Multiplicative decomposition and infinite divisibility of the mandel distribution
Infinite divisibility of the Mandel distribution that arises in quantum optics is proved. On the basis of this fact, the multiplicative decomposition of this distribution into the countable convolution of some Poisson distributions is constructed.
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| Zitieren: | Multiplicative decomposition and infinite divisibility of the mandel distribution / Yu.P. Virchenko, N.N. Vitokhina // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 131–139. — Бібліогр.: 9 назв.— англ. |
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| author | Virchenko, Yu.P. Vitokhina, N.N. |
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| citation_txt | Multiplicative decomposition and infinite divisibility of the mandel distribution / Yu.P. Virchenko, N.N. Vitokhina // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 131–139. — Бібліогр.: 9 назв.— англ. |
| collection | DSpace DC |
| description | Infinite divisibility of the Mandel distribution that arises in quantum optics is proved. On the basis of this fact, the multiplicative decomposition of this distribution into the countable convolution of some Poisson distributions is constructed.
|
| first_indexed | 2025-12-07T15:35:52Z |
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 131–139
UDC 519.21
YU. P. VIRCHENKO AND N. N. VITOKHINA
MULTIPLICATIVE DECOMPOSITION AND INFINITE
DIVISIBILITY OF THE MANDEL DISTRIBUTION
Infinite divisibility of the Mandel distribution that arises in quantum optics is proved.
On the basis of this fact, the multiplicative decomposition of this distribution into
the countable convolution of some Poisson distributions is constructed.
1. Introduction
The registration process of low-intensity electromagnetic radiation represents the ab-
sorption of some energy portions named photons. The number ñ of registered photons
during the time T is random according to its physical nature1). Therefore, the descrip-
tion of the registration process is based on the probability distribution Pn ≡ Pr{ñ = n}
of this random value. The determination of this distribution is the quantum optics prob-
lem. In this section of theoretical physics, it is considered [1], [2] that, under some certain
physical conditions, the distribution Pn is the composite Poissonian one,
Pn =
1
n!
E
(
J̃n
T exp[−J̃T ]
)
, (1)
where J̃T is the random variable representing the energy of the electromagnetic field
absorbed by a photon counter during the registration time T . In quantum optics, it is
referred to as the Mandel distribution [3].
In the work, we study the probability distribution (1) from the mathematical point
of view in the physically most simple case where the electromagnetic radiation is the
so-called one-mode and completely polarized. In addition, we consider that the electro-
magnetic radiation is completely noisy. Mathematically, this case is described by the
random variable J̃T which is defined by the formula [1]
J̃T ≡ J [ζ̃] =
T∫
0
∣∣∣ζ̃(s)
∣∣∣2 ds ,
where ζ̃(s) = ξ̃(s) + iη̃(s), s ∈ R, are trajectories of the complex process with real
and imaginary parts ξ̃ = {ξ̃(t); t ∈ R}, η̃ = {η̃(t); t ∈ R} corresponding to Ornstein–
Uhlenbeck processes which are stochastically equivalent and independent.
The Ornstein–Uhlenbeck processes are Markovian and Gaussian ones, and they are
completely characterized by these properties together with the stationarity condition.
This class of processes is parametrized by two numbers ν > 0, σ > 0. In view of
Markovian and Gaussian properties, each Ornstein–Uhlenbeck process is completely de-
termined by the conditional probability density w(x0, t0|x, t) of the transition from the
2000 AMS Mathematics Subject Classification. Primary 60G35, 81K05.
Key words and phrases. Infinite divisibility, Mandel distribution, Poisson distribution, Ornstein–
Uhlenbeck process.
This research has been partially supported by RFFI and Belgorod State University.
1The sign ”tilde” points out further that the corresponding mathematical object is random.
131
132 YU. P. VIRCHENKO AND N. N. VITOKHINA
point x0 ∈ R at t0 ∈ R to the point x ∈ R at t ∈ R (see, for example, [4], III, §8, and
also [5]). It depends on the parameters ν and σ and has the following form:
w(x0, t0|x, t) =
(
ν
πσ
(
1 − e−2ν|t−t0|
)
)1/2
exp
(
−ν
[
x − x0e
−ν|t−t0|]2
σ
(
1 − e−2ν|t−t0|
)
)
. (2)
Then, fixing the parameters ν and σ completely determines the probability distributions
of random variables JT [ξ̃], JT [η̃] and, thus, in view of independence and equivalence of
the processes {ξ(t); t ∈ R}, {η(t); t ∈ R}, it defines the probability distribution density
of the random variable JT [ζ̃].
