Accurate computation of the MGF of the lognormal distribution and its application to sum of lognormals

IEEE Transactions on Communications

Volumn 58, Issue 5, 2010, Pages 1568-1577

  Tellambura, C ^a   Senaratne, D ^a  

^a UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Characteristic function; Moment generating function; Sum of lognormals

Indexed keywords

ACCURACY LEVEL; ANALYTICAL EXPRESSIONS; CHARACTERISTIC FUNCTIONS; CLOSED FORM; COMPUTATIONAL LOADS; CONVERGENCE ACCELERATION; CUMULATIVE DISTRIBUTION FUNCTION; EPSILON-ALGORITHM; GENERATING FUNCTIONS; INTEGRATION RULE; INTEGRATION TECHNIQUES; LAMBERT FUNCTION; LOG-NORMAL; LOG-NORMAL DISTRIBUTION; LOG-NORMAL RANDOM VARIABLE; LOGNORMAL SUMS; MOMENT GENERATING FUNCTION; SADDLE POINT; SCIENCE AND ENGINEERING; SIGNIFICANT DIGITS; TRAPEZOIDAL RULES; WIRELESS COMMUNICATIONS;

CONVERGENCE OF NUMERICAL METHODS; DISTRIBUTION FUNCTIONS; FUNCTION EVALUATION; RANDOM VARIABLES; WIRELESS TELECOMMUNICATION SYSTEMS;

STEEPEST DESCENT METHOD;

EID: 77952562336     PISSN: 00906778     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCOMM.2010.05.080640     Document Type: Article

References (43)

