Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\) that has density \(f\), characteristic function \(\psi(u)\), and cumulant generating function \(c(u)\). Assume that the characteristic function is real valued in this case and use the alternate definition of the cumulant generating function given by \(c(u)=\log [\psi(u)]\). Define the density \(f_{\lambda}(t)=\exp (\lambda t) f(t) / \psi(\lambda)\) and let \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following the density \(f_{\lambda}\). Let \(f_{n}\) denote the density of \(n \bar{X}_{n}\) and \(f_{n, \lambda}\) denote the density of \(n \bar{Y}_{n}\). Using characteristic functions, prove that \(f_{n, \lambda}(t)=\exp [\lambda t-n c(\lambda)] f_{n}(t)\).