The characteristic function Qξ(−iλ) = E exp(iλJT [ξ̃]), λ ∈ R, of the random variable
JT [ξ̃] has been calculated in [6]:
Qξ̃(λ) = E exp(−λJT [ξ̃]) =
(
4rν exp(νT )
(r + ν)2 exp(rT ) − (r − ν)2 exp(−rT )
)1/2
, (3)
where r =
√
ν2 + 2λσ. From this, the characteristic function is determined as a function
of complex variable on a two-list Riemann surface. For the complex process {ζ(t); t ∈ R},
the similar function is determined by the formula
Q(λ) = Qξ̃(λ)Qη̃(λ) =
[
Qξ̃(λ)
]2
(4)
in view of independence and equivalence of the processes ξ̃, η̃. Whence it follows that
Q(λ) is meromorphic (see, Corollary 1 below).
The availability of the explicit formula of the function Q(λ) allows us to use some
methods of the theory of complex variable functions to study the properties of the rather
complicated Mandel distribution Pn corresponding to this characteristic function. The
purpose of the present work is to construct the multiplicative decomposition of the Man-
del distribution on Poisson distributions and, in particular, to give the proof of its infinite
divisibility.
2. Properties of the function Q(λ)
According to (3) and (4), the function Q(λ) has the form
Q(λ) = 4νr [F (r)]−1
eνT , (5)
F (r) = (ν + r)2erT − (ν − r)2e−rT . (6)
Lemma 1. The function [Q(λ)]−1 is an entire function of λ ∈ C.
Proof. The function F (r) is obviously an entire function of the variable r. In addition, one
can directly verify that it is odd. Hence, it can be represented in the form F (r) = rG(r2).
Then G(x) is also an entire function of x ∈ C. Really, for the coefficients 〈ak; k ∈ Z+〉 of
the decomposition
F (r) =
∞∑
k=0
akr2k+1
determining the entire function F , we have
lim sup
k→∞
|ak|1/(2k+1) = 0 .
Then the following decomposition of the function G(x) is valid:
G(x) =
∞∑
k=0
akxk
MULTIPLICATIVE DECOMPOSITION 133
. Here, the coefficients possess the property
lim sup
k→∞
|ak|1/k =
(
lim sup
k→∞
|ak|1/2k+1
)2
= 0 .
Hence, the last power decomposition has the infinite radius of convergence.
Let us notice that r2 = ν2 + 2νλ is the entire function of λ. Then, according to (5),
[Q(λ)]−1 =
1
4ν
G(r2)e−νT , (7)
i.e. [Q(λ)]−1 is the superposition of entire functions and, hence, it is the entire function
of λ.
Lemma 2. The zeros {λn; n ∈ Z+} of the function [Q(λ)]−1 are real and negative. They
are defined by the formula
λn = −ν2 + x2
n
2σ
, n ∈ Z+, (8)
where xn are positive solutions of the equation
tg xT =
2νx
x2 − ν2
, x ∈ R . (9)
They satisfy the inequalities
π
T
(
n − 1
2
)
< xn ≤ π
T
(
n +
1
2
)
, n ∈ N, (10)
and x0 satisfies the inequality ν ≤ x0 ≤ π/2T (x0 exists only if ν ≤ π/2T ).
Proof. According to (7), the zeros of the function [Q(λ)]−1 are solutions of the equation
G(r2) = 0. Then the set of zeros coincides with the set of solutions of the equation
e2rT =
(
ν − r
ν + r
)2
(11)
except the point r = 0 since
lim
r→0
[Q(λ)]−1 =
e−νT
4ν
lim
r→0
F (r)
r
=
e−νT
4ν
F ′(0) =
νT
2
eνT �= 0
is valid at this point. Equating the modules of both parts of (11), we obtain
ν2 + |r|2 − 2νRe r
ν2 + |r|2 + 2νRe r
= e2νT Re r .
This equality is impossible at Re r �= 0 since, at Re r > 0, the left-hand side is less than 1,
and the right-hand side is more than 1, ν �= 0. At Re r < 0, we have opposite inequalities.