1. E. Limpert, W. A. Stahel, and M. Abbt, "Log-normal distributions across the sciences: keys and clues, " BioScience, vol. 51, no. 5, pp. 341-352, May 2001. [Online]. Available: http://www.ingentaconnect.com/content/ aibs/bio/2001
2. J. Aitchison and J. A. C. Brown, The Lognormal Distribution, with Special Reference to Its Uses in Economics. Cambridge University Press, 1957.
3. E. L. Crow and K. Shimizu, Eds., Lognormal Distributions: Theory and Application. CRC, 1987.
4. G. L. Stüber, Principles of Mobile Communication (2nd ed.). Norwell, MA: Kluwer Academic Publishers, 2001.
5. C. Fischione, F. Graziosi, and F. Santucci, "Approximation for a sum of on-off lognormal processes with wireless applications, " IEEE Trans. Commun., vol. 55, no. 10, pp. 1984-1993, Oct. 2007.
6. M. Pratesi, F. Santucci, and F. Graziosi, "Generalized moment matching for the linear combination of lognormal rvs: application to outage analysis in wireless systems, " IEEE Trans. Wireless Commun., vol. 5, no. 5, pp. 1122-1132, May 2006.
7. M. Pratesi, F. Santucci, F. Graziosi, and M. Ruggieri, "Outage analysis in mobile radio systems with generically correlated log-normal interferers, " IEEE Trans. Commun., vol. 48, no. 3, pp. 381-385, Mar. 2000.
8. C. Tellambura, "Bounds on the distribution of a sum of correlated lognormal random variables and their application, " IEEE Trans. Commun., vol. 56, no. 8, pp. 1241-1248, Aug. 2008.
9. J. Zhang and V. Aalo, "Effect of macrodiversity on average-error probabilities in a Rician fading channel with correlated lognormal shadowing, " IEEE Trans. Commun., vol. 49, no. 1, pp. 14-18, Jan. 2001.
10. T. Piboongungon and V. A. Aalo, "Outage probability of L-branch selection combining in correlated lognormal fading channels, " Electron. Lett., vol. 40, pp. 886-888, July 2004.
11. L. Fenton, "The sum of log-normal probability distributions in scatter transmission systems, " IRE Trans. Commun. Syst., vol. 8, no. 1, pp. 57-67, Mar. 1960.
12. S. C. Schwartz and Y. S. Yeh, "On the distribution function and moments of power sums with log-normal components, " Bell Syst. Tech. J., vol. 61, no. 7, pp. 1441-1462, 1982.
13. D. Schleher, "Generalized Gram-Charlier series with application to the sum of log-normal variates (corresp.), " IEEE Trans. Inf. Theory, vol. 23, no. 2, pp. 275-280, Mar. 1977.
14. N. Marlow, "A normal limit theorem for power sums of independent random variables, " B. S. T. J., vol. 46, pp. 2081-2090, Nov. 1967.
15. I. Nasell, "Some properties of power sums of truncated normal random variables, " B. S. T. J., vol. 46, pp. 2091-2110, Nov. 1967.
16. J. C. S. S. Filho, P. Cardieri, and M. D. Yacoub, "Simple accurate lognormal approximation to lognormal sums, " Electron. Lett., vol. 41, pp. 1016-1017, Sep. 2005.
17. N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, "Approximating a sum of random variables with a lognormal, " IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2690-2699, July 2007.
18. A. Ligeti, "Outage probability in the presence of correlated lognormal useful and interfering components, " IEEE Commun. Lett., vol. 4, no. 1, pp. 15-17, Jan. 2000.
19. N. C. Beaulieu, A. A. Abu-Dayya, and P. J. McLane, "Estimating the distribution of a sum of independent lognormal random variables, " IEEE Trans. Commun., vol. 43, no. 12, p. 2869, Dec. 1995.
20. S. Szyszkowicz and H. Yanikomeroglu, "On the tails of the distribution of the sum of lognormals, " in Proc. IEEE International Conf. Commun. ICC '07, June 2007, pp. 5324-5329.
21. Z. Liu, J. Almhana, and R. McGorman, "Approximating lognormal sum distributions with power lognormal distributions, " IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 2611-2617, July 2008.
22. Z. Liu, J. Almhana, F. Wang, and R. McGorman, "Mixture lognormal approximations to lognormal sum distributions, " IEEE Commun. Lett., vol. 11, no. 9, pp. 711-713, Sept. 2007.
23. N. C. Beaulieu and F. Rajwani, "Highly accurate simple closed-form approximations to lognormal sum distributions and densities, " IEEE Commun. Lett., vol. 8, no. 12, pp. 709-711, Dec. 2004. (Pubitemid 40085291)
24. C. L. J. Lam and T. Le-Ngoc, "Estimation of typical sum of lognormal random variables using log shifted gamma approximation, " IEEE Commun. Lett., vol. 10, no. 4, pp. 234-235, Apr. 2006.
25. -, "Log-shifted gamma approximation to lognormal sum distributions, " IEEE Trans. Veh. Technol., vol. 56, pp. 2121-2129, July 2007.
26. H. Nie and S. Chen, "Lognormal sum approximation with type IV Pearson distribution, " IEEE Commun. Lett., vol. 11, no. 10, pp. 790-792, Oct. 2007.
27. L. Zhao and J. Ding, "Least squares approximations to lognormal sum distributions, " IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 991-997, Mar. 2007. (Pubitemid 46562967)
28. -, "A strict approach to approximating lognormal sum distributions, " in Proc. Canadian Conf. Electrical Computer Eng. CCECE '06, May 2006, pp. 916-919.
29. N. C. Beaulieu and Q. Xie, "An optimal lognormal approximation to lognormal sum distributions, " IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 479-489, Mar. 2004.
30. A. Gil, J. Segura, and N. M. Temme, Numerical Methods for Special Functions. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2007.
31. J. A. Gubner, "A new formula for lognormal characteristic functions, " IEEE Trans. Veh. Technol., vol. 55, no. 5, pp. 1668-1671, Sep. 2006.
32. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, "On the Lambert W function, " Adv. Comput. Math., vol. 5, pp. 329-359, 1996.
33. T. Ooura and M. Mori, "A robust double exponential formula for Fourier-type integrals, " J. Comput. Appl. Math., vol. 112, no. 1-2, pp. 229-241, 1999, numerical evaluation of integrals.
34. I. M. Longman, "A method for the numerical evaluation of finite integrals of oscillatory functions, " Math. Comput., vol. 14, pp. 53-59, 1960.
35. -, "Note on a method for computing infinite integrals of oscillatory functions, " Proc. Cambridge Philos. Soc., vol. 52, pp. 764-768, 1956.
36. D. Shanks, "Non-linear transformations of divergent and slowly convergent sequences, " J. Math. Phys., vol. 34, pp. 1-42, 1955.
37. n) tranformation, " Math. Tables Aids Comput., vol. 10, pp. 91-96, 1956.
38. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1970.
39. J. D. Murray, Asymptotic Analysis, 2nd edition, Ser. Applied Mathematical Sciences. New York: Springer-Verlag, 1984, vol. 48.
40. C. W. Helstrom, "Evaluating the detectability of Gaussian stochastic signals by steepest descent integration, " IEEE Trans. Aerosp. Electron. Syst., vol. 19, no. 3, pp. 428-437, May 1983.
41. C. W. Helstrom and J. A. Ritcey, "Evaluating radar detection probabilities by steepest descent integration, " IEEE Trans. Aerosp. Electron. Syst., vol. 20, no. 5, pp. 624-634, Sep. 1984.
42. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd edition. Cambridge University Press, 2007, The Art of Scientific Computing.
43. F. Bornemann, D. Laurie, S. Wagon, and J. Waldvogel, The SIAM 100-digit Challenge. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2004, A Study in High-Accuracy Numerical Computing, with a foreword by David H. Bailey.