Probability density function (pdf) for sum of n

correlated lognormal variables is deducted as a special

convolution integral. Pdf for weighted sums (where weights can

be any real numbers) is also presented. The result for four

dimensions was checked by Monte Carlo simulation

Nagy, Tamás

English

Corvinus Research Archive

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1 
  
Abstract— Probability density function (pdf) for sum of n 
correlated lognormal variables is deducted as a special 
convolution integral. Pdf for weighted sums (where weights can 
be any real numbers) is also presented. The result for four 
dimensions was checked by Monte Carlo simulation. 
 
Index Terms— amount of fading, cochannel interference, 
lognormal distribution  
 
I. INTRODUCTION 
HE original problem as stated by Fenton [1] in 1960 is the 
following: “Given several random variables, each with a 
log-normal probability distribution, what is the probability 
distribution of the sum of the random variables?” 
By mathematical notation: let we have n lognormal 
variables, defined by the corresponding correlated 
multidimensional normal distribution (X), where   contains 
expected values and Σ is the covariance matrix. We try to find 
the probability density function (θ) of sum of (correlated) 
lognormal variables (S(C)LN): 












=
)exp(
...
)exp(
)exp(
2
1
nX
X
X
Y
 ( )Σ,N~ µX  ?
1
=





∑
=
n
i
iYθ       
Examples for S(C)LN can be found in almost every science, 
in wireless communications applications include [2]fading 
and shadowing modeling, and assessing cochannel 
interference. In finance the evaluation of exotic (basket, Asian) 
options [3], portfolio level Value at Risk and optimal portfolio 
selection are the most relevant S(C)LN problems. 
Fenton [1] has written about (without detailed 
specifications) a numerical integration solution in case of 
uncorrelated summands. He concentrated on finding a 
approximate solution for uncorrelated case for avoiding (with 
his own words) the “tedious” work of numerical integration. 
He presented an approximation of distribution of uncorrelated 
SLN by moment matching. After Fenton’s initial work almost 
 
Manuscript received January 17, 2011.  
T. Nagy is with Corvinus University of Budapest, Budapest, H-1093 
Hungary (e-mail: tamasnagy1976@gmail.com). 
 
all of the efforts were done trying to define better 
approximations [4]-[7], or lower and/or upper bounds for 
cumulative density function [8-9], [2] and less researches were 
done on numerical integral solution for problem of S(C)LN. 
In the literature there are some direct integral expr ssions 
formulating the pdf or cdf of SCLN. For 2 dimensional 
correlated case Zacks and Tsokos [10] presented a formula of 
characteristic function. Leipnik [11] demonstrated he pdf of n 
dimensional uncorrelated SLN. 
In the following, an exact, convolution integral solution for 
probability density function is presented for n-dimensional 
correlated case, which - based on my best knowledge - was not 
published before. Pdf of weighted sum of n correlated 
lognormals is also provided. 
II.  CONVOLUTION INTEGRAL SOLUTION 
In a general, 2 dimensional independent case, the pdf of two 
convoluted variables is the following: 
( ) gfdxxKgxfKh ⊗=−⋅= ∫
∞
∞−
)()(  (1) 
In n dimensional case there are n-1 integrals, the argument 
of last probability density function is  ∑
−
=
−
1
1
n
i
ixK  
For getting the right pdf of SCLN, few problems areneeded 
to be handled: 
• The random variables are correlated (the correlation 
structure is defined on the level of corresponding normal 
factors) 
• The integral variable is dx, but the argument in lognormal 
pdf is ln(x) 
Based on the linear transformation property [12] any 
multidimensional normal distribution (Σ is the covariance 
matrix) can be built form uncorrelated standard normal 
distribution (Z): 
( )
( ) Tn
T
LLI,0N~
LL,N~
⋅=Σ
⋅+⋅
Z
ZL µµ  (3) 
The formula for reversed way transformation is: 
( )[ ] ZL =−Σ⋅− µµ,N1  (4) 
Let Yi denote the i-th dependent lognormal variable, and zi 
 
