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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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