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Deviation probabilities for randomized decomposable statistics in a polynomial scheme. (English. Russian original) Zbl 0662.60041
Sov. Math., Dokl. 36, 583-585 (1988); translation from Dokl. Akad. Nauk SSSR 297, 1062-1064 (1987).
Consider a randomized decomposable statistic, i.e. \(R_ N=\sum^{N}_{1}f_{mN}(\nu_ m)\), where \(\nu =(\nu_ 1,...,\nu_ N)\) has a polynomial distribution \(M(n;p_ 1,...,p_ n)\) and \(f_{mN}(x)\) is a random function of the nonnegative integer argument x, \(m=1,...,N.\)
The authors present four theorems (without proofs) establishing large deviation asymptotics of \(P(R_ N>x(Var R_ N)^{1/2})\) in the zones \(x=O((\log N)^{1/2})\), \(x=o(N^{1/6})\), \(x=o(N^{1/2})\), and \(x\sim const N^{1/2}\), under the corresponding conditions on the functions \(f_{mN}(\cdot)\).
Reviewer: J.Steinebach
60F10 Large deviations
62E20 Asymptotic distribution theory in statistics