Numerical approximation of probability mass functions via the inverse discrete Fourier transform (Q2513667)

Let \(T\) be a random variable with distribution \(F\). The characteristic function and Fourier transform of \(T\) are defined as \[ \varphi_T(s)=\int_{-\infty}^{\infty}e^{ist}dF(t) \] and \[ \hat{f}(\omega)=\int_{-\infty}^{\infty}e^{-2\pi i \omega}dF(t), \] respectively. The author suggests using the inverse fast Fourier transform for the inversion of the characteristic function to probability mass functions. To make it the author derives error bounds for lattice distributions. For example, the next statement is proved. { Lemma 1.} For a nonnegative lattice random variable \(T\), defined on the support \(n\Delta t\), \(n=0,1,\dots,\infty\), the pointwise error of the forward discrete Fourier transform is less or equal to \(\operatorname{P}(T\geq N\Delta t)\).