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Article

Keywords:
genetic entropy; α-entropy; random partitions; complete convergence
Summary:
We prove the complete convergence of Shannon’s, paired, genetic and α-entropy for random partitions of the unit segment. We also derive exact expressions for expectations and variances of the above entropies using special functions.
References:
[1] Baum L. E., Katz M.: Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 (1968), 108–123 MR 0198524
[2] Bieniek M., Szynal D.: A contribution to results on random partitions of the segment. Internat. J. Pure and Appl. Math. 13 (2004), 3, 337–378 MR 2057117 | Zbl 1059.62051
[3] Burbea N., Rao N.: Entropy differential metrics and divergence measures in probability spaces: a unified approach. J. Multivariate Anal. 12 (1982), 575–596 MR 0680530
[4] Darling D. A.: On a class of problems related to the random division of an interval. Ann. Math. Statist. 24 (1953), 239–253 MR 0058891 | Zbl 0053.09902
[5] Erdős P.: On a theorem of Hsu and Robbins. Ann. Math. Statist. 20 (1949), 286–291 MR 0030714
[6] Feller W.: An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York 1966 MR 0210154 | Zbl 0598.60003
[7] Goldstein S.: On entropy of random partitions of the segment $[0,1]$. Bull. Soc. Sci. Lett. Łódź XXIV 4 (1974), 1–7 MR 0448500
[8] Gradstein I. S., Ryzyk I. M.: Tables of Integrals, Sums, Series and Products. Fourth edition. Academic Press, New York – London 1965
[9] Graham R. L., Knuth D. E., Patashnik O.: Concrete Mathematics. Addison–Wesley Publishing Company Advanced Book Program, Reading, MA 1989 MR 1001562 | Zbl 0836.00001
[10] Ekstörm M.: Sum-functions of spacings of increasing order. J. Statist. Plann. Inference 136 (2006), 2535–2546 MR 2279820
[11] Hall P.: Limit theorems for sums of general functions of $m$-spacings. Math. Proc. Cambridge Philos. Soc. 96 (1984), 517–532 MR 0757846 | Zbl 0559.62013
[12] Hall P.: On power distributional tests based on sample spacings. J. Multivariate Anal. 19 (1986), 201–224 MR 0853053
[13] Hansen E. R.: A Table of Series and Products. Prentice-Hall, Englewood Clifts, N. J. 1975 Zbl 0438.00001
[14] Havrda J., Charvát F.: Quantification method in classification process: Concept of structural $\alpha $-entropy. Kybernetika 3 (1967), 30–35 MR 0209067
[15] Heyde C. C.: A suplement to the strong law of large numbers. J. Appl. Probab. 12 (1975), 173–175 MR 0368116
[16] Hsu P. L., Robbins H.: Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 25–31 MR 0019852 | Zbl 0030.20101
[17] Latter B. D. H.: Measures of genetic distance between indviduals and populations. Publ. Univ. Hawai, Honolulu, Genetic Structure of Populations (1973), 27–39
[18] Menendez M. L., Morales D., Pardo, L., Salicrú M.: Asymptotic distribution of $\lbrace h,\phi \rbrace $-entropies. Comm. Statist. – Theory Methods 22 (1993), 7, 2015–2031 MR 1238377
[19] Misra N.: A new test if uniformity based on overlapping sample spacings. Comm. Statist. – Theory Methods 30 (2001), 7, 1435–1470 MR 1861865
[20] Renyi A.: New nonadditive measures of entropy for discrete probability distributions. In: Proc. 4th Berkeley Symp. Math. Statist. and Prob. Vol. 1, 1961, pp. 547–561 MR 0132570
[21] Shannon C. E.: A mathematical theory of communications. Bell System Tech. J. 27 (1948), 379–425, 623–656 MR 0026286
[22] Shao Y., Jimenez R.: Entropy for random partitons and its applications. J. Theoret. Probab. 11 (1998), 417–433 MR 1622579
[23] Slud E.: Entropy and maximal spacings for random partitions. Z. Warsch. verw. Gebiete 41 (1978), 341–352 MR 0488242 | Zbl 0353.60019
[24] Temme N. M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York 1996 MR 1376370 | Zbl 0856.33001
[25] Srivastava H. M., Tu S.-T., Wu T.-C.: Some combinatorial series identities associated with the Digamma function and harmonic numbers. Appl. Math. Lett. 13 (2000), 101–106 MR 1755751 | Zbl 0953.33001
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