 "This work has been submitted to the IEEE for possible  publication. Copyright may be transferred without 
notice, after  which this version may no longer be accessible.” 
(Weighted) Sum of n Correlated Lognormals: 
convolution integral solution (Jan 2011) 
Tamas Nagy 
T 
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2 
the i-th independent standard normal variable: 
( ) ( )












==












=Σ
n
2
1
n
2
1
z
...
z
z
,0NZ
lnY
...
lnY
l
,N I
nY
µ  (5) 
Uncorrelated standard normal variables can be created by 
first decreasing the logarithm of correlated lognormal variables 
by their expected values and then creating linear combination 
by corresponding elements of L-1 matrix. The i-th independent 
standard normal variable can be calculated as follows (whereas  
L-1 matrix is lower triangular, only k ≤ i elements have non 
zero weights): 
( ) ( )[ ] ( ) ( )[ ]∑∑
=
−
=
− −⋅=−⋅=
i
k
kk
n
k
kki YkiLYkiLz
1
1
1
1 ln,ln, µµ  (6) 
A. Joint probability density function 
In the multidimensional dependent lognormal factor space 
the probability density related to a given point equals to the 
probability density related to the corresponding point in the 
multidimensional independent standard normal factor space. 
Joint probability density function of multidimensional 
correlated lognormal variables (f) can be expressed by one 
dimensional standard normal distributions (denoting h 
multivariate standard normal pdf, φ is one dimensional 
standard normal pdf, 1_
−
iL   the ith row of matrix 
1−L  ): 
( ) [ ]( ) ( )[ ]( )∏
=
−− −⋅=−⋅=
n
i
i YLYLhYf
1
1
_
1 lnln µϕµ  (7)  
B. Modifications related to integral variables 
For integrating function f, we have to make modificat ons 
related to integral variables. The relevant question is the 
following: if we change the correlated lognormal variables 
infinitesimally, what will be the n-volume of the subspace 
determined by changing of corresponding uncorrelated 
standard normal variables? 
Let us change the lognormal variables one by one, let iY∆  
denote the change vector, where all of items are 0 but i-th 
element is ∆Y. Vector ig  means the corresponding change 
vector in uncorrelated standard normal factor space c used by 
iY∆ , it can be calculated as follows: 
( )( ) ( )( )µµ −⋅−−∆+⋅= −− YLYYLg ii lnln 11  (8) 
In continuous case (∆Y→0), if 1_
−
iL  is the i-th column of   
matrix, and ∗  denoting element wise multiplication, then: 





∗= −
Y
dY
Lg ii
1
_
 (9) 
Based on g vectors an M matrix can be built, storing the 
changes in space of independent standard normal variables 
caused by one by one changing of correlated lognormal 
variables. M determines an n-dimensional hyper-
parallelepiped, its n-volume (nV) can be calculated as absolute 
value of M’s determinant. Whereas M is triangular matrix, its 
determinant is the product of diagonal elements: 
∏
=
−
⋅=
n
i
i
i
ii dY
Y
L
nV
1
1
 (10) 
C. Probability density function of SCLN  
Based on the result we can construct the pdf of SCLN in n 
dimensional case. First in 3 dimensions (for the sake of 
illustration): 
 
( )
( ) ( )( )( ) ( )( ) ( )( )( )
( )( ) ( )( ) ( )( )( )
0
lnlnln
lnlnln
21
0 0
21321
1
3322
1
3211
1
31
22
1
2211
1
2111
1
11
2121
1
33
1
22
1
11
>−−
⋅⋅−−−⋅+−⋅+−⋅⋅
⋅−⋅+−⋅⋅−⋅⋅
−−⋅⋅
⋅⋅
=
∫ ∫
∞ ∞
−−−
−−−
−−−
YYKfor
dYdYYYKLYLYL
YLYLYL
YYKYY
LLL
KSCLN
µµµϕ
µµϕµϕ
θ
 (11) 
 In n dimensional case the pdf of SCLN will be the 
following (for getting simpler structure let us use a b vector 
containing notations of variables): 
( ) ( )( )
0
,...,,,
ln
1
1
1
1
121
0
1
10 0 1 1
1
1
>−





