Quantile constructions for Khinchin's and Pruitt's theorems and for Doeblin's universal laws (Q1261860)

We say that a distribution \(F\) is in the domain of partial attraction of the distribution \(\sigma\) if there are sequences \(\{A_ n\}\), \(\{C_ k\}\), and \(\{n_ k\}\) such that if \(X_ 1,X_ 2,\ldots\) are independent random variables with common distribution \(F\), then the random variables \({1\over A_{n_ k}}\left\{\sum^ n_{k=1}X_ k- C_ k\right\}\) have asymptotic distribution \(\sigma\). It is known, that then \(\sigma\) must be an infinitely divisible distribution, and Khinchin proved that conversely, the domain of partial attraction of an infinitely divisible distribution is never empty. The first part of the paper under review gives a direct proof for Khinchin's theorem by constructing \(F\) and the sequences \(\{A_ n\}\), \(\{C_ k\}\), and \(\{n_ k\}\) explicitly. This construction is then modified to give a probabilistic proof for the famous Doeblin theorem, according to which there is a universal distribution lying in the domain of partial attraction of every infinitely divisible distribution. The proofs are based on a stochastic representation of all infinitely divisible distributions via the Poisson process, due to the first author, \textit{E. Haeusler} and \textit{D. M. Mason} [ibid. 9, No. 3, 259-333 (1988; Zbl 0657.60029)].