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Large deviations for random power moment problem. (English) Zbl 1062.60023

Authors’ summary: We consider the set \(M_n\) of all \(n\)-truncated power moment sequences of probability measures on \([0,1]\). We endow this set with the uniform probability. Picking randomly a point in \(M_n\), we show that the upper canonical measure associated with this point satisfies a large deviation principle. Moderate deviations are also studied completing earlier results on asymptotic normality given by F.-C. Chang, J. H. B. Kemperman and W. J. Studden [Ann. Probab. 21, 1295-1309 (1993; Zbl 0778.60010)]. Surprisingly, our large deviations results allow us to compute explicitly the \((n+1)\)th moment range size of the set of all probability measures having the same \(n\) first moments. The main tool to obtain these results is the representation of \(M_n\) on canonical moments [see the book of H. Dette and W. J. Studden, “The theory of canonical moments with applications in statistics, probability, and analysis” (1997; Zbl 0886.62002)].

MSC:

60F10 Large deviations
60F05 Central limit and other weak theorems
60E05 Probability distributions: general theory
60B10 Convergence of probability measures
60A10 Probabilistic measure theory
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