 −=
⋅










 −⋅⋅=
∑
∑
∫ ∏∫ ∫∏ ∑
−
=
−
=
−
∞ −
=
∞ ∞
= =
−
−
n
i
i
n
i
in
n
i
i
n
i
i
k
kkik
i
ii
SCLN
YKfor
YKYYYbwhere
dYbL
b
L
K µϕθ K
 (12) 
D. Probability density function of weighted SCLN 
(wSCLN) 
We can extend the problem of SCLN to weighted case 
(where weights (α) can be any real numbers). The weighted 
sum of correlated lognormals (wSCLN) problem is typical in 
finance (e.g. a portfolio where we can hold negative quantity 
of asset, and the prices are lognormally distributed): 
?
1
=





∑
=
n
i
iiYαθ  
Pdf of wSCLN can be expressed as follows: 
( ) ( )( )






⋅−=











 ⋅−
=
>




 ⋅−
⋅










 −⋅⋅=
∑
∑
∑
∫ ∏∫ ∫∏ ∑
−
=
−
−
=
−
−
=
∞ −
=
∞ ∞
= =
−
−
1
1
121
1
1
121
1
1
0
1
10 0 1 1
1
1
,...,,,
,...,,,
0
ln
n
i
iin
n
n
i
ii
n
n
n
i
ii
n
i
i
n
i
i
k
kkik
i
ii
wSCLN
YKYYYcwhere
YK
YYYbwhere
YKfor
dYbL
c
L
K
α
α
α
αα
µϕθ K
 (13) 
III.  NUMERICAL VALIDATION  
The numerical validation was done for wSCLN (weighted 
SLCN). The convolution integral was computed by own built 
algorithm. When designing such an algorithm significant 
amount of computational time can be spared by taking the 
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3 
condition in (13) into account. 
For potential values of underlying 
lognormal variables (Yi) symmetric intervals 
were supposed, defined by d times the given 
standard deviation: 
)exp()exp( iiiii dYd σµσµ ⋅+≤≤⋅−  (14) 
An initial lower (ilbi) and upper bound 
(iubi) for the given αiYi can be calculated 
(the min and max functions are required for 
handling the potential negative sign of α):
[ ]
[ ])exp(),exp(max
)exp(),exp(min
iiiiiii
iiiiiii
ddiub
ddilb
σµασµα
σµασµα
⋅+⋅⋅−⋅=
⋅+⋅⋅−⋅=
 (15) 
For further purposes a reversed-way cumulated sums of the 
iubi and iubi vectors were used: 





=





=
∑∑
∑∑
==
==
1
2
11
1
2
11
,....,,
,....,,
iubiubiubrcub
ilbilbilbrclb
i
i
n
i
i
i
i
n
i
i  (16) 
Based on these vectors the potential values of K can be 
restricted (by increasing d, the intervals widen): 
( ) ( )11 rcubKrclb ≤≤  (17) 
If we calculate value of θ(K) by multi-level nested loops, 
and the order of calculation is determined by i, conditional 
(narrowed) bounds for Yi can be determined based on: 
• Known variables (denoting with underlyings): K, Y1, Y2, 
…, Yi-1 
• Potential maximum and minimum values of yet unknow  
variables (Yi+1, Yi+2, ….Yn-1), their cumulated sum stored by 
rclb and rcub vectors  
• Initial bounds for Yi (ilbi and iubi) 
The conditional (narrowed) bounds for a given Yi will be 
the following: 






















−⋅−





−⋅−
≤≤






















−⋅−





−⋅−
∑∑
∑∑
−
=
+
−
=
+
−
=
+
−
=
+
i
i
j
ijji
i
j
ijji
i
i
i
j
ijji
i
j
ijji
rcubYKrclbYKiub
Y
rcubYKrclbYKilb
αααα
αααα
1
1
1
1
1
1
1
1
1
1
1
1
,max,min
,min,max
 (18) 
For comparing the theoretical and simulation result the 
calculation was done in 4 dimensions with the following 
parameters: 












−
−−
−
=












=












=












−
−
=
0.11.02.05.0
1.00.12.03.0
2.02.00.14.0
5.03.04.00.1
1.0
08.0
2.0
10.0
07.0
22.0
08.0
10.0
3.0
8.1
8.0
2.1
ρσµα
  
