# Bounds and Approximations for the Distribution of a Sum of Lognormal Random
Variables

Fredrik Berggren The author is with Huawei Technologies Sweden AB, Stockholm,
Sweden. (e-mail: fredrik.b@huawei.com).

###### Abstract

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.

###### Index Terms:

Bound, shadow fading, Mellin transform, sum of lognormal random variables,
tangential mean.

##  I Introduction

The sum of lognormal random variables (RVs) appears in various problems, e.g.,
in cellular radio communications where it is used to model the received co-
channel interference power, or the received signal power from different
diversity branches, subject to lognormal shadow fading. Based on the
distribution, it is possible to determine the probability of link outage,
i.e., that the signal power, or signal-to-interference ratio (SIR) falls below
a given threshold. No closed-form expression has been presented for its
cumulative distribution function (CDF) and plenty of research papers
concerning the properties of the CDF have been published for this long-
standing problem. For example, some methods approximate the sum with another
lognormal RV whose moments or cumulants are matched to those of the sum of RVs
[1], [2],[3]. Furthermore, moment matching by numerical integration of the
moment generating function (MGF) [4] [5] or of the characteristic function
(CF) [6] has been proposed. Direct numerical integration of the Laplace
transform [7] has also been exploited. A good fit to the CDF by approximations
based on other probability distributions has also been reported, e.g.,
[8],[9],[10],[11], [12],[13].

Another direction is to bound the CDF. In [14], order statistics was used to
derive upper- and lower bounds on the CDF for a sum of $N$ independent
lognormal RVs, where the bounds were given either by Q-functions or on
integral form. This methodology was extended in [15] to bounds for the cases
of $N=2$ or $N=3$ arbitrarily correlated lognormal RVs, or of $N$ equally
correlated lognormal RVs, where the bounds were given on integral form. The
geometric mean-arithmetic mean inequality was utilized in [16] to obtain an
upper bound on the CDF, described by a Q-function, for any $N$ and for RVs
with arbitrary correlation. Lower bounds on the CDF have been given for the
special case $N=2$ expressed by the Q-function [17],[18] or by the Marcum
Q-function [19]. In [20], a bound was given on the error between the CDF of an
approximation distribution obtained by moment matching and the CDF of the sum
of lognormal RVs.

In this work, we will derive a new bound on the CDF, and from this bound we
will also derive new approximations to the CDF. The contributions of this work
are summarized as:

  * •

We utilize the lesser-known tangential mean-arithmetic mean inequality to
derive a novel upper bound on the CDF for a sum of $N$ independent lognormal
RVs. Our method replaces the sum of lognormal RVs with a product of shifted
lognormal RVs. Remarkably, the bound is tight in the sense that it will
converge to the desired CDF as the shift goes to infinity.

  * •

We use the Mellin transform together with numerical integration in order to
compute the bound. At the left tail of the CDF, we find that the bound can be
expressed by a Q-function.

  * •

We propose novel simple approximations to the CDF expressed by a product of
Q-functions. The approximations are more accurate than existing ones, e.g.,
Farley’s method.

Sec. II contains the derivation of the bound and a discussion about its
tightness. Approximations to the CDF are contained in Sec. III and numerical
evaluations are given in Sec. IV, while Sec. V concludes the paper.

##  II Bound on the CDF and its Left Tail Distribution

###  II-A Preliminaries

Let $X_{i}$ be a normal RV with probability density function (pdf)

$f_{X_{i}}(x)=\frac{1}{\sqrt{2\pi}\sigma_{i}}e^{-\frac{(x-\mu_{i})^{2}}{2\sigma_{i}^{2}}},\,-\infty<x<\infty,\,\sigma>0$
(1)

which implies that $e^{X_{i}}$ is a lognormal RV. Our first objective is to
give a bound for the CDF of the sum of $N$ lognormal RVs,

<EMAIL_ADDRESS>(2)

The CDF is as ususal defined by
$F_{S_{N}}(\gamma)=\int_{0}^{\gamma}f_{S_{N}}(x)dx$, where the pdf
$f_{S_{N}}(x)$ does not exist on closed-form. In wireless communications, the
outage probability is the probability that the SIR falls below a threshold,
$\gamma_{\mathrm{th}}$. For lognormal shadow fading, where the received power
is modelled by $e^{X_{i}}$, this can be expressed as

$\displaystyle\mathrm{Pr}\left[\frac{e^{X_{0}}}{\tsum\slimits@_{i=1}^{N}e^{X_{i}}}\leq\gamma_{\mathrm{th}}\right]$
$\displaystyle=\mathrm{Pr}\left[\tsum\slimits@_{i=1}^{N}e^{X_{i}}\geq\frac{e^{X_{0}}}{\gamma_{\mathrm{th}}}\right]$
(3)
$\displaystyle=1-F_{S_{N}}\left(\frac{e^{X_{0}}}{\gamma_{\mathrm{th}}}\right).$
(4)

The l.h.s. is the complementary CDF (CCDF) of $S_{N}$.

