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Sum of uniformly distributed random variables and a family of nonanalytic \(C^{\infty}\)-functions. (English) Zbl 0625.60054

Let \(\{U_ n\}^{\infty}_{n=1}\) be a sequence of independent random variables, each uniformly distributed on [-1,1], and let \(0<\alpha <1\) be fixed. In consideration of the rounding errors which may arise in very lengthy numerical calculations, the authors are led to study the sequence \[ X_ n=[U_ n+(1-\alpha)U_{n-1}+...+(1-\alpha)^{n-2} U_ 2+(1- \alpha)^{n-1} U_ 1],\quad n=1,2,.... \] If \(p_ n(x)\) denotes “the” probability density function of \(X_ n\) (presumably the derivative of the distribution function), the authors show that the \(p_ n\) converge uniformly on \({\mathbb{R}}\) to a \(C^{\infty}\) (density) function \(p_{\infty}\) which is nonanalytic at infinitely many points in [-1,1], and nowhere analytic on [-1,1] when \(\alpha =\).
Reviewer: B.Horkelheimer

MSC:

60G50 Sums of independent random variables; random walks
60E05 Probability distributions: general theory
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References:

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