Bounds on the distribution of a sum of independent lognormal random variables

Abstract

The distribution function of a sum of lognormal random variables appears in several communication problems. Approximations are usually used for such distribution as no closed form nor bounds exist. Bounds can be very useful in assessing the performance of any given system. In this paper, we derive upper and lower bounds on the distribution function of a sum of independent lognormal random variables. These bounds are given in a closed form and can be used in studying the performance of cellular radio and broadcasting systems. KeywordsShadow fading channels, lognormal random variable, cochannel interference, upper bound, lower bound, outage probability, cumulative distribution function.

Fitting the Modified-Power-Lognormal to the Sum of Independent Lognormals Distribution

“…We can also categorise the methods based on their complexity: most methods require extensive numerical integration, and many are iterative. Only [9], [10], [16] are closed-form, and [18] is mostly closed-form, similarly to our method.…”

“…In Figures 1-3, we present Monte-Carlo simulations of the SLN cdf, along with three methods for approximating it: the proposed MPLN method, moment-matching [2], [9], [29] (using the first and second moments), and Ben Slimane's method (7) [16]. In the context of inter-cell co-channel interference, typical values of σ i are between 6 and 12 dB.…”

“…Farley's method was extended by Ben Slimane [16] to include non-identical independent summands. Like Farley's, it is based on the fact that the sum of positive random variables is always greater or equal than the greatest of these random variables.…”

“…These methods can be categorised by their scope into three classes: generally correlated [9]- [15], independent [16]- [21], independent and identically distributed [22]- [24] Y i 's. We can also categorise the methods based on their complexity: most methods require extensive numerical integration, and many are iterative.…”

“…We use the 3-parameter Modified-Power-Lognormal (MPLN) distribution as our approximating function, which has been previously proposed for approximating the SLN cdf in various ways [3], [5], [13], [16], [22]. In Section II, we summarise the many arguments, some novel, for the choice of this particular distribution.…”