The following results will be utilized for deriving the bound. For any set of
positive integers, $\mathbf{y}=(y_{1},y_{2},\ldots,y_{N})$, the arithmetic
mean (AM), the geometric mean (GM) and the tangential mean111The general
definition is
$\mathrm{TM}(\delta)=(\delta+y_{1})^{\alpha_{1}}\ldots(\delta+y_{N})^{\alpha_{N}}-\delta$
where $\alpha_{1}+\alpha_{2}+\ldots+\alpha_{N}=1$. (TM) are defined as:

$\displaystyle\mathrm{AM}(\mathbf{y})$
$\displaystyle=\frac{1}{N}\tsum\slimits@_{i=1}^{N}y_{i}$ (5)
$\displaystyle\mathrm{GM}(\mathbf{y})$
$\displaystyle=\left(\tprod\slimits@_{i=1}^{N}y_{i}\right)^{\frac{1}{N}}$ (6)
$\displaystyle\mathrm{TM}(\mathbf{y},\delta)$
$\displaystyle=\left(\tprod\slimits@_{i=1}^{N}(\delta+y_{i})\right)^{\frac{1}{N}}-\delta,\,\delta>0$
(7)

It has been shown that the following inequality holds [21],

$\mathrm{GM}(\mathbf{y})\le\mathrm{TM}(\mathbf{y},\delta)\le\mathrm{AM}(\mathbf{y})$
(8)

where equality is achieved only when $y_{1}=y_{2}=\ldots=y_{N}$. Moreover, it
was proven in [21] that TM$(\mathbf{y},\delta)$ is a monotonically increasing
function of $\delta$, and it can be shown that (see Appendix)

$\lim_{\delta\rightarrow\infty}\mathrm{TM}(\mathbf{y},\delta)=\mathrm{AM}(\mathbf{y}).$
(9)

Since $\mathrm{AM}=S_{N}/N$, the RV $N\cdot\mathrm{TM}(e^{\mathbf{X}},\delta)$
with $\mathbf{X}=(X_{1},X_{2},\ldots,X_{N})$, will approach $S_{N}$ for large
$\delta$.

###  II-B A Bound on the CDF

Now, let us define the shifted lognormal RV, $Y_{i}=\delta+e^{X_{i}}$, which
has a pdf given by

$f_{Y_{i}}(y,\delta)=\frac{1}{\sqrt{2\pi}\sigma_{i}(y-\delta)}e^{-\frac{(\ln(y-\delta)-\mu_{i})^{2}}{2\sigma_{i}^{2}}},\,y>\delta,\delta\ge
0.$ (10)

The shift $\delta$ describes a linear translation of the standard lognormal
pdf. By definition
$F_{Y_{i}}(\gamma,\delta)=\int_{\delta}^{\gamma}f_{Y_{i}}(y,\delta)dy$ and the
CDF becomes

$F_{Y_{i}}(\gamma,\delta)=\begin{cases}0,&\gamma\le\delta\\\
1-Q\left(\frac{\ln(\gamma-\delta)-\mu_{i}}{\sigma_{i}}\right),&\gamma>\delta\end{cases}$
(11)

where $Q(x)=1/(\sqrt{2\pi}\sigma)\int_{x}e^{-\frac{y^{2}}{2}}dy$. The moments
can be defined from dB units by ${\sigma=\lambda\sigma_{\mathrm{dB}}}$ and
${\mu=\lambda\mu_{\mathrm{dB}}}$, with ${\lambda=\ln(10)/10\approx 0.23026}$
[6].

Moreover, define the product of shifted lognormal RVs,

$Z_{N}=\tprod\slimits@_{i=1}^{N}Y_{i},$ (12)

and its CDF,

$F_{Z_{N}}(\gamma,\delta)=\int_{\delta^{N}}^{\gamma}f_{Z_{N}}(x,\delta)dx.$
(13)

By using (7) and ${\mathrm{AM}=S_{N}/N}$, our proposed upper bound on the CDF
follows.

$F_{S_{N}}(\gamma)\le
F_{Z_{N}}\left(\left(\frac{\gamma}{N}+\delta\right)^{N},\delta\right),\,\gamma\geq
0$ (14)

It should be remarked that we have not used any independence assumption for
the RVs to arrive at (14). Hence, the bound holds also for correlated
lognormal RVs. However, numerical evaluation of (14) with correlated RVs
appears to be a nontrivial task and we do not consider this case further
herein.

###  II-C Tightness of the Bound

Obviously due to (8), (14) is tighter than the bound in [16], which is based
on the GM. Interestingly, (14) is asymptotically tight, since (9) implies that

$\lim_{\delta\rightarrow\infty}F_{Z_{N}}\left(\left(\frac{\gamma}{N}+\delta\right)^{N},\delta\right)=F_{S_{N}}(\gamma).$
(15)

The CDF can be obtained through (16), which by variable substitution reduces
to (17).

