Some uniformity classes in Blackwell's renewal theorem for distributions with atomic or singular component (Q915275)

Let F be a nonarithmetic d.f. on \([0,\infty)\) with positive and finite mean \(\mu_ F\) and let \(\nu\) be its distribution. Furthermore let \(U_ F(B)=\sum^{\infty}_{n=0}\nu^{*n}(B),\) where B is a Borel set in \([0,\infty)\) and \(\nu^{*n}\) is the n-th convolution of \(\nu\). The Blackwell theorem says that \[ (1)\quad U_ F[t,t+A]\to A/\mu_ F\quad as\quad t\to \infty \text{ for } any\quad A>0. \] The paper gives uniform (over some class \({\mathcal F}\) of distribution functions) explicit bounds for the convergence rate of (1) for distributions having an atomic or a singular component. For example, a function f(A,t) is given such that \(f(A,t)\to 0\) as \(t\to \infty\) and \[ | U_ F[t,t+A]-A/\mu_ F| \leq f(A,t). \]