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Section of Euler summability method. (English) Zbl 1100.65005
A connection between Euler’s means and expressions for the distribution function of a sum of independent random variables is established. The Euler mean $$E(y,S| n)$$ of order $$n$$ derived from the constants $$c$$ and sequence $$S$$ has the form $$\{\sum c(n,k). S(k)\;|\;0\leq k\leq n\}$$; the $$c(n,k)$$ involve binomial coefficients and integer powers of a parameter, say $$y$$. The independent random variables $$f(1),\dots,f(n)$$ ($$f(k)$$ having a distribution depending on the real parameters $$q(k)$$ ($$1\leq k\leq n$$)), the random variable $$f(1)+\cdots+f(n)$$ has a distribution function $$F(f| n)$$ whose value $$F(f| n)(t)$$ is expressible as an Euler mean, the parameter $$y$$ in which is simply related to the $$q(k)$$. In the case in which all $$q(k)$$ are equal to a common value $$q$$, various special expressions, depending on the relationship between $$q$$ and $$n$$, are given for the mean. Making use of Euler mean theory, it is shown that with $$t>0$$, $$F(f| n)(t)$$ tends, as $$n$$ increases, to a limit.
##### MSC:
 65B15 Euler-Maclaurin formula in numerical analysis 62H10 Multivariate distribution of statistics 65C60 Computational problems in statistics (MSC2010) 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
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