$\displaystyle\lim_{\delta\rightarrow\infty}F_{Z_{N}}\left(\left(\frac{\gamma}{N}+\delta\right)^{N},\delta\right)$
$\displaystyle=\lim_{\delta\rightarrow\infty}\int_{\delta^{N}}^{\left(\frac{\gamma}{N}+\delta\right)^{N}}f_{Z_{N}}(x,\delta)dx$
(16)
$\displaystyle=\lim_{\delta\rightarrow\infty}\int_{0}^{\gamma}\left(\frac{x}{N}+\delta\right)^{N-1}f_{Z_{N}}\left(\left(\frac{x}{N}+\delta\right)^{N},\delta\right)dx$
(17)

By taking the derivative w.r.t. to $\gamma$, it follows that the integrand in
(17) will converge to $f_{S_{N}}(x)$ as $\delta\rightarrow\infty$, i.e.,

$f_{S_{N}}(x)=\lim_{\delta\rightarrow\infty}\left(\frac{x}{N}+\delta\right)^{N-1}f_{Z_{N}}\left(\left(\frac{x}{N}+\delta\right)^{N},\delta\right).$
(18)

Furthermore, it is straightforward to show that for all the means

$\min_{i}y_{i}\le\mathrm{GM}(\mathbf{y}),\mathrm{TM}(\mathbf{y},\delta),\mathrm{AM}(\mathbf{y})\le\max_{i}y_{i}.$
(19)

Therefore, we have

$\mathrm{AM}(\mathbf{y})-\mathrm{GM}(\mathbf{y})\le\max_{i}y_{i}-\min_{i}y_{i}$
(20)

which implies that the gap between the AM and the GM (and therefore also the
gap between the AM and the TM) closes when the maximum difference between the
elements in $\mathbf{y}$ approaches 0. Thus, firstly, the bound (14) will
become tighter with smaller variances $\sigma_{i}^{2}$, since the r.h.s of
(20) will then on average become smaller. Secondly, for small $\gamma$, all
$y_{i}$ will be small, i.e., $\epsilon=\max_{i}y_{i}-\min_{i}y_{i}$ will be
small. Thus using (8), it follows from the squeeze theorem in calculus that

$\displaystyle\lim_{\epsilon\rightarrow
0}\mathrm{AM}(\mathbf{y})-\mathrm{GM}(\mathbf{y})$
$\displaystyle=\lim_{\epsilon\rightarrow
0}\mathrm{AM}(\mathbf{y})-\mathrm{TM}(\mathbf{y},\delta)$ (21)
$\displaystyle=0.$ (22)

Hence, (14) will approach the bound from [16] for small $\gamma$, regardless
of the $\delta$ value. If we define $V_{i}=e^{X_{i}}$ and
$W_{N}=\tprod\slimits@_{i=1}^{N}V_{i}$, we have the bound from [16]

$F_{W_{N}}(x)=1-Q\left(\frac{\ln x-\bar{\mu}}{\bar{\sigma}}\right)$ (23)

where
$\bar{\mu}=\mathbb{E}[\tsum\slimits@_{i=1}^{N}X_{i}]=\tsum\slimits@_{i=1}^{N}\mu_{i}$
and
$\bar{\sigma}^{2}=\mathbb{E}[\left(\tsum\slimits@_{i=1}^{N}X_{i}-\bar{\mu}\right)^{2}]$,
where $\mathbb{E}$ is the expectation value operator. Taking the limit, due to
(21), we obtain

$\displaystyle\lim_{\gamma\rightarrow
0}F_{Z_{N}}\left(\left(\frac{\gamma}{N}+\delta\right)^{N},\delta\right)$
$\displaystyle=\lim_{\gamma\rightarrow
0}F_{W_{N}}\left(\left(\frac{\gamma}{N}\right)^{N}\right)$ (24)
$\displaystyle=\lim_{\gamma\rightarrow
0}1-Q\left(\frac{N\ln(\gamma/N)-\bar{\mu}}{\bar{\sigma}}\right).$ (25)

We shall therefore expect that the bound (14) is well approximated by (25) for
small $\gamma$, i.e., at the left tail of the CDF. We can then rewrite (25)
for small $\gamma$ as

$F_{Z_{N}}(\gamma,\delta)\approx
1-Q\left(\frac{\ln(\gamma)-\ln(N)-\bar{\mu}/N}{\bar{\sigma}/N}\right).$ (26)

and identify that the r.h.s. of (26) describes the mean and variance of a
lognormal RV with parameters

$\displaystyle\hat{\mu}$
$\displaystyle=\ln(N)+\tsum\slimits@_{i=1}^{N}\mu_{i}/N$ (27)
$\displaystyle\hat{\sigma}^{2}$
<EMAIL_ADDRESS>(28)

###  II-D Computation of the Bound

The bound can be computed for the case of independent RVs as follows. The pdf
for a product of independent RVs can be determined by multiplicative
convolution (aka Mellin convolution) or by Mellin transforms [22]. Since the
$Y_{i}$ are independent, the pdf $f_{Z_{N}}(x)$ follows from $N$-times
repeated Mellin convolution, and by defining $f_{Z_{1}}(y)=f_{Y}(y,\delta)$,
and using the definition of Mellin convolution [22], we obtain:

$\displaystyle f_{Z_{n}}(x)$
$\displaystyle=\int_{\delta^{n-1}}^{\frac{x}{\delta}}f_{Y}\left(\frac{x}{y},\delta\right)\frac{f_{Z_{n-1}}(y)}{y}dy,x\ge\delta^{n},\,n\ge
2$ (29)

