Structural properties of generalised Planck distributions. (English) Zbl 1486.60028
Summary: A family of generalised Planck (GP) laws is defined and its structural properties explored. Sometimes subject to parameter restrictions, a GP law is a randomly scaled gamma law; it arises as the equilibrium law of a perturbed version of the Feller mean reverting diffusion; the density functions can be decreasing, unimodal or bimodal; it is infinitely divisible. It is argued that the GP law is not a generalised gamma convolution. Characterisations are obtained in terms of invariance under random contraction of a weighted version of a related law. The GP law is a particular instance of equilibrium laws obtained from a recursion suggested by a genetic mutation-selection balance model. Some related infinitely divisible laws are exhibited.
MSC:
| 60E05 | Probability distributions: general theory |
| 62E10 | Characterization and structure theory of statistical distributions |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60J60 | Diffusion processes |
| 92D10 | Genetics and epigenetics |
Keywords:
Planck distribution; mean reverting diffusion; modal properties; infinite divisibility and self-decomposability; length biased and weighted laws; characterisation; mutation-selection balanceSoftware:
DLMFReferences:
| [1] | Berg, C., On powers of Stieltjes moment sequences, I. J. Theor. Prob., 18, 871-889 (2005) · Zbl 1086.44003 · doi:10.1007/s10959-005-7530-6 |
| [2] | Brigo, D.; Mercurio, F., Interest Rate Models - Theory and Practice (2001), Berlin: Springer, Berlin · Zbl 1038.91040 · doi:10.1007/978-3-662-04553-4 |
| [3] | Bondesson, L., On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative half line, Scand. Actuar. J., 3-4, 225-247 (1987) · Zbl 0649.60013 · doi:10.1080/03461238.1987.10413830 |
| [4] | Bondesson, L., Generalized Gamma Convolutions and Related Classes of Distributions and Densities (1992), New York: Springer-Verlag, New York · Zbl 0756.60015 · doi:10.1007/978-1-4612-2948-3 |
| [5] | Bondesson, L., A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables, J. Theor. Probab, 28, 1063-1081 (2015) · Zbl 1375.60053 · doi:10.1007/s10959-013-0523-y |
| [6] | Cox, J.; Ingersoll, J.; Ross, S., A theory of the term structure of interest rates, Econometrica, 53, 385-408 (1985) · Zbl 1274.91447 · doi:10.2307/1911242 |
| [7] | Cropper, W. H., The Quantum Physicists (1970), New York: O.U.P., New York |
| [8] | Davis, H. T., The Theory of Econometrics (1941), Bloomington: Principia Press, Bloomington · Zbl 0060.32104 |
| [9] | Dennis, B.; Patil, G. P., The gamma distribution and weighted multimodal gamma distributions as models of population abundance, Math. Biosci., 68, 187-212 (1984) · Zbl 0536.92025 · doi:10.1016/0025-5564(84)90031-2 |
| [10] | Dharmadhikari, S.; Joag-Dev, K., Unimodality, Convexity, and Applications (1998), New York: Academic Press, New York · Zbl 0646.62008 |
| [11] | Erdélyi, A, Functions, HigherTranscendental, Vol. III, (Eds). McGraw-Hill, New York (1955). |
| [12] | Feller, W., Two singular diffusion problems, Ann. Math., 54, 173-182 (1951) · Zbl 0045.04901 · doi:10.2307/1969318 |
| [13] | Garsia, A. M.; Orey, S.; Rodemich, E., Asymptotic behaviour of successive coefficients of some power series, Illinois J. Math., 6, 620-629 (1962) · Zbl 0107.28103 · doi:10.1215/ijm/1255632709 |
