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Ulam’s redistribution of energy problem: Collision transformations. (English) Zbl 0582.60035

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

MSC:

60F05 Central limit and other weak theorems
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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.
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