Thus, the CDF could be obtained by computing $N$ nested finite range integrals
according to (29). Alternatively, the computation could be done in the
transform domain. The Mellin transform for (10) is defined for complex values
$s=\alpha+\mathrm{j}\beta$, where $\mathrm{j}=\sqrt{-1}$, by

$\phi_{Y}(s)=\int_{\delta}y^{s-1}f_{Y}(y,\delta)dy.$ (30)

The largest interval $a<\alpha<b$ where (30) converges is denoted as the
fundamental strip. When $y\rightarrow\delta^{+}$ we have

$f_{Y}(y,\delta)\sim((y-\delta)(y-\delta)^{\ln(y-\delta)})^{-1}$ (31)

and when $y\rightarrow\infty$, we have

$f_{Y}(y,\delta)\sim(yy^{\ln(y)})^{-1}$ (32)

so

$f_{Y}(y,\delta)\underset{y\rightarrow\delta^{+}}{=}\mathcal{O}((y-\delta)^{-\ln(y-\delta)}),\,f_{Y}(y,\delta)\underset{y\rightarrow\infty}{=}\mathcal{O}(y^{-\ln(y)})$
(33)

and since $\lim_{y\rightarrow\delta^{+}}\ln(y-\delta)=-\infty$ and
$\lim_{y\rightarrow\infty}\ln(y)=\infty$, we have $a=-\infty$ and $b=\infty$
(cf. Lemma 1 [23]). The Mellin transform of the $N$-times Mellin convolution
(29) is the product of the $N$ Mellin transforms [22]. The inversion of the
Mellin transform is unique and is given by the line integral for any $\alpha$
where $a<\alpha<b$

$f_{Z_{N}}(x)=\frac{1}{2\pi\mathrm{j}}\int_{\alpha-\mathrm{j}\cdot\infty}^{\alpha+\mathrm{j}\cdot\infty}x^{-s}\phi_{Y}(s)^{N}ds$
(34)

which is thus admitted for any real value $\alpha$. There is no closed-form
expression for $\phi_{Y}(s)$ and numerical integration of (30) and (34) will
be required to determine
$F_{Z_{N}}\left(\left(\frac{\gamma}{N}+\delta\right)^{N}\right)$. For
convenience of notation, (29) and (34) are given for $\mu_{i}=\mu$ and
$\sigma_{i}=\sigma$ but it is straightforward to generalize to non-uniform
parameters $\mu_{i}$ and $\sigma_{i}$.

Fig. 1 depicts the Mellin transform222In our evaluations, numerical
integration by an adaptive quadrature method is used for (30) and a
trapezoidal method is used for (34). as function of $\beta$ for a fixed value
of $\alpha$. The decay of $\phi(s)$ becomes slower as $\delta$ increases,
which implies that a larger range of $\beta$ values needs to be evaluated in
order to have sufficiently small integration error in (34).

Figure 1: The real part (solid blue) and imaginary part (dashed red) of the
Mellin transform $\phi(\alpha+\mathrm{j}\beta)$ for $\alpha=1$, for $\delta=2$
and $\delta=10$, $\sigma=1$ and $\sigma=2$, with $\mu=0$.

##  III Approximations of the CDF

We next seek to find simple approximations to the r.h.s. of (14). These
expressions cannot be formally proven to be upper bounds on the CDF, although
they may practically become so for a large range of $\gamma$ values. Hence,
they are referred to as approximations of the CDF.

###  III-A Approximation for $N=2$

For $N=2$, we have $Z_{2}=Y_{1}Y_{2}$ and the CDF becomes

$F_{Z_{2}}(x)=\int_{\delta}^{\frac{x}{\delta}}F_{Y}\left(\frac{x}{y},\delta\right)f_{Y}(y,\delta)dy,\,x\ge\delta^{2}$
(35)

where we for convenience of notation have assumed $\sigma_{i}=\sigma$ and
$\mu_{i}=\mu,\,\forall i$. Therefore, we obtain

$\displaystyle F_{Z_{2}}\left(\left(\frac{\gamma}{2}+\delta\right)^{2}\right)$
$\displaystyle=\int_{\delta}^{\frac{\left(\frac{\gamma}{2}+\delta\right)^{2}}{\delta}}F_{Y}\left(\frac{1}{y}\left(\frac{\gamma}{2}+\delta\right)^{2},\delta\right)f_{Y}(y,\delta)dy$
(36)
$\displaystyle=\int_{0}^{\frac{\left(\frac{\gamma}{2}+\delta\right)^{2}}{\delta}-\delta}\left(1-Q\left(\frac{\ln\left(\frac{\left(\frac{\gamma}{2}+\delta\right)^{2}}{\delta+y}-\delta\right)-\mu}{\sigma}\right)\right)f_{Y}(y,0)dy.$
(37)

where (11) was used in (37). Since

$\lim_{\delta\rightarrow\infty}\frac{\left(\frac{\gamma}{2}+\delta\right)^{2}}{y+\delta}-\delta=\gamma-y$
(38)

it is straightforward to verify that (37) asymptotically reduces to
$F_{S_{2}}(\gamma)=\int_{0}^{\gamma}F_{Y}(\gamma-y,0)f_{Y}(y,0)dy$, i.e., the
CDF resulting from the integration for the CDF obtained by a convolution of
$f_{Y}(y,0)$. By change of integration variable, we have the equivalent
representation

