Fractional order error estimates for the renewal density (Q267135)

In this paper, the author assumes that \(X\) is a real random variable with finite and positive mean \(\mu \) and with density function \(f(.)\). The author assumes that \(\operatorname{E} | X | ^{\alpha}<\infty\) with \( \alpha >3/2\). The renewal density is given by \(u(x)=\sum_{n=1}^{\infty }f^{\otimes n}(x)\), where \(f^{\otimes n}\) is the \(n\)-fold convolution of \(f\). In the main theorems, the author obtains precise estimates for the error \( u(x)-\mu ^{-1}I_{(0,\infty )}\). The proofs are based on an intelligent use of Fourier transforms.