A Law Limit Theorem for a sequence of random variables

An application of Levy's continuity theorem and Hankel transform allow us to establish a law limit theorem for the sequence $V_n=f(U)\sin(n U)$, where $U$ is uniformly distributed in $(0,1)$ and $f$ a given function. Further, we investigate the inverse problem by specifying a limit distribution and look for the suitable function $f$ ensuring the convergence in law to the specified distribution. Our work recovers and extends existing similar works, in particular we make it possible to sample from known laws including Gaussian and Cauchy distributions.