$\lim_{\delta\rightarrow\infty}F_{Z_{2}}\left(\left(\frac{\gamma}{2}+\delta\right)^{2}\right)=\int_{-\infty}^{\ln(\gamma)}\mathop{\mathchoice{\vtop{\halign{#\cr$\hfil\displaystyle\left(1-Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma}\right)\right)\hfil$\crcr\kern
2.0pt\nointerlineskip\cr\@mathmeasure\displaystyle{ \left(1-
Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma} \right)
\right)}\@mathmeasure\displaystyle{\upbrace}\@mathmeasure\displaystyle{\upbraceg}\@mathmeasure\displaystyle{\upbracegg}\@mathmeasure\displaystyle{\upbraceggg}\@mathmeasure\displaystyle{\upbracegggg}$\displaystyle\bracelu\leaders\hbox{$\bracemid$}\hfill\bracemu\leaders\hbox{$\bracemid$}\hfill\braceru$\crcr}}}{\vtop{\halign{#\cr$\hfil\textstyle\left(1-Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma}\right)\right)\hfil$\crcr\kern
2.0pt\nointerlineskip\cr\@mathmeasure\textstyle{ \left(1-
Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma} \right)
\right)}\@mathmeasure\textstyle{\upbrace}\@mathmeasure\textstyle{\upbraceg}\@mathmeasure\textstyle{\upbracegg}\@mathmeasure\textstyle{\upbraceggg}\@mathmeasure\textstyle{\upbracegggg}$\textstyle\bracelu\leaders\hbox{$\bracemid$}\hfill\bracemu\leaders\hbox{$\bracemid$}\hfill\braceru$\crcr}}}{\vtop{\halign{#\cr$\hfil\scriptstyle\left(1-Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma}\right)\right)\hfil$\crcr\kern
2.0pt\nointerlineskip\cr\@mathmeasure\scriptstyle{ \left(1-
Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma} \right)
\right)}\@mathmeasure\scriptstyle{\upbrace}\@mathmeasure\scriptstyle{\upbraceg}\@mathmeasure\scriptstyle{\upbracegg}\@mathmeasure\scriptstyle{\upbraceggg}\@mathmeasure\scriptstyle{\upbracegggg}$\scriptstyle\bracelu\leaders\hbox{$\bracemid$}\hfill\bracemu\leaders\hbox{$\bracemid$}\hfill\braceru$\crcr}}}{\vtop{\halign{#\cr$\hfil\scriptscriptstyle\left(1-Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma}\right)\right)\hfil$\crcr\kern
2.0pt\nointerlineskip\cr\@mathmeasure\scriptscriptstyle{ \left(1-
Q\left(\frac{\ln(\gamma-e^{x})-\mu}{\sigma} \right)
\right)}\@mathmeasure\scriptscriptstyle{\upbrace}\@mathmeasure\scriptscriptstyle{\upbraceg}\@mathmeasure\scriptscriptstyle{\upbracegg}\@mathmeasure\scriptscriptstyle{\upbraceggg}\@mathmeasure\scriptscriptstyle{\upbracegggg}$\scriptscriptstyle\bracelu\leaders\hbox{$\bracemid$}\hfill\bracemu\leaders\hbox{$\bracemid$}\hfill\braceru$\crcr}}}}\limits_{g(x)}f_{X}(x)dx.$
(39)

Focusing on the case $\delta\rightarrow\infty$, where the bound is tight, an
approximation to the CDF can be obtained on closed-form (i.e., not requiring
integration) from (39) by the conventional approximation
$\mathbb{E}g(x)\approx g(\mathbb{E}x)$, where the expectation operator
$\mathbb{E}$ is with respect to the RV $x$ with pdf
$h_{2}(x)=f_{X}(x)/C_{2}(\gamma)$. The normalization factor

$\displaystyle C_{2}(\gamma)$
$\displaystyle=\int_{-\infty}^{\ln(\gamma)}f_{X}(x)dx$ (40)
$\displaystyle=1-Q\left(\frac{\ln(\gamma)-\mu}{\sigma}\right)$ (41)

is needed such that $h_{2}(x)$ is a pdf on the interval
$-\infty<x\le\ln(\gamma)$. Then, we get the approximation

$\displaystyle F_{S_{2}}(\gamma)\approx
C_{2}(\gamma)\left(1-Q\left(\frac{\ln(\gamma-e^{\mu_{2}(\gamma)})-\mu}{\sigma}\right)\right)$
(42)

where $\mu_{2}=\mathbb{E}x$, which equals

$\displaystyle\mu_{2}(\gamma)$
$\displaystyle=\int_{-\infty}^{\ln(\gamma)}xh_{2}(x)dx$ (43)
$\displaystyle=\frac{1}{C_{2}(\gamma)}\left(\mu\left(1-Q\left(\frac{\ln(\gamma)-\mu}{\sigma}\right)\right)-\frac{\sigma}{\sqrt{2\pi}}e^{-\frac{(\ln(\gamma)-\mu)^{2}}{2\sigma^{2}}}\right)$
(44)
$\displaystyle=\mu-\frac{\sigma}{\sqrt{2\pi}C_{2}(\gamma)}e^{-\frac{(\ln(\gamma)-\mu)^{2}}{2\sigma^{2}}}$
(45)

An interpretation of (42) is that the CDF is the product between the CDF of
the lognormal RV and the CDF of a shifted lognormal RV, where the shift
$e^{\mu_{2}(\gamma)}$ is not fixed but depends on $\gamma$.