Thus, for the validity of (11), it is necessary that Re r = 0. Putting r = ix, x ∈ R, in
(11) yields
±eixT =
ν − ix
ν + ix
,
whence
± cosxT =
ν2 − x2
ν2 + x2
, ± sinxT = − 2νx
ν2 + x2
,
which is equivalent to (9). The solution x = 0 of this equation, i.e. r = 0, may not be
taken into account. Further, if x is the solution of Eq. (9), then (−x) is also its solution.
But they give the same value λ = −(x2 + ν2)/2σ. Then, to find zeros λn, n ∈ N, of
the function [Q(λ)]−1, it is necessary to choose only one of them. Therefore, we consider
further only positive solutions of Eq. (9).
The function tg xT grows on x, and the function [−2νx/(ν2 − x2)] decreases since it
has the derivative [−2ν(ν2 + x2)/(ν2 − x2)2]. Since the function tgxT is periodic with
134 YU. P. VIRCHENKO AND N. N. VITOKHINA
period π/2T and varies from −∞ to ∞ on the period, there exists only one intersection
of the graphs of these functions in each interval (π(2n − 1)/2T, π(2n + 1)/2T ]. Each
intersection gives one solution of Eq. (9). It is a simple solution since the derivatives of
both the above-mentioned functions are finite and not equal to zero. Hence, their graphs
are intersected transversally. Hence, Eq. (9) has the infinite set of simple solutions
xn, n ∈ N, satisfying inequalities (10). In addition, this equation has the solution x0 in
the interval (0, π/2T ] satisfying the inequalities ν < x0 ≤ π/2T provided ν ≤ π/2T .
From this analysis, it follows that Eq. (8) has the infinite set of simple zeros λn,
n ∈ Z+, λn = −(2σ)−1(ν2 + x2
n), n ∈ Z+ and the zero λ0 is realized only when the
condition ν ≤ π/2T is valid. Thus, |λ0| > ν2/σ.
Corollary 1. The simple zeros of the function [Q(λ)]−1 are real and negative poles of
the function Q(λ). Since they satisfy the condition |λn| → ∞ as n → ∞ in view of (8)
and (10), the function Q(λ) is meromorphic.
Since
xn > a = min{ν, π/2T } , n ∈ Z+ ,
the function Q(λ) is analytic in a circle centered at λ = 0 with a radius not less than
(ν2 + a2)/2σ.
Lemma 3. The growth order of the entire function [Q(λ)]−1 does not exceed 1/2.
Proof. The growth order θ of the entire function is defined by
θ = lim
x→∞ sup
ln ln max{| [Q(λ)]−1 |; |λ| = x}
ln x
. (12)
Due to (5) and (6), we get
| [Q(λ)]−1 | ≤ e−νT
2ν|r| (ν + |r|)2e|r|T ,
whence
lim
|r|→∞
ln ln | [Q(λ)]−1 |
ln |r| ≤ lim
|r|→∞
(
1 +
ln T
ln |r|
)
= 1 .
Since |r| ≤ (ν2 + 2σ|λ|)1/2 is valid, i.e.
lim
|λ|→∞
ln |r|
ln |λ| ≤
1
2
,
relation (12) yields
θ =
[
lim
|r|→∞
ln ln | [Q(λ)]−1 |
ln |r|
] [
lim
|λ|→∞
ln |r|
ln |λ|
]
≤ 1
2
.
Corollary 2. Since the growth order of the function [Q(λ)]−1 is less than 1, the Hada-
mard decomposition
[Q(λ)]−1 = [Q(0)]−1
∏
n
(
1 − λ
λn
)
, (13)
where Q(0) = 1, is valid for it (see, for example, [7]).
3. The multiplicative decomposition
of the composite Poisson distribution
In this section, we demonstrate that, under certain conditions, the composite Poisson
distribution can be represented as the convolution of an infinite sequence of Poisson
distributions p(l), l ∈ N with step l.
MULTIPLICATIVE DECOMPOSITION 135
Definition. The lattice probability distribution p(l) = 〈p(l)
n ; n ∈ Z+〉 with step l ∈ N is
called the Poisson distribution with parameter αl ∈ R+ and step l ∈ N if
p(l)
n = {(αm
l e−αl/m!
)
, if n = ml, m ∈ Z+ ; 0; if n �= ml}. (14)
The generator function of the probability distribution p(l) is presented in the following
form:
Hl(x) = e−αl
∞∑
m=0
αm
l
m!
xlm = exp
(
αl(xl − 1)
)
. (15)
Let ñ be the lattice random variable with values in Z+. It is distributed according to
a composite Poisson distribution, i.e. the probabilities Pr{ñ = n}, n ∈ Z+, of its values
are determined by the formula
pn =
1
n!
