Blackwell, David; Mauldin, R. Daniel Ulam’s redistribution of energy problem: Collision transformations. (English) Zbl 0582.60035 Lett. Math. Phys. 10, 149-153 (1985). Main result: Let \(U,X_ 1,X_ 2\) be independent random variables, where \(X_ 1,X_ 2\) are identically distributed. Under the assumption that all moment of U and \(X_ i(i=1,2)\) exist, it is shown that \(\lim_{k\to \infty}E[(T^ kX)^ n]=a_ n\) exists and satisfies Carleman’s condition \(\sum^{\infty}_{n=1}a_ n^{-1/2n}<\infty,\) where X is a random variable with the same distribution like \(X_ i\) \((i=1,2)\) and \(TX=U(X_ 1+X_ 2).\) In particular, \(T^ kX\) converges in distribution to a distribution which is uniquely determined by its moments. Reviewer: D.Plachky Cited in 1 ReviewCited in 4 Documents MSC: 60F05 Central limit and other weak theorems Keywords:convergence in distribution; Carleman’s condition PDFBibTeX XMLCite \textit{D. Blackwell} and \textit{R. D. Mauldin}, Lett. Math. Phys. 10, 149--153 (1985; Zbl 0582.60035) Full Text: DOI References: [1] SohatJ. A. and TamarkinJ. D., ?The Problem of Moments?, Amer. Math. Soc. Math. Surveys, Vol. 1, Amer. Math. Soc., New York, 1943, p. 20. [2] BillingsleyP., Probability and Measure, Wiley, New York, 1979, p. 344. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.