To assess whether the r.h.s. of (42) may become an upper bound, we inspect the
derivatives of $g(x)$,

$\displaystyle\frac{dg(x)}{dx}$
$\displaystyle=-\frac{e^{x-\frac{(\ln(\gamma-e^{x})-\mu)^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma(\gamma-e^{x})}$
(46) $\displaystyle\frac{d^{2}g(x)}{dx^{2}}$
$\displaystyle=-\frac{e^{x-\frac{(\ln(\gamma-e^{x})-\mu)^{2}}{2\sigma^{2}}}(\gamma\sigma^{2}+e^{x}\ln(\gamma-e^{x})-\mu
e^{x}))}{\sqrt{2\pi}\sigma^{3}(\gamma-e^{x})^{2}}$ (47)

leading to that $dg(x)/dx$ is negative on the interval
$-\infty<x\le\ln(\gamma)$ and that it is a monotonically non-increasing
function for $-\infty<x\le x_{0}$, where $x_{0}$ is the solution to

$\gamma\sigma^{2}+e^{x_{0}}\ln(\gamma-e^{x_{0}})-\mu e^{x_{0}}=0.$ (48)

Thus, $g(x)$ is a concave function on the interval $-\infty<x\le x_{0}$.
Therefore, if $\ln(\gamma)-x_{0}$ is relatively small, the r.h.s. of (42) may
practically become an upper bound for a large range of $\gamma$ values since
Jensen’s inequality implies that $\mathbb{E}g(x)\le g(\mathbb{E}x)$ for
$-\infty<x\le x_{0}$, albeit $g(x)$ is not concave on $x_{0}<x\le\ln(\gamma)$.

###  III-B Approximation for $N>2$

The approximation method of Sec. III.A can be applied repeatedly for $N>2$ and
we will show this explicitly for $N=3$. By defining
$\gamma_{2}=\gamma-e^{\mu_{2}(\gamma)}$ and using $S_{3}=S_{2}+e^{X_{3}}$ with
(42), it follows that

$\displaystyle F_{S_{3}}(\gamma)$ $\displaystyle\approx
C_{2}(\gamma)\\!\\!\\!\\!\\!\\!\int\displaylimits_{-\infty}^{\ln(\gamma_{2})}\left(1-Q\left(\frac{\ln(\gamma_{2}-e^{x})-\mu}{\sigma}\right)\right)f_{X}(x)dx$
(49) $\displaystyle\approx
C_{3}(\gamma)C_{2}(\gamma)\left(1-Q\left(\frac{\ln(\gamma_{2}-e^{\mu_{3}(\gamma)})-\mu}{\sigma}\right)\right)$
(50)

where the approximation in (49) is due to the approximation of
$F_{S_{2}}(\gamma)$ and (50) is due to $\mathbb{E}g(x)\approx g(\mathbb{E}x)$,
where the RV $x$ has a pdf $h_{3}(x)=f_{X}(x)/C_{3}(\gamma)$, and the
normalization factor

$\displaystyle C_{3}(\gamma)$
$\displaystyle=\int\displaylimits_{-\infty}^{\ln(\gamma_{2})}f_{X}(x)dx$ (51)
$\displaystyle=1-Q\left(\frac{\ln(\gamma_{2})-\mu}{\sigma}\right)$ (52)

is needed such that $h_{3}(x)$ is a pdf on the interval
$-\infty<x\le\ln(\gamma_{2})$. Furthermore, we obtain

$\displaystyle\mu_{3}(\gamma)$
$\displaystyle=\int\displaylimits_{-\infty}^{\ln(\gamma_{2})}xh_{3}(x)dx$ (53)
$\displaystyle=\frac{1}{C_{3}(\gamma)}\left(\mu\left(1-Q\left(\frac{\ln(\gamma_{2})-\mu}{\sigma}\right)\right)-\frac{\sigma}{\sqrt{2\pi}}e^{-\frac{(\ln(\gamma_{2})-\mu)^{2}}{2\sigma^{2}}}\right)$
(54)
$\displaystyle=\mu-\frac{\sigma}{\sqrt{2\pi}C_{3}(\gamma)}e^{-\frac{(\ln(\gamma_{2})-\mu)^{2}}{2\sigma^{2}}}$
(55)

The error in the approximation is due to reusing the pdf for $N=2$ in (49) and
thereafter using $\mathbb{E}g(x)\approx g(\mathbb{E}x)$ again. Thus, the
approximation may be less accurate as $N$ increases. The above steps can be
applied repeatedly to obtain $\gamma_{N}$, $\mu_{N}$ and $C_{N}$ for $N>3$,
which could be used to derive an approximation as for (50).