EJ̃n exp(−J̃) , (16)
where J̃ is a random variable which takes its values on [0,∞). Let its characteristic
function EeitJ̃ be analytic in a circle with the center at t = 0. Then the power series of
this function converging in this circle is determined as
EeitJ̃ =
∞∑
l=0
(it)l
l!
Ml , (17)
where Ml = EJ̃ l, l ∈ Z+, M0 = 1 are the moments of the random variable J̃ . In addition,
the power series of the logarithm of the characteristic function is defined, i.e.
lnEeitJ̃ =
∞∑
l=0
(it)l
l!
Kl . (18)
It converges in a circle that has, generally speaking, a smaller radius with the same
center. The coefficients Kl, l ∈ Z+, K0 = 0 of this series are named cumulants of the
random variable J̃ . All of them are finite. If there are no zeroes of the function EeitJ̃ ,
then the convergence radii of both decompositions (17) and (18) coincide.
Thus, in the common convergence circle of both series, the following formula is valid:
∞∑
l=0
(it)l
l!
Ml = exp
( ∞∑
l=0
(it)l
l!
Kl
)
. (19)
Let
H(z) =
∞∑
n=0
znpn (20)
be the generator function of the random variable ñ. It is obviously analytic in the unit
circle. According to the transformations
H(z) =
∞∑
n=0
zn
n!
EJ̃ne−J̃ = E exp
(
(z − 1)J̃
)
=
= E
∞∑
n=0
(z − 1)n
n!
J̃n =
∞∑
n=0
(z − 1)n
n!
Mn
and to the analyticity property of the function Q(λ), the generator function H(z) is
analytic in a circle with the center at the point z = 1. Then there exists a negative real
λ such that the series
H(e−λ) =
∞∑
n=0
e−λnpn (21)
136 YU. P. VIRCHENKO AND N. N. VITOKHINA
converges and, hence, the convergence radius of the power series (20) is greater than
unity due to the fact that∣∣∣∣∣
∞∑
n=0
pnzn
∣∣∣∣∣ ≤
∞∑
n=0
pne−λn < ∞ , z ≤ e−λ.
In particular, it follows from the above inequality that all the moments Eñl, l ∈ N, of
the random variable ñ are finite,
Eñl =
∞∑
n=0
pnnl < ∞ .
Since H(0) = Ee−J̃ �= 0 and H ′(0) = EJ̃e−J̃ < M1, the analytic function ln H(z) is
defined in the neighborhood of the point z = 0. Therefore, in this neighborhood, the
decomposition
ln H(z) =
∞∑
l=0
K̄l
l!
zl (22)
is valid. We name the coefficients K̄l, l ∈ N as reduced cumulants.
Further, let all the reduced cumulants K̄l with numbers l ∈ N be nonnegative. Then,
from (22), we have the series
∞∑
l=1
K̄l
l!
zl−1 =
1
z
(ln H(z) − K̄0)
having nonnegative coefficients and converging in a circle with a nonzero radius. The
modulus maximum of this analytic function is attained on the positive part of the real axis
(see, for example, [9]). Then, if z∗ > 0 is the divergence point of this series, H(z∗) = ∞
is valid. Hence, z∗ > e−λ and, therefore, its convergence radius is not less than the
convergence radius of series (20). From here, it follows that series (22) converges at the
point z = 1, too. Since H(1) = 1, the identity
∞∑
l=0
1
l!
K̄l = 0 (23)
follows from (22) at z = 1. This means that
−
∞∑
l=1
1
l!
K̄l = K̄0 .
Then formula (22) takes the form
ln H(z) =
∞∑
l=1
zl − 1
l!
K̄l . (24)
Putting z = eit in (22), we obtain
lnEeitñ =
∞∑
l=1
eitl − 1
l!
K̄l . (25)
This decomposition gives us the Kolmogorov representation
lnEeitñ = itEñ +
∞∫
0
(
eitx − 1 − itx
) dμ(x)
x2
of the characteristic function of the infinitely divisible distribution corresponding to the
nonnegative random variable with the finite second moment. Here, the measure μ(x)
is a monotone nondecreasing function determining the finite measure on [0,∞) (see, for
MULTIPLICATIVE DECOMPOSITION 137
example, [8]). In our case, this measure is determined by the formula
μ(x) =
∞∑
l=1
lK̄l
(l − 1)!
