A sum of lognormal random variables (RVs) appears in many problems of science and engineering. For example, it is invloved in computing the distribution of recevied signal and interference powers for radio channels subject to lognormal shadow fading. Its distribution has no closed-from expression and it is typically characterized by approximations, asymptotes or bounds. We give a novel upper bound on the cumulative distribution function (CDF) of a sum of $N$ lognormal RVs. The bound is derived from the tangential mean-arithmetic mean inequality. By using the tangential mean, our method replaces the sum of $N$ lognormal RVs with a product of $N$ shifted lognormal RVs. It is shown that the bound can be made arbitrarily close to the desired CDF, and thus it becomes more accurate than any other bound or approximation, as the shift approaches infinity. The bound is computed by numerical integration, for which we introduce the Mellin transform, which is applicable to products of RVs. At the left tail of the CDF, the bound can be expressed by a single Q-function. Moreover, we derive simple new approximations to the CDF, expressed as a product $N$ Q-functions, which are more accurate than the previous method of Farley.

翻译：对数正态随机变量之和出现在许多科学与工程问题中。例如，它涉及计算受对数正态阴影衰落影响的无线信道中接收信号与干扰功率的分布。该分布无闭式表达式，通常通过近似、渐近线或界来表征。本文给出了$N$个对数正态随机变量之和的累积分布函数的一个新上界。该界基于切线均值-算术平均不等式推导得出。通过使用切线均值，我们的方法将$N$个对数正态随机变量之和替换为$N$个平移对数正态随机变量的乘积。结果表明，当平移参数趋于无穷时，该界可任意接近目标累积分布函数，从而比任何其他界或近似更精确。该界通过数值积分计算，为此我们引入适用于随机变量乘积的梅林变换。在累积分布函数的左尾，该界可由单个Q函数表示。此外，我们推导了新的简单近似，表示为$N$个Q函数的乘积，其精度优于Farley的先前方法。