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Limit theorems for logarithmic means. (English) Zbl 0618.26015

The author examines his logarithmic means \[ L_ r(x_ 0,...,x_ n) = \begin{cases} (\int_{A_ n}n!(\sum^{n}_{k=0}x_ kv_ k)^ rdv_ 0...dv_{n-1})^{1/r} & (r\neq 0), \\ \exp\int_{A_ n}n! \log(\sum^{n}_{k=0} x_ kv_ k)dv_ 0... dv_{n-1} & (r=0), \end{cases} \] where \(A_ n=\{(v_ 0,...,v_{n-1}):\) \(0\leq v_ k\leq 1\); \(k=0,...,n-1\); \(0\leq \sum^{n-1}_{k=0}v_ k\leq 1\}\) and \(v_ n:=1- \sum^{n-1}_{k=0}v_ k\), their limits and possible applications to probability theory. Similar computations are done for the Leach-Sholander means [E. B. Leach and C. Sholander, J. Math. Anal. Appl. 104, 390-407 (1984; Zbl 0558.26014)]: \[ \begin{split} E(r,s;x_ 0,...,x_ n)= \\ (\int_{A_ n}(\sum^{n-1}_{k=0}x_ kv_ k)^{s-n}dv_ 0...dv_{n-1}/\int_{A_ n}(\sum^{n-1}_{k=0}x_ kv_ k)^{r- n}dv_ 0...dv_{n-1})^{1/(s-r)}\qquad (r\neq s),\end{split} \]
\[ \begin{split} E(r,r;x_ 0,...,x_ n)= \\ \exp (\int_{A_ n}(\sum^{n}_{k=0}x_ kv_ k)^{r-n}\log (\sum^{n}_{k=0}x_ kv_ k)dv_ 0...dv_{n-1}/\int_{A_ n}(\sum^{n}_{k=0}x_ kv_ k)^{r-n}dv_ 0...dv_{n-1}),\qquad (r=s). \end{split} \]
Reviewer: J.Aczél

MSC:

26D15 Inequalities for sums, series and integrals
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A51 Convexity of real functions in one variable, generalizations
33B15 Gamma, beta and polygamma functions
60E05 Probability distributions: general theory
60F15 Strong limit theorems

Citations:

Zbl 0558.26014
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References:

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