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Generalized binomial law and regularly varying moments. (English) Zbl 1053.62018

The author defines a random variable \(X_n\) having the generalized binomial law in the following way: \(P\{X_n=k\}=p_{nk}c^k/P_n(c)\), \(k\leq n\), \(k,n\in N\cup\{0\}\), where \(P_n(c)=\sum_{k\leq n}p_{nk}c^k\) is a polynomial with non-positive zeros. An interesting asymptotic behavior of regularly varying moments for this law is proved, i.e., that the asymptotic behavior of the first moment determines the behavior of all subsequent moments: \(E(K_\rho(X_n))\sim K_\rho(E(X_n))\), as \(E(X_n)\to\infty\) (\(n\to\infty\)), where \(\rho>0\), and \(K_\rho\) is a regularly varying function with index \(\rho\).

MSC:

62E10 Characterization and structure theory of statistical distributions
62E20 Asymptotic distribution theory in statistics
60E99 Distribution theory
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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