| [14] | Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products (1980), New York: Academic Press, New York · Zbl 0521.33001 |
| [15] | Gupta, P. L.; Gupta, R. C.; Ong, S. -H.; Srivasta, H. M., A class of Hurwitz-Lerch distributions and their applications in reliability., Appl. Math. Comput., 196, 521-531 (2008) · Zbl 1131.62093 |
| [16] | van Harn, K.; Steutel, F., Infinite divisibility and the waiting time paradox, Stoch. Model., 11, 527-540 (1995) · Zbl 0843.60020 · doi:10.1080/15326349508807358 |
| [17] | Horn, R. A., On moment sequences and renewal sequences, J. Math. Anal. Appl., 31, 130-135 (1970) · Zbl 0175.16701 · doi:10.1016/0022-247X(70)90123-X |
| [18] | Horsthemke, W.; Lefever, R., Noise Induced Transitions (1984), Berlin: Springer, Berlin · Zbl 0529.60085 |
| [19] | Hu, C. -Y.; Iksanov, A.; Lin, G. D.; Zakusylo, O. K., The Hurwitz zeta distribution, Austral. N.Z.J. Statist., 48, 1-6 (2006) · Zbl 1118.60011 · doi:10.1111/j.1467-842X.2006.00420.x |
| [20] | Johnson, N. L.; Kotz, S., Continuous Univariate Distributions - 2 (1970), Boston: Houghton-Mifflin Co., Boston · Zbl 0213.21101 |
| [21] | Johnson, N. L.; Kemp, A. W.; Kotz, S., Univariate Discrete Distributions, 3rd ed (2005), New York: Wiley, New York · Zbl 1092.62010 · doi:10.1002/0471715816 |
| [22] | Kingman, J. F. C., Regenerative Phenomena (1972), London: Wiley, London · Zbl 0236.60040 |
| [23] | Kingman, J. F. C., A simple model for the balance between selection and mutation, J. Appl. Prob., 15, 1-12 (1978) · Zbl 0382.92003 · doi:10.2307/3213231 |
| [24] | Kittel, C., Thermal Physics (1969), New York: Wiley, New York |
| [25] | Klar, B.; Parthasarathy, P. R.; Henze, N., Zipf and Lerch limit of birth and death processes, Prob. Eng. Info. Sci., 24, 129-144 (2010) · Zbl 1192.60096 · doi:10.1017/S0269964809990179 |
| [26] | Kleiber, C.; Kotz, S., Statistical Size Distributions in Economics and Actuarial Sciences (2003), New York: Wiley-Interscience, New York · Zbl 1044.62014 · doi:10.1002/0471457175 |
| [27] | Kotz, S.; Kozubowski, T. J.; Podgórski, K., The Laplace Distribution and Generalizations (2001), Boston: Birkhäuser, Boston · Zbl 0977.62003 · doi:10.1007/978-1-4612-0173-1 |
| [28] | Lamberton, D.; Lapeyre, B., Introduction to Stochastic Calculus Applied to Finance (1996), London: Chapman-Hall, London · Zbl 1167.60001 |
| [29] | Li, Z., Measure-Valued Branching Processes (2011), Berlin: Springer, Berlin · Zbl 1235.60003 · doi:10.1007/978-3-642-15004-3 |
| [30] | Lin, G. D.; Hu, C. -Y., The Riemann zeta distribution, Bernoulli, 7, 817-828 (2001) · Zbl 0996.60013 · doi:10.2307/3318543 |
| [31] | Longair, M. S., Theoretical Concepts in Physics (1984), Cambridge: C.U.P., Cambridge · Zbl 1433.70001 |
| [32] | Lukacs, E., A characterization of the gamma distribution, Ann. Math. Statist., 26, 319-324 (1965) · Zbl 0065.11103 · doi:10.1214/aoms/1177728549 |
| [33] | Nadarajah, S.; Kotz, S., A generalized Planck distribution, TEST, 15, 361-374 (2006) · Zbl 1110.62015 · doi:10.1007/BF02607057 |
| [34] | Olver, F. W.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), New York: C.U.P., New York · Zbl 1198.00002 |
| [35] | Pais, A., Subtle is the Lord (1982), New York: O.U.P., New York |