###  III-C Approximation for large $N$

Let us define the RV $\tilde{Z}_{N}=Z_{N}^{\frac{1}{N}}$ and assume that $N$
is large. Since

$\displaystyle\ln\left(\tilde{Z}_{N}\right)=\frac{1}{N}\tsum\slimits@_{i=1}^{N}\ln(Y_{i}),$
(56)

a central limit theorem (CLT) implies that, as $N\rightarrow\infty$,
$\ln(\tilde{Z}_{N})$ will be a normal RV with mean $\tilde{\mu}$ and variance
$\tilde{\sigma}^{2}$. Therefore, $\tilde{Z}_{N}$ will converge to a lognormal
RV. The mean and variance can be determined according to the CLT through

$\displaystyle\tilde{\mu}$
$\displaystyle=\int_{-\infty}\ln(\delta+e^{x})f_{X}(x)dx$ (57)
$\displaystyle\approx\frac{1}{\sqrt{\pi}}\tsum\slimits@_{m=1}^{M}w_{m}\ln(\delta+e^{\sqrt{2}\sigma
x_{m}+\mu})$ (58) $\displaystyle\tilde{\sigma}^{2}$
$\displaystyle=\frac{1}{N}\left(\int_{-\infty}\ln(\delta+e^{x})^{2}f_{X}(x)dx-\tilde{\mu}^{2}\right)$
(59)
$\displaystyle\approx\frac{1}{N}\left(\frac{1}{\sqrt{\pi}}\tsum\slimits@_{m=1}^{M}w_{m}\ln(\delta+e^{\sqrt{2}\sigma
x_{m}+\mu})^{2}-\tilde{\mu}^{2}\right)$ (60)

where (58) and (60) are due to Gauss-Hermite integration of order $M$. The
weights $w_{m}$ and abscissas $x_{m}$ can be found in [24]. By using (7) and
AM=$S_{N}/N$, we can therefore make the following approximation for large $N$.

$\displaystyle F_{Z_{N}}\left(\left(\frac{\gamma}{N}+\delta\right)^{N}\right)$
$\displaystyle\approx F_{\tilde{Z}_{N}}\left(\frac{\gamma}{N}+\delta\right)$
(61)
$\displaystyle=1-Q\left(\frac{\ln\left(\frac{\gamma}{N}+\delta\right)-\tilde{\mu}}{\tilde{\sigma}}\right)$
(62)

##  IV Numerical Examples

###  IV-A Evaluation of the Left Tail

Fig. 2 shows the l.h.s. of (26) and the CDF, which is obtained by numerical
integration, for the case of $N=2$, $\sigma=1$ and $\mu=0$. The distance
between the curves is almost constant, but since the plot is in logarithmic
scale, it implies that the actual difference between the two curves decreases
as $\gamma$ decreases. Hence, it confirms the bound (14) can for small
$\gamma$ be approximated by the bound of [16]. However, asymptotically as
$\gamma\rightarrow 0$, the distribution of $S_{N}$ may not be that of a
lognormal RV [25]. As a reference case the lower bound [(10a)-(10c) of [19]],
which applies for $N=2$ and is based on the Marcum-Q function is included,
which is even tighter.

Figure 2: The CDF obtained by numerical integration and the lognormal
approximation for $N=2$, $\sigma=1$ and $\mu=0$.

###  IV-B Evaluation of the Bound

We plot the CCDF, hence the hitherto derived upper bounds on the CDF are
displayed as lower bounds on the CCDF. Additionally, the CDF is plotted in
Fig. 5 in order to be able to compare bounds at low $\gamma$. For comparison,
we use previously published lower bounds on the CCDF. For example, the r.h.s.
of the following inequality is plotted

$1-F_{S_{N}}(\gamma)\ge
1-\left(1-Q\left(\frac{\ln(\gamma)-\mu}{\sigma}\right)\right)^{N}$ (63)

which is a special case of [14] and is also referred to as Farley’s method
[1]. Notably, the bound of [16] is based on the GM and it will be worse than
(14), so it is not included. For $N=6$, we also include the improved bound
[(eq. (11) in [14]], which is a two-dimensional integral that is evaluated
numerically. The distribution of $S_{N}$ obtained by Monte-Carlo simulations
is thereto contained in the plots.

We select $\delta=10$ and $\delta=100$ in order to demonstrate the bounds, and
apply $\sigma=1\,(\sigma_{\mathrm{dB}}=4.34)$ and
$\sigma=2\,(\sigma_{\mathrm{dB}}=8.69)$. Typical vaules of
$\sigma_{\mathrm{dB}}$ in radio channels are in the range 4–12 dB. From Fig. 3
and Fig. 4, as expected, it can be observed that the bound is tighter for
smaller $\sigma$. Morover, as $N$ increases, a larger value of $\delta$ is
required in order to close the gap to the desired CCDF, especially for large
$\sigma$. For small $\gamma$, the bound is accurate in general. For the
evaluated values of $\delta$, the new bounds are shown to be better than (63)
for small $\gamma$, while (63) is better for large $\gamma$. Fig. 5 shows that
the proposed bound is very accurate at the left tail of the CDF, whereas the
previous bounds are not good at all.

