I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a random variable $X$, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise to some function $\varphi$ which is continuous at $0$, with $\varphi$ not the characteristic function of a well-know law (Binomial, Cauchy, Normal, etc).



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