For calculating the theoretical pdf, own algorithm was used 
(d = 5, number of steps for K = 400, number of step for Y1, 
Y2, Y3 =  80). The simulation method was Monte Carlo (with 
6.000.000 repetitions). 
The simulated histogram and theoretical frequencies w re 
the followings: 
 
 
 
 
As we can see the simulated and theoretical frequencies are 
very close to each other. 
IV.  CONCLUSION 
The exact form of pdf of SCLN and wSCLN can be 
calculated based on convolution integrals.  
In numerical validation a 4 dimension weighted case was 
presented (which can not be handled by simple lognormal 
approximation (logarithm of negatives are not valid)). By 
special designing of numerical integral algorithm significant 
computational time can be spared, the total computation by 
suggested algorithm required 11.5 minutes in a single laptop. 
By using more sophisticated algorithm and powerful computer, 
the exact convolution solution can be a feasible way for also 
high dimensions. 
REFERENCES 
[1] L. F. Fenton, “The sum of lognormal probability distributions in scatter 
transmission systems,” IRE Trans. Commun. Syst., vol. CS-8, pp. 57–
67, Mar. 1960. 
[2] 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. 
[3] M. A. Milevsky and S. E. Posner, “Asian options, the sum of 
lognormals and the reciprocal gamma distribution,” J. Financ. Quant. 
Anal., vol. 33, no. 3, pp. 409–422, Sep. 1998. 
[4] S. C. Schwartz and Y. S.Yeh, “On the distribution fu ction and 
moments of power sums with lognormal components,” Bell Syst. Tech. 
J., vol. 61, pp. 1441–1462, Sept. 1982. 
[5] N. C. Beaulieu and Q. Xie, “An optimal lognormal approximation to 
lognormal sum distribution,” IEEE Trans. Veh. Technol., vol. 53, no. 2, 
pp. 479–489, Mar. 2004. 
[6] 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, Jul. 2007. 
[7] Q. T. Zhang and S. H. Song, “A Systematic Procedure for Accurately 
Approximating Lognormal-Sum Distributions,” IEEE Trans. Veh. 
Technol., vol. 57, no. 1, pp. 663–666, Jan. 2008. 
[8] Ben Slimane, “Bounds on the distribution of a sum of independent 
lognormal random variables,” IEEE Trans. Commun., vol. 49, no. 6, pp. 
975–978, Jun. 2001. 
[9] F. Berggren and S. Slimane, “A simple bound on the outage probability 
with lognormally distributed interferers,” IEEE Commun. Lett., vol. 8, 
no. 5, pp. 271–273, May. 2004. 
[10] S. Zacks and C. P. Tsokos, “The distribution of sum of dependent log-
normal variables,” Technical Report No 31, Defense Technical 
Information Center, Apr. 1978. 
[11] R. B. Leipnik, “On lognormal random variables-I: The characteristic 
function,” J. Aust. Math. Soc. Ser. B, no. 3, pp. 327-347, 1991. 
[12] P. Glasserman, “Generating Random Numbers and Random Variables” 
in Monte Carlo Methods in Financial Engineering, 1st ed. New York, 
Springer-Verlag, 2003, ch. 2, sec. 2.3.1, pp. 65. 
 
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
2
4
6
8
10
x 10
4
wSCLN
Frequency
 
 
Simulated histogram
Convolution integral PDF
 
Fig. 1.  Comparison of theoretical (convolution integral) and simulated frequencies 
  


