,x_u]$.  Thus the distribution of $X_i$ belongs to the [maximum domain of attraction][2] of the Weibull distribution, meaning that, for appropriately chosen sequence $c_n$, $c_n^{-1}(M_n-x_u)\xrightarrow{D}W_\alpha$ where $W_\alpha$ is Weibull random variable with distribution function $\Psi_\alpha(x)=\exp[-(-x)^\alpha]$, $\alpha>0$.  A some kind of a converse to Levy continuity theorem *might* allow me to use the Weibull characteristic function to obtain the desired moments.  However, I would appreciate other suggestions -- perhaps there are more elegant ways to get there.


  [1]: http://en.wikipedia.org/wiki/L%C3%A9vy%27s_continuity_theorem
  [2]: http://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem