25th Biennial Symposium on Communications
A Power Series Expansion for the Truncated
Lognormal Characteristic Function
Norman C. Beaulieu, Fellow, IEEE
Department of Electrical and Computer Engineering
University of Alberta
Edmonton, Alberta, Canada
Email: [email protected]
Abstract—An infinite series expansion for the characteristic
function of the lognormal distribution does not exist, and no
infinite series representation of the characteristic function of
any modified form of the lognormal distribution is found in
the literature. A power series expansion is derived for the
characteristic function of the truncated lognormal distribution.
The series is proved to converge absolutely for any level of
truncation. Equivalently, the series converges absolutely for any
nonzero value of probability in the missing tail, and the truncated
lognormal can be made arbitrarily close, but not equal to, the
lognormal while retaining convergence. The behaviours of the
moments of the truncated lognormal and lognormal distributions
are examined in detail.
I. I NTRODUCTION
The characteristic function (CF) of a probability density
function (PDF) has fundamental importance in probability
theory. It is universally used as a mathematical tool for determining the PDF of a sum of independent random variables
(RVs). The CF of a sum of independent RVs is the product
of the CFs of the individual summands, and the standard
approach to determining the PDF of a sum of independent
RVs is to find the CFs of the summand RVs, form the product
to obtain the CF of the sum PDF, and then obtain the sum
PDF by inverse transformation of the sum CF. This approach
is usually much more efficient and computationally robust than
determining the sum PDF by the alternative approach, multiple
convolutions of the summand PDFs. In the best case, the CFs
of the summand PDFs are known in closed-form. Even when
the CFs are not known in closed-form, numerical computation
of the CFs is usually preferable to determining the sum PDF
by iterated convolutions.
The lognormal distribution is important in many areas
including wireless communications, where it is used to describe shadowing effects in radio transmission. No closedform expression is known for the lognormal characteristic
function. Numerical computation of lognormal characteristic
functions is notoriously difficult because the defining integral
formulas are not suited to common numerical integration
techniques [1], [2]. Different approaches have been reported.
Barakat [3] derived a modified Hermite polynomial series
for the lognormal CF, but this method fails to converge for
practical values of dB-spread used to model shadowing in
wireless networks. Leipnik [4] defined an ancillary logarithmic
variable leading to a Hermite function infinite series in the
978-1-4244-5711-3/10/$26.00 ©2010 IEEE
defined variable, but the result is complex and unwieldy
for application; for instance, the series coefficients must be
computed recursively in terms of Riemann Zeta functions.
Beaulieu and Xie [5] found that a modified Clenshaw-Curtis
method of numerical quadrature [6] is more efficient for
computing lognormal CFs than common methods of numerical
integration (trapezoidal rule, Simpson’s rule, modified Hermite
polynomial method, fast Fourier transform method). Gubner
[1] derived an improved integral formula for the CF that
removed the dependence of the oscillations in the integrand on
the CF domain argument. Beaulieu [2] followed Gubner’s approach but in addition transformed the semi-infinite improper
integral result into a definite integral over a finite interval.
Infinite series expansions have ubiquity in mathematics, so
much so that infinite series expansions for most functions are
widely tabulated in books, tables and reference works. It is
probably fair to state that understanding or knowledge of a
function is considering lacking if its infinite series expansion
is not known. In some cases, the infinite series can directly be
used to compute the function, or to approximate the function.
The infinite series expansion may offer the only knowledge
of the derivative or integral of a particular function (one can
integrate or differentiate a convergent infinite power series
easily by operating on each term separately and the radius of
convergence is preserved [7, (1.3.2.7)] ). Many mathematical
proofs are based on expanding the function(s) involved into
infinite series for further examination, comparison or manipulation (e.g., [8, Appendix B]).
It is well known that an infinite series representation of
the lognormal characteristic function (CF) does not exist [9].
The reason for this is the rapid and unbounded growth of
the nth order moment of the lognormal distribution. Letting
mn denote the nth lognormal moment, it can be shown that
lim mn → ∞. That is, the lognormal distribution moments
n→∞ n!
grow faster than the factorial.
Note that the mathematical lognormal distribution is defined
on [0, ∞). In all practical systems, the lognormal parameter
cannot approach infinity without limit. It is, therefore, both
reasonable and relevant to consider a truncated lognormal
distribution defined on [0, B], where 0 < B < ∞ is finite.
It is immediate that the nth order moment of the truncated
lognormal distribution is bounded as, mn ≤ B n and, hence,
284
25th Biennial Symposium on Communications
n
lim mn ≤ lim Bn! = 0.
n→∞ n!
n→∞
In this paper, we derive an infinite series representation for
the CF of a truncated lognormal distribution and prove that
the infinite series converges for an arbitrary, finite, value of
B. To the best of the author’s knowledge, no infinite series
representation for the truncated lognormal distribution, or
any other modified version of the lognormal distribution, has
been previously reported. Some interesting observations about
approximating the lognormal distribution by the truncated
lognormal distribution, and about approximating the lognormal
sum distribution by the truncated lognormal sum distribution
are also presented.
The infinite series is derived in Section II, and a proof
of the convergence of the series for arbitrary values of B,
B > 0 is also given. In Section III, we derive some properties
of the moments of the lognormal and truncated lognormal
distributions. Section IV concludes the paper.
the PDF gT (x) in (2), one has
mn
where F (t) is the cumulative Gaussian distribution function,
Z t − r2
e 2
√ dr.
(2b)
F (t) =
2π
−∞
The term F lnB
scales the truncated lognormal PDF so
σ
R∞
RB
that 0 gT (x)dx = 0 gT (x)dx = 1. We wish to find the
CF of random variable (R.V.) X where X has the truncated
lognormal PDF given in (2). Let E[·] denote the expectation
operation. Then the CF of RV X is defined as [10]
ΦX (ω) = E[ejωX ].
(3)
Formally, using an infinite Taylor series expansion for ejωX
in (3), one obtains
#
"∞
X (jωX)n
(4a)
ΦX (ω) = E
n!
n=0
=
where
∞
X
(jω)n
mn
n!
n=0
mn = E[X n ]
th
=e
(4b)
(4c)
is the n moment of the RV X, and where the linearity
property of the expectation operation has been used. Using
n2 σ 2
2
Z
lnB
−∞
=e
=e
and a truncated version of the lognormal PDF given by


x>B

0, (lnx)2

−
2
gT (x) = √ e 2σ lnB , 0 < x ≤ B
(2a)

2πxσF ( σ )


0,
x≤0
(5a)
(5b)
(5c)
where the transformation x = ey has been used. Completing
the square in the exponential arguments in the numerator of
the integrand in (5c) yields
II. D ERIVATION
We consider the lognormal probability density function
(PDF) given by
 (lnx)2
 e− 2σ2
√
, x>0
(1)
g(x) =
2πxσ

0,
x≤0
= E[X n ]
(lnx)2
Z B
xn e− 2σ2
√
dx
=
2πxσF lnB
0
σ
y2
Z lnB
eny e− 2σ2
√
dy
=
2πσF lnB
−∞
σ
mn
n2 σ 2
2
n2 σ 2
2
(y−nσ 2 )2
e− 2σ2
√
dy
2πσF lnB
σ
z2
lnB−nσ 2
e− 2σ2
√
dz
2πσF lnB
−∞
σ
nσ
F lnB
σ −
lnB
F σ
Z
(6a)
(6b)
(6c)
where z has been used as a dummy variable of integration.
Using (6c) in (4b) gives the final result formally,
n2 σ 2
∞
X
nσ
e 2 F lnB
σ −
(jω)n .
(7)
ΦX (ω) =
lnB
n!F
σ
n=0
We use the ratio test [11, Ch. 3] to prove that the infinite series
(7) converges for all ω, for all values of B < ∞. Setting
n
n2 σ 2
e 2 F lnB
σ − nσ j
(8)
an =
n!F lnB
σ
one has
an+1 L = lim n→∞
an = lim
n→∞
(n+1)2 σ 2
2
lnB
σ
− (n + 1)σ
.
n2 σ 2
(n + 1)e 2 F lnB
σ − nσ
e
F
(9)
Let Q(·) denote the usual Q-function [12, p. 931]. Then
F (−x) = Q(x) and, hence, (9) can be written as
(n+1)2 σ 2
e 2
Q (n + 1)σ − lnB
σ
L = lim
(10)
.
n2 σ 2
n→∞
(n + 1)e 2 Q nσ − lnB
σ
lnB
Now as n → ∞, (n + 1)σ − lnB
σ → ∞ and nσ − σ → ∞
for any fixed value of B < ∞ for any σ > 0. Since [12, p.
932]
x2
e− 2
(11)
lim Q(x) = √
x→∞
2πx
the limit L in (10) becomes,
2
" (n+1)2 σ2 ( lnB
(n+1)2 σ 2
σ )
e 2
e− 2 e(n+1)lnB e− 2
L = lim
2
n→∞
( lnB
σ )
n2 σ 2
n2 σ 2
(n + 1)e 2 e− 2 enlnB e− 2
285
25th Biennial Symposium on Communications
#
(n + 1)σ − lnB
σ
× nσ − lnB
σ
(12a)
140
10
n
B
m (∞)
n
lnB
e
=0<1
(12b)
= lim
n→∞ n + 1
for any value of B < ∞. Therefore, the infinite series (7)
converges absolutely for B < ∞ [11, Ch. 3].
lim mn (∞) = lim e
n→∞
n→∞
n2 σ 2
2
= ∞.
Regarding the truncated lognormal moments, one has
lnB
nσ
n2 σ 2 F
σ −
2
lim mn (B)=lim e
n→∞
n→∞
F lnB
σ
lnB
n2 σ 2 Q nσ −
σ
2
=lim e
n→∞
F lnB
σ
2
−(lnB )
2 2
σ
n2 σ 2 −n σ
e 2 e 2 enlnBe 2
=lim √
n→∞ 2πF lnB
nσ− lnB
σ
σ
lnB
elnB (n− 2σ2 )
=lim √
n→∞
2πF lnB
nσ − lnB
σ
σ
= ∞
(13)
n
10
10
80
10
σ = 0.25
n
Bn, m (∞), and m (B) for σ=0.25
100
60
10
n
III. M OMENTS
It is of interest to examine the moments of the truncated
lognormal distribution and compare them to the moments of
the lognormal distribution. It is well known that the moments
of the lognormal distribution do not uniquely determine the
lognormal distribution [9]. However, the uniqueness theorem
of the Taylor series expansion [11, Ch. 8] guarantees that
the series representation (7) is the unique expansion of its
form for the truncated lognormal distribution. Observe that
the moments mn of the truncated lognormal distribution
given in (6c) are functions of the truncation value, that is,
mn = mn (B), n = 1, · · · . The lognormal moments are given
by mn (∞), n = 0, 1, · · · . Note that although the lognormal
moments are all finite,
m (B)
120
40
10
20
10
0
10
0
10
20
30
40
50
n
60
70
80
90
100
Fig. 1. The lognormal moments, mn (∞), and truncated
lognormal moments,
mn (B), versus n for σ = 0.25 and Q lnσB = 10−10 . For comparison,
B n is also shown.
of the ratio of the nth order moment to n! that determines
convergence of the infinite serries representation of the CF.
One has from (14d) that
elnB (n− 2σ2 )
mn (B)
= lim √
n→∞
n→∞
n!
nσ −
2πn!F lnB
σ
lnB
lim
(14a)
(lnB)2
(15a)
e− 2σ2
enlnB
. (15b)
= lim √
n→∞
n! nσ − lnB
2πF lnB
σ
σ
(14b)
(14c)
lnB
σ
We use the inequality [14]
√
1
1
2πnn+ 2 e−n+ 12n+1 < n!
(16)
with (15b) and n = elnn to obtain
−1
(lnB)2
enlnB nσ − lnB
mn (B)
e− 2σ2
σ
×
lim
< lim √
√
1
1
n→∞
n→∞ 2πF lnB
n!
2πe(n+2)lnn e−n+ 12n+1
σ
(17a)
−1
(lnB)2
−
lnB
nlnB
2
e
nσ − σ
e 2σ
×
= lim √
1
n(lnn−1)+ lnn
n→∞
2 + 12n+1
2πF lnB
e
σ
(17b)
(14d)
(14e)
for all B ≥ 0 where (11) has been used to go from (14b) to
(14c). The truncated lognormal moments are also all finite,
but tend to infinity as do the lognormal moments as n
increases without bound. Fig. 1 and Fig. 2 show the lognormal
moments, mn (∞), and the truncated lognormal moments,
mn (B), versus n for σ = 0.25 and σ = 6 dB, respectively.
The former value of σ is of interest in optical transmission
[3], while the latter is a typical value for modeling shadowing
in wireless
networks [13]. The value of B is chosen such that
Q lnσB = 10−10 , that is, the probability of the truncated
tail is 10−10 . Figs. 1 and 2 also show the value B n for
comparison. One sees that as expected, all of B n , mn (∞),
and mn (B) tend to infinity as n increases without bound. The
lognormal moments overtake and dominate the other moments
for sufficiently large n.
The important difference in the behaviours of the lognormal
and truncated lognormal moments lies in the ratio of mnn!(B)
and mnn!(∞) . As can be seen from (4b), it is the behaviour
(lnB)2
e− 2σ2
×0
= lim √
n→∞
2πF lnB
σ
= 0
for all 0 < B < ∞. Hence,
lognormal distribution.
lim mnn!(B)
n→∞
(17c)
(17d)
= 0 for the truncated
Considering the lognormal distribution, we use the inequality [14]
√
1
1
n! < 2πnn+ 2 e−n+ 12n
(18)
to obtain
286
mn (∞)
n→∞
n!
lim
=
lim
n→∞
e
n2 σ 2
2
n!
(19a)
25th Biennial Symposium on Communications
300
300
10
10
mn (∞)
n!
mn (B)
n!
Bn
mn(∞)
mn(B)
250
250
10
10
mn(∞)/n! and mn(B)/n! for σ=0.6
Bn, mn(∞), and mn(B) for σ= 6 dB
200
200
10
σ = 6 dB
150
10
100
10
10
150
10
σ = 0.6
100
10
50
10
50
10
0
10
0
10
−50
0
1
2
3
4
5
6
7
8
10
9
n
60
80
100
120
140
160
IV. C ONCLUSION
40
An infinite power series expansion was derived for the
characteristic function of the truncated lognormal distribution.
The series converges absolutely for all values of truncated
tail probability except zero. The moment-to-factorial ratio of
the truncated lognormal distribution utimately tends to zero,
whereas that of the lognormal distribution grows without
bound as the order of the moment increases without limit.
20
10
σ = 0.25
0
10
n
40
Fig. 4.
The moment-to-factorial ratio for the lognormal and truncated
lognormal distributions with σ = 0.6 and Q lnσB = 10−10 .
10
n
20
n
Fig. 2. The lognormal moments, mn (∞), and truncated
lognormal moments,
mn (B), versus n for σ = 6 dB and Q lnσB = 10−10 . For comparison,
B n is also shown.
m (∞)/n! and m (B)/n! for σ=0.25
0
−20
10
−40
10
−60
10
−80
10
R EFERENCES
−100
10
−120
10
mn (∞)
n!
mn (B)
n!
−140
10
−160
10
0
20
40
60
80
100
120
140
n
Fig. 3.
The moment-to-factorial ratio for the lognormal
and truncated
lognormal distributions with σ = 0.25 and Q lnσB = 10−10 .
>
=
lim √
n→∞
lim √
n→∞
= ∞
e
n2 σ 2
2
1
1
2πe(n+ 2 )lnn e−n+ 12n
e
n2 σ 2
2
2πen(lnn−1)+
lnn
1
2 + 12n
(19b)
(19c)
(19d)
and, hence, lim mnn!(∞) = ∞. Thus, a series of the form
n→∞
(4b) for the lognormal distribution does not exist. Fig. 3 and
Fig. 4 show mnn!(∞) versus n for σ = 0.25 and σ = 0.6,
respectively (numerical difficulties preclude using σ = 6 dB
in this figure). Again, B is chosen to set the probability
of the truncated tail to 10−10 . Observe that the lognormal
moment/factorial ratio grows without bound, whereas the
truncated lognormal/factorial ratio goes to zero, both as n
increases without bound.
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287
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