Some characterizations of the normal distribution (Q1962220)

Based on the properties of two i.i.d. continuous random variables \(X\), \(Y\), three characterizations of the normal distribution are given. First it is shown that \(X\) is standard normal (\(N(0,1)\)) if \(2XY/(\sqrt{X^{2} +Y^{2}})\) is \(N(0,1)\). Assuming \[ E|X|^{2n+1} = 2nE|X|^{2n-1},\quad n \geq 1, \] and the pdf of \(X\) equals its MacLaurin series expansion around the origin, it is shown that if \(X/Y\) is standard Cauchy (\(C(0,1)\)), then \(X\) is \(N(0,1)\). Under similar conditions, it is shown that \(X\) must be normal if \((X+Y)/|X-Y|\) is \(C(0,1)\).