Figure 3: CCDF for the proposed bound (14) with $\delta=10$ and $\delta=100$,
$N=2$ and $N=6$, for $\sigma=1$ with $\mu=0$. Figure 4: CCDF for the proposed
bound (14) with $\delta=10$ and $\delta=100$, $N=2$ and $N=6$, for $\sigma=2$
with $\mu=0$. Figure 5: CDF for the proposed bound (14) with $\delta=10$ and
$\delta=100$, $N=2$ and $N=6$, for $\sigma=1$ with $\mu=0$.

###  IV-C Evaluation of the Approximations

Fig. 6 contains the approximation for $N=2$ and shows that (42) appears as a
lower bound and that it is better than Farley’s method (63). In particular,
the difference is significant for small $\sigma$. We solve for $x_{0}$
numerically and the subplot includes $\epsilon=\ln(\gamma)-x_{0}$, which shows
that $\epsilon$ decreases with increasing $\gamma$ and increasing $\sigma$. If
$\epsilon$ is fairly small, $g(x)$ is concave over most of the integration
interval which leads to that (42) practically appears as an upper bound. Fig.
7 shows that the approximation (50) for $N=3$ is accurate for large $\gamma$,
i.e., the right tail, and for large $\sigma$. The inset plot shows that for
small $\gamma$ and small $\sigma$, the approximation (50) intersects with the
CDF and it is thus not a lower bound. These results show that the suggested
approximation works well, at least when $N$ is moderately large.

For the case of large $N$, since the product of $N$ RVs is replaced with a
single RV, the inequality (8) cannot be used and the approximation (62) is not
necessarily a lower bound on the CCDF for an arbitrary $\delta$. This can be
observed from Fig. 8, wherein the curve for $\delta=100$ intersects with the
desired CCDF, where we have used $N=30$. The inset plot shows that, for the
fixed value $\gamma=70$, (62) is an increasing function of $\delta$ but it
does not converge to the actual value of the CCDF. Notably, the previous
bounds are not accurate at all for such large $N$ and do not work well.

Figure 6: CCDF of approximation (42) for $N=2$ for $\sigma=0.5,1$ and 2 with
$\mu=0$. The inset shows the difference $\epsilon=\ln(\gamma)-x_{0}$ for
different $\sigma$. Figure 7: CCDF of approximation (42) for $N=3$ for
$\sigma=0.5,1$ and 2 with $\mu=0$. The inset shows a zoom in for the case of
$\sigma=0.5$. Figure 8: CCDF of approximation (62) for $\delta=10$ and
$\delta=100$, for $N=30$, $\sigma=1$ and $\mu=0$. The inset shows the CCDF for
$\gamma=70$ as function of $\delta$ and where the dashed line is the value
obtained by simulation.

##  V Conclusions

We have derived an upper bound on the CDF of a sum of lognormal RVs which
becomes tight and converges to the CDF for large values of the shift $\delta$.
Thus, it is more accurate as $\delta$ increases and will outperform previously
suggested upper bounds and approximation methods. Evaluation is done by
numerical integration. The price of its accuracy is that it may require a
larger computational cost in terms of numerical integration effort, than
approximation methods, e.g., the ones based on moment matching or those using
other pdfs than the lognormal, or other bounds. The bound can be approximated
by a single Q-function at the left tail. Furthermore, we gave simple
approximations to the bound on closed-form by means of products of
Q-functions. These approximations were shown to be more accurate than, e.g.,
Farley’s method, at least for moderately large $N$. An implication of this
work is that the classical problem of the sum of lognormal RVs could
alternatively be viewed as a problem of a product of shifted lognormal RVs
with large shifts.

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First, rewrite the $N$th root:

$\displaystyle\left(\tprod\slimits@_{i=1}^{N}(\delta+y_{i})\right)^{\frac{1}{N}}$
$\displaystyle=\delta\left(1+\frac{y_{1}}{\delta}\right)^{\frac{1}{N}}\left(1+\frac{y_{2}}{\delta}\right)^{\frac{1}{N}}\ldots\left(1+\frac{y_{N}}{\delta}\right)^{\frac{1}{N}}$
(64)

Furthermore, the Taylor series at $t=0$ can be obtained as

$\left(1+t\right)^{\frac{1}{N}}=1+\frac{t}{N}+\frac{1}{N}\left(\frac{1}{N}-1\right)\frac{t^{2}}{2}+\mathcal{O}(t^{3}).$
(65)

Utilizing the first two terms of (65) and expanding (64), we obtain

$\displaystyle\lim_{\delta\rightarrow\infty}\left(\tprod\slimits@_{i=1}^{N}(\delta+y_{i})\right)^{\frac{1}{N}}\\!\\!\\!\\!-\delta=$
$\displaystyle\lim_{\delta\rightarrow\infty}\delta\left(1+\frac{y_{1}}{N\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right)\left(1+\frac{y_{2}}{N\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right)\ldots\left(1+\frac{y_{N}}{N\delta}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)\right)-\delta$
(66) $\displaystyle=$
$\displaystyle\lim_{\delta\rightarrow\infty}\frac{y_{1}+y_{2}+\ldots+y_{N}}{N}+\mathcal{O}\left(\frac{1}{\delta^{2}}\right)$
(67) $\displaystyle=$ $\displaystyle\frac{y_{1}+y_{2}+\ldots+y_{N}}{N}.$ (68)