Weighted Sum of Correlated Lognormals: Convolution Integral Solution

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  1   Abstract— Probability density function (pdf) for sum of n correlated lognormal variables is deducted as a special convolution integral. Pdf for weighted sums (where weights can be any real numbers) is also presented. The result for four dimensions was checked by Monte Carlo simulation.  Index Terms— amount of fading, cochannel interference, lognormal distribution   I. INTRODUCTION HE original problem as stated by Fenton [1] in 1960 is the following: “Given several random variables, each with a log-normal probability distribution, what is the probability distribution of the sum of the random variables?” By mathematical notation: let we have n lognormal variables, defined by the corresponding correlated multidimensional normal distribution (X), where   contains expected values and Σ is the covariance matrix. We try to find the probability density function (θ) of sum of (correlated) lognormal variables (S(C)LN): =)exp(...)exp()exp(21nXXXY ( )Σ,N~ µX  ?1=∑=niiYθ       Examples for S(C)LN can be found in almost every science, in wireless communications applications include [2] fading and shadowing modeling, and assessing cochannel interference. In finance the evaluation of exotic (basket, Asian) options [3], portfolio level Value at Risk and optimal portfolio selection are the most relevant S(C)LN problems. Fenton [1] has written about (without detailed specifications) a numerical integration solution in case of uncorrelated summands. He concentrated on finding an approximate solution for uncorrelated case for avoiding (with his own words) the “tedious” work of numerical integration. He presented an approximation of distribution of uncorrelated SLN by moment matching. After Fenton’s initial work almost  Manuscript received January 17, 2011.  T. Nagy is with Corvinus University of Budapest, Budapest, H-1093 Hungary (e-mail: tamasnagy1976@gmail.com).  all of the efforts were done trying to define better approximations [4]-[7], or lower and/or upper bounds for cumulative density function [8-9], [2] and less researches were done on numerical integral solution for problem of S(C)LN. In the literature there are some direct integral expressions formulating the pdf or cdf of SCLN. For 2 dimensional correlated case Zacks and Tsokos [10] presented a formula of characteristic function. Leipnik [11] demonstrated the pdf of n dimensional uncorrelated SLN. In the following, an exact, convolution integral solution for probability density function is presented for n-dimensional correlated case, which - based on my best knowledge - was not published before. Pdf of weighted sum of n correlated lognormals is also provided. II. CONVOLUTION INTEGRAL SOLUTION In a general, 2 dimensional independent case, the pdf of two convoluted variables is the following: ( ) gfdxxKgxfKh ⊗=−⋅= ∫∞∞−)()(  (1) In n dimensional case there are n-1 integrals, the argument of last probability density function is  ∑−=−11niixK  For getting the right pdf of SCLN, few problems are needed to be handled: • The random variables are correlated (the correlation structure is defined on the level of corresponding normal factors) • The integral variable is dx, but the argument in lognormal pdf is ln(x) Based on the linear transformation property [12] any multidimensional normal distribution (Σ is the covariance matrix) can be built form uncorrelated standard normal distribution (Z): ( )( ) TnTLLI,0N~LL,N~⋅=Σ⋅+⋅ZZL µµ (3) The formula for reversed way transformation is: ( )[ ] ZL =−Σ⋅− µµ ,N1  (4) Let Yi denote the i-th dependent lognormal variable, and zi   "This work has been submitted to the IEEE for possible  publication. Copyright may be transferred without notice, after  which this version may no longer be accessible.” (Weighted) Sum of n Correlated Lognormals: convolution integral solution (Jan 2011) Tamas Nagy T > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  2 the i-th independent standard normal variable: ( ) ( )===Σn21n21z...zz,0NZlnY...lnYl,N InYµ  (5) Uncorrelated standard normal variables can be created by first decreasing the logarithm of correlated lognormal variables by their expected values and then creating linear combination by