A Law Limit Theorem for a sequence of random variables

TL;DR

This paper establishes a law limit theorem for a sequence involving a function and a sine term of a uniform random variable, using Levy's theorem and Hankel transform, and explores inverse problems for sampling from known distributions.

Contribution

It introduces a new limit theorem for sequences of the form V_n=f(U)sin(nU) and extends methods for sampling from Gaussian and Cauchy distributions.

Findings

Proves convergence in law for the sequence V_n

Provides a method to sample from Gaussian and Cauchy distributions

Extends existing limit theorems to new functional forms

Abstract

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.