Log-Shifted Gamma Approximation to Lognormal Sum Distributions

This paper proposes the log-shifted gamma (LSG) approximation to model the sum of M lognormally distributed random variables (RVs). The closed-form probability density function of the resulting LSG RV is presented, and its parameters are directly derived from those of the M individual lognormal RVs by using an iterative moment-matching technique without the need for curve fitting of computer-generated distributions. Simulation and analytical results on the cumulative distribution function (cdf) of the sum of M lognormal RVs in different conditions indicate that the proposed LSG approximation can provide better accuracy than other lognormal approximations over a wide cdf range, especially for large M and/or standard deviation.

[1]  S. Schwartz,et al.  On the distribution function and moments of power sums with log-normal components , 1982, The Bell System Technical Journal.

[2]  Qi Tu Zhang,et al.  Co-channel interference analysis for mobile radio suffering lognormal shadowed Nakagami fading , 1999 .

[3]  M. Romeo,et al.  Broad distribution effects in sums of lognormal random variables , 2002, physics/0211065.

[4]  Norman C. Beaulieu,et al.  Estimating the distribution of a sum of independent lognormal random variables , 1995, IEEE Trans. Commun..

[5]  Slimane Ben Slimane,et al.  Bounds on the distribution of a sum of independent lognormal random variables , 2001, IEEE Trans. Commun..

[6]  Norman C. Beaulieu,et al.  Comparison of methods of computing correlated lognormal sum distributions and outages for digital wireless applications , 1994, Proceedings of IEEE Vehicular Technology Conference (VTC).

[7]  Yoshihiro Ishikawa,et al.  Capacity Design and Performance of Call Admission Control in Cellular CDMA Systems , 1997, IEEE J. Sel. Areas Commun..

[8]  L. Fenton.  The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems , 1960 .

[9]  R. Leipnik,et al.  On lognormal random variables: I-the characteristic function , 1991, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[10]  P. Cardieri,et al.  Statistics of the sum of lognormal variables in wireless communications , 2000, VTC2000-Spring. 2000 IEEE 51st Vehicular Technology Conference Proceedings (Cat. No.00CH37026).

[11]  R. Barakat.  Sums of independent lognormally distributed random variables , 1976 .

[12]  A. Abu-Dayya,et al.  Outage probabilities in the presence of correlated lognormal interferers , 1994 .

[13]  S. Posner,et al.  Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution , 1998 .

[14]  Norman C. Beaulieu,et al.  An optimal lognormal approximation to lognormal sum distributions , 2004, IEEE Transactions on Vehicular Technology.

[15]  Vijay K. Bhargava,et al.  Simple and accurate methods for outage analysis in cellular mobile radio systems-a unified approach , 2001, IEEE Trans. Commun..

[16]  D. C. Schleher,et al.  Generalized Gram-Charlier series with application to the sum of log-normal variates (Corresp.) , 1977, IEEE Trans. Inf. Theory.

[17]  William A. Janos,et al.  Tail of the distribution of sums of log-normal variates , 1970, IEEE Trans. Inf. Theory.

[18]  A. Safak,et al.  Statistical analysis of the power sum of multiple correlated log-normal components , 1993 .

[19]  P. G. Bradley,et al.  A Stochastic Model for Determining the Economic Prospects of Petroleum Exploration over Large Regions , 1970 .