Ulam's redistribution of energy problem: Collision transformations
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.
scientific article; zbMATH DE number 3932106