corresponding elements of L-1 matrix. The i-th independent standard normal variable can be calculated as follows (whereas  L-1 matrix is lower triangular, only k ≤ i elements have non zero weights): ( ) ( )[ ] ( ) ( )[ ]∑∑=−=−−⋅=−⋅=ikkknkkki YkiLYkiLz1111 ln,ln, µµ  (6) A. Joint probability density function In the multidimensional dependent lognormal factor space the probability density related to a given point equals to the probability density related to the corresponding point in the multidimensional independent standard normal factor space. Joint probability density function of multidimensional correlated lognormal variables (f) can be expressed by one dimensional standard normal distributions (denoting h multivariate standard normal pdf, φ is one dimensional standard normal pdf, 1_−iL   the ith row of matrix 1−L  ): ( ) [ ]( ) ( )[ ]( )∏=−−−⋅=−⋅=nii YLYLhYf11_1 lnln µϕµ  (7)  B. Modifications related to integral variables For integrating function f, we have to make modifications related to integral variables. The relevant question is the following: if we change the correlated lognormal variables infinitesimally, what will be the n-volume of the subspace determined by changing of corresponding uncorrelated standard normal variables? Let us change the lognormal variables one by one, let iY∆  denote the change vector, where all of items are 0 but i-th element is ∆Y. Vector ig  means the corresponding change vector in uncorrelated standard normal factor space caused by iY∆ , it can be calculated as follows: ( )( ) ( )( )µµ −⋅−−∆+⋅= −− YLYYLg ii lnln 11  (8) In continuous case (∆Y→0), if 1_−iL  is the i-th column of   matrix, and ∗  denoting element wise multiplication, then: ∗=−YdYLg ii1_ (9) Based on g vectors an M matrix can be built, storing the changes in space of independent standard normal variables caused by one by one changing of correlated lognormal variables. M determines an n-dimensional hyper-parallelepiped, its n-volume (nV) can be calculated as absolute value of M’s determinant. Whereas M is triangular matrix, its determinant is the product of diagonal elements: ∏=−⋅=niiiii dYYLnV11 (10) C. Probability density function of SCLN  Based on the result we can construct the pdf of SCLN in n dimensional case. First in 3 dimensions (for the sake of illustration):  ( )( ) ( )( )( ) ( )( ) ( )( )( )( )( ) ( )( ) ( )( )( )0lnlnlnlnlnln210 02132113322132111312212211121111112121133122111>−−⋅⋅−−−⋅+−⋅+−⋅⋅⋅−⋅+−⋅⋅−⋅⋅−−⋅⋅⋅⋅=∫ ∫∞ ∞−−−−−−−−−YYKfordYdYYYKLYLYLYLYLYLYYKYYLLLKSCLNµµµϕµµϕµϕθ (11)  In n dimensional case the pdf of SCLN will be the following (for getting simpler structure let us use a b vector containing notations of variables): ( ) ( )( )0,...,,,ln11111210110 0 1 111>−−=⋅−⋅⋅=∑∑∫ ∏∫ ∫∏ ∑−=−=−∞−=∞ ∞= =−−niiniinniiniikkkikiiiSCLNYKforYKYYYbwheredYbLbLK µϕθ K (12) D. Probability density function of weighted SCLN (wSCLN) We can extend the problem of SCLN to weighted case (where weights (α) can be any real numbers). The weighted sum of correlated lognormals (wSCLN) problem is typical in finance (e.g. a portfolio where we can hold negative quantity of asset, and the prices are lognormally distributed): ?1=∑=niiiYαθ  Pdf of wSCLN can be expressed as follows: ( ) ( )( )⋅−=⋅−=>⋅−⋅−⋅⋅=∑∑∑∫ ∏∫ ∫∏ ∑−=−−=−−=∞−=∞ ∞= =−−1112111121110110 0 1 111,...,,,,...,,,0lnniiinnniiinnniiiniiniikkkikiiiwSCLNYKYYYcwhereYKYYYbwhereYKfordYbLcLKαααααµϕθ K (13) III. NUMERICAL VALIDATION The numerical validation was done for wSCLN (weighted SLCN). The convolution integral was computed by own built algorithm. When designing such an algorithm significant amount of computational time can be spared by taking the > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  3 condition in (13) into account. For potential values of underlying lognormal variables (Yi) symmetric intervals were supposed, defined by d times the given standard deviation: )exp()exp( iiiii dYd σµσµ ⋅+≤≤⋅−  (14) An initial lower (ilbi) and upper bound (iubi) for the given αiYi can be calculated (the min and max functions are required for handling the potential negative sign of α): [ ][ ])exp(),exp(max)exp(),exp(miniiiiiiiiiiiiiiddiubddilbσµασµασµασµα⋅+⋅⋅−⋅=⋅+⋅⋅−⋅= (15) For further purposes a reversed-way cumulated sums of the iubi and iubi vectors were used: ==∑∑∑∑====12111211,....,,,....,,iubiubiubrcubilbilbilbrclbiiniiiinii (16) Based on these vectors the potential values of K can be restricted (by increasing d, the intervals widen): ( ) ( )11 rcubKrclb ≤≤  (17) If we calculate value of θ(K) by multi-level nested loops, and the order of calculation is determined by i, conditional (narrowed) bounds for Yi can be determined based on: • Known variables (denoting with underlyings): K, Y1, Y2, …, Yi-1 • Potential maximum and minimum values of yet unknown variables (Yi+1, Yi+2, ….Yn-1), their cumulated sum stored by rclb and rcub vectors  • Initial bounds for Yi (ilbi and iubi) The conditional (narrowed) bounds for a given Yi will be the following: −⋅−−⋅−≤≤−⋅−−⋅−∑∑∑∑−=+−=+−=+−=+iijijjiijijjiiiijijjiijijjircubYKrclbYKiubYrcubYKrclbYKilbαααααααα111111111111,max,min,min,max  (18) For comparing the theoretical and simulation result the calculation was done in 4 dimensions with the following parameters: −−−−===−−=0.11.02.05.01.00.12.03.02.02.00.14.05.03.04.00.11.008.02.010.007.022.008.010.03.08.18.02.1ρσµα  For calculating the theoretical pdf, own algorithm was used (d = 5, number of steps for K = 400, number of steps for Y1, Y2, Y3 =  80). The simulation method was Monte Carlo (with 6.000.000 repetitions). The simulated histogram and theoretical frequencies were the followings:     As we can see the simulated and theoretical frequencies are very close to each other. IV. CONCLUSION The exact form of pdf of SCLN and wSCLN can be calculated based on convolution integrals.  In numerical validation a 4 dimension weighted case was presented (which can not be handled by simple lognormal approximation (logarithm of negatives are not valid)). By special designing of numerical integral algorithm significant computational time can be spared, the total computation by suggested algorithm required 11.5 minutes in a single laptop. By using more sophisticated algorithm and powerful computer, the exact convolution solution can be a feasible way for also high dimensions. REFERENCES [1] L. F. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,” IRE Trans. Commun. Syst., vol. CS-8, pp. 57–67, Mar. 1960. [2] 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. [3] M. A. Milevsky and S. E. Posner, “Asian options, the sum of lognormals and the reciprocal gamma distribution,” J. Financ. Quant. Anal., vol. 33, no. 3, pp. 409–422, Sep. 1998. [4] S. C. Schwartz and Y. S.Yeh, “On the distribution function and moments of power sums with lognormal components,” Bell Syst. Tech. J., vol. 61, pp. 1441–1462, Sept. 1982. [5] N. C. Beaulieu and Q. Xie, “An optimal lognormal approximation to lognormal sum distribution,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 479–489, Mar. 2004. [6] 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, Jul. 2007. [7] Q. T. Zhang and S. H. Song, “A Systematic Procedure for Accurately Approximating Lognormal-Sum Distributions,” IEEE Trans. Veh. Technol., vol. 57, no. 1, pp. 663–666, Jan. 2008. [8] Ben Slimane, “Bounds on the distribution of a sum of independent lognormal random variables,” IEEE Trans. Commun., vol. 49, no. 6, pp. 975–978, Jun. 2001. [9] F. Berggren and S. Slimane, “A simple bound on the outage probability with lognormally distributed interferers,” IEEE Commun. Lett., vol. 8, no. 5, pp. 271–273, May. 2004. [10] S. Zacks and C. P. Tsokos, “The distribution of sums of dependent log-normal variables,” Technical Report No 31, Defense Technical Information Center, Apr. 1978. [11] R. B. Leipnik, “On lognormal random variables-I: The characteristic function,” J. Aust. Math. Soc. Ser. B, no. 3, pp. 327-347, 1991. [12] P. Glasserman, “Generating Random Numbers and Random Variables” in Monte Carlo Methods in Financial Engineering, 1st ed. New York, Springer-Verlag, 2003, ch. 2, sec. 2.3.1, pp. 65.  -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40246810x 104wSCLNFrequency  Simulated histogramConvolution integral PDF Fig. 1.  Comparison of theoretical (convolution integral) and simulated frequencies   

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Asian options, the sum of lognormals and the reciprocal gamma distribution,”

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Weighted Sum of Correlated Lognormals: Convolution Integral Solution

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