| [36] | Pakes, A. G., Length biasing and laws equivalent to the log-normal, J. Math. Anal. Appl., 197, 825-854 (1996) · Zbl 0852.60013 · doi:10.1006/jmaa.1996.0056 |
| [37] | Pakes, A. G., Characterization by invariance under length-biasing and random scaling, J. Statist. Planning Inf., 63, 285-310 (1997) · Zbl 0892.60022 · doi:10.1016/S0378-3758(97)00024-4 |
| [38] | Pakes, A. G.; Navarro, J., Distributional characterizations through scaling relations, Aust. N.Z. J. Stat., 49, 115-135 (2007) · Zbl 1117.62015 · doi:10.1111/j.1467-842X.2007.00468.x |
| [39] | Pakes, A. G.; Sapatinas, T.; Fosam, E. B., Characterizations, length-biasing, and infinite divisibility, Statist. Pap., 37, 53-69 (1996) · Zbl 0847.60010 · doi:10.1007/BF02926159 |
| [40] | Puertas-Centeno, D.; Toranzo, I.; Dehasa, J., Biparametric complexities and generalized Planck radiation law, J. Phys. A: Math. Theor., 50, 505001 (22pp.) (2017) · Zbl 1381.81180 · doi:10.1088/1751-8121/aa95f4 |
| [41] | Sato, K. -I., Lévy Processes and Infinitely Divisible Distributions, Revised ed. (2013), Cambridge: C.U.P, Cambridge · Zbl 1287.60003 |
| [42] | Schilling, R. L.; Song, R.; Vondrac̆ek, Z., Bernstein Functions, 2nd. ed (2012), Berlin: De Gruyter, Berlin · Zbl 1257.33001 · doi:10.1515/9783110269338 |
| [43] | Seshadri, V., Inverse-Gaussian Distributions: A Case Study in Natural Exponential Families (1993), Oxford: Clarendon Press, Oxford |
| [44] | Steutel, F. W.; van Harn, K., Infinite Divisibility of Probability Distributions on the Real Line (2004), New York: Marcel Dekker Inc., New York · Zbl 1063.60001 |
| [45] | Stewart, S. M., Blackbody radiation functions and polylogarithms, J. Quant. Spectrosc. Radiative Trans., 113, 232-238 (2012) · doi:10.1016/j.jqsrt.2011.10.010 |
| [46] | Stuart, A., Gamma-distributed products of independent random variables, Biometrika, 49, 564-565 (1962) · Zbl 0111.34201 · doi:10.1093/biomet/49.3-4.564 |
| [47] | Tomovski, Z̆.; Saxena, R.; Pogány, T., Probability distributions associated with Mathieu type series, ProbStat. Forum, 05, 86-96 (2012) · Zbl 1260.60020 |
| [48] | Turelli, M., Heritable genetic variation via mutation-selection balance: Lerch’s zeta meets the abdominal bristle, Theor. Popul. Biol., 25, 138-193 (1984) · Zbl 0541.92015 · doi:10.1016/0040-5809(84)90017-0 |
| [49] | Varrò, S., Irreducible decomposition of Gaussian distributions and the spectrum of black-body radiation, Physica Scripta, 75, 160-169 (2007) · Zbl 1119.82307 · doi:10.1088/0031-8949/75/2/008 |
| [50] | Valluri, S.; Gil, M.; Jeffrey, J.; Basu, S., The Lambert W function and quantum statistics, J. Math. Phys., 50, 102103 (2009) · Zbl 1248.82006 · doi:10.1063/1.3230482 |
| [51] | Weinberg, S., Lectures on Quantum Mechanics (2013), New York: C.U.P., New York · Zbl 1264.81010 |
| [52] | Wolfe, S., On a continuous analogue of the stochastic difference equation X_n=ρX_n−1+B_n, Stoch. Processes Appl, 12, 301-312 (1982) · Zbl 0482.60062 · doi:10.1016/0304-4149(82)90050-3 |
| [53] | Wong, E., The construction of a class of stationary Markoff processes, Stochastic Processes in Mathematics, Physics and Engineering, Proc. Symp Appl. Math. XVI (1984), Providence: American Mathematical Society, Providence |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
