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A triptych of discrete distributions related to the stable law. (English) Zbl 0794.60007

Summary: We derive useful distributional representations for three discrete laws: the discrete stable distribution of F. W. Steutel and K. Van Harn [Ann. Probab. 7, 893-899 (1979; Zbl 0418.60020)], the discrete Linnik distribution introduced by A. Pakes (unpublished) and a distribution of M. Sibuya [Ann. Inst. Stat. Math. 31, 373-390 (1979; Zbl 0448.62008)]. These representations may be used to obtain simple uniformly fast random variate generators.

MSC:

60E07 Infinitely divisible distributions; stable distributions
65C10 Random number generation in numerical analysis
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[1] Ahrens, J. H.; Dieter, U., Sampling from binomial and Poisson distributions: a method with bounded computation times, Computing, 25, 193-208 (1980) · Zbl 0424.60018
[2] Ahrens, J. H.; Dieter, U., Computer generation of Poisson deviates from modified normal distributions, ACM Trans. Math. Software, 8, 163-179 (1982) · Zbl 0485.65006
[3] Ahrens, J. H.; Dieter, U., Generating gamma variates by a modified rejection technique, Comm. ACM, 25, 47-54 (1982) · Zbl 0472.65005
[4] Ahrens, J. H.; Kohrt, K. D.; Dieter, U., Algorithm 599. Sampling from gamma and Poisson distributions, ACM Trans. Math. Software, 9, 255-257 (1983) · Zbl 0513.65097
[5] Best, D. J., Letter to the editor, Appl. Statist., 27, 181 (1978)
[6] Cheng, R. C.H.; Feast, G. M., Some simple gamma variate generators, Appl. Statist., 28, 290-295 (1979) · Zbl 0433.65005
[7] Cheng, R. C.H.; Feast, G. M., Gamma variate generators with increased shape parameter range, Comm. ACM, 23, 389-393 (1980) · Zbl 0439.60029
[8] Devroye, L., Non-Uniform Random Variate Generation (1986), Springer: Springer New York · Zbl 0593.65005
[9] Devroye, L., Random variate generation for the digamma and trigamma distributions, J. Statist. Comput. Simul., 43, 197-216 (1992)
[10] Irwin, J. O., The generalized Waring distribution, J. Roy. Statist. Soc. Sec. A, 138, 18-31 (1975), part I
[11] Johnson, N. L.; Kotz, S., Developments in discrete distributions, Internat. Statist. Rev., 50, 71-101 (1982), 1969-1980 · Zbl 0497.62020
[12] Kanter, M., Stable densities under change of scale and total variation inequalities, Ann. Probab., 3, 697-707 (1975) · Zbl 0323.60013
[13] Linnik, Yu. V., Linear forms and statistical criteria: I, II, Selected Transl. Math. Statist. Probab., 3, 1-40 (1962)
[14] Linnik, Yu. V., Linear forms and statistical criteria: I, II, Selected Transl. Math. Statist. Probab., 3, 41-90 (1962)
[15] Marsaglia, G., The squeeze method for generating gamma variates, Comput. Math. Appl., 3, 321-325 (1977) · Zbl 0384.65005
[16] Minh, D. Le, Generating gamma variates, ACM Trans. Math. Software, 14, 261-266 (1988) · Zbl 0648.65003
[17] Pakes, A. Personal communication (1993).; Pakes, A. Personal communication (1993).
[18] Schmeiser, B. W.; Kachitvichyanukul, V., (Poisson random variate generation. Poisson random variate generation, Res. Mem. (1981), School of Indust. Engrg., Purdue Univ: School of Indust. Engrg., Purdue Univ West Lafayette, IN), 81-84
[19] Shimizu, R., Generalized hypergeometric distribution, Proc. Inst. Statist. Math., 16, 147-165 (1968) · Zbl 0207.18802
[20] Sibuya, M., Generalized hypergeometric, digamma and trigamma distributions, Ann. Inst. Statist. Math., 31, 373-390 (1979) · Zbl 0448.62008
[21] Sibuya, M.; Shimizu, R., The generalized hypergeometric family of distributions, Ann. Inst. Statist. Math., 33, 177-190 (1981) · Zbl 0474.62021
[22] Stadlober, E., The ratio of uniforms approach for generating discrete random varietes, J. Comput. Appl. Math., 31, 181-189 (1990) · Zbl 0701.65007
[23] Steutel, F. W.; Van Harn, K., Discrete analogues of self-decomposability and stability, Ann. Probab., 7, 893-899 (1979) · Zbl 0418.60020
[24] Zolotarev, V. M., One-Dimensional Stable Distributions (1986), Amer. Math. Soc: Amer. Math. Soc Providence, RI · Zbl 0589.60015
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