χ(x − l) ,
where χ(x) is the indicator function of the set [0,∞). The finiteness of the measure μ,
i.e. the convergence of the series
μ(∞) =
∞∑
l=1
lK̄l
(l − 1)!
< ∞
, is guaranteed by the finiteness of the second moment Eñ2 of the random variable ñ.
At last, let us consider the arbitrary pair of distributions (14) p(l) and p(m), l, m ∈ N,
with the generator functions (15) which have αl =
(
K̄l/l!
)
, αm =
(
K̄m/m!
)
. The lattice
distribution which is formed by the operation ◦ of convolution applying to this pair lattice
distributions has components determined by the formula
(p(l) ◦ p(m))n =
n∑
k=0
p
(l)
k p
(m)
n−k , n ∈ Z+ .
Therefore, its characteristic function is the product Hl(x)Hm(x). Then the distribution
p corresponds to the generator function
H(x) = exp
( ∞∑
l=1
xl − 1
l!
K̄l
)
=
∞∏
l=1
Hl(x) .
Thus, this distribution is performed as the infinite convolution
p = p(1) ◦ p(2) ◦ ... ◦ p(l) ◦ ... (26)
of the Poisson distributions (14), where p(l) has step l and parameter αl =
(
K̄l/l!
)
.
The infinite convolution (26) converges component-wise, since the Poisson distribu-
tions p(l) with step l > n have the first nonzero component (p(l))k at such k > 0 that
k = l. Then, for each fixed n ∈ Z+, the following formula is valid:
pn =
(
p(1) ◦ ... ◦ p(n)
)
n
(
p
(n+1)
0 p
(n+2)
0 ...
)
.
The expression in the first bracket is a finite sum, and the infinite product of null com-
ponents in the second bracket converges since
∞∏
l=n+1
e−αl = exp
(
−
∞∑
l=n+1
αl
)
= exp
(
−
∞∑
l=n+1
K̄l
l!
)
.
We summarize the above analysis in the following statement.
Theorem 1. Let the random variable J̃ be concentrated on R+, let its characteristic
function be analytic in a circle with a nonzero radius, and let its reduced cumulants K̄l,
l ∈ N be nonnegative. Then the composite Poisson distribution p defined by (16) and
constructed on the basis of the random variable J̃ is infinitely divisible. It is represented
as the component-wise converging infinite convolution (26) constructed on the basis of
Poisson distributions p(l) which have steps l and parameters αl =
(
K̄l/l!
)
, l ∈ N, respec-
tively.
4. The multiplicative decomposition
of the Mandel distribution
In this section, we demonstrate that Theorem 1 may be applied to the Mandel distri-
bution that will lead us to the basic result of the work.
138 YU. P. VIRCHENKO AND N. N. VITOKHINA
Theorem 2. If the parameters ν, σ of the random variable J̃T satisfy the condition
ν2/2σ > 1, then its reduced cumulants are positive, and they are determined by the
formula
K̄l =
∞∑
n=1
(l − 1)!
(1 + |λn|)l
> 0 .
Proof. From expression (13) for the function [Q(λ)]−1, taking into account the negativity
of zeros λn, n ∈ Z+, we obtain
lnEeλJ̃ = lnQ(−λ) = −
∞∑
n=1
ln
(
1 − λ
|λn|
)
=
∞∑
m=1
1
m
∞∑
n=1
(
λ
|λn|
)m
.
Hence, according to (18), the cumulants of the random variable J̃ are
Km =
∞∑
n=1
(m − 1)!
|λn|m . (27)
Let the condition ν2/2σ > 1 be fulfilled. According to Corollary 1, all the moduli of
the zeros λn, n ∈ Z+ exceed 1, and therefore the function Q(λ) is analytic in the circle
centered at λ = 0 and with radius being more than 1. Then the decomposition
∞∑
n=0
(z − 1)n
n!
Mn = H(z)
converges at |z − 1| ≤ 1 + ε at an ε > 0 being sufficiently small. Since there are no zeros
of the function Q(λ), the power series of ln Q(λ) has the convergence radius greater than
1. Applying the logarithm operation to both sides of the last equality and using the
definition of reduced cumulants, we obtain the equality of decompositions converging as
|z − 1| ≤ 1 + ε:
∞∑
m=0
(z − 1)m
m!
Km =
∞∑
l=0
zl
l!
K̄l .
The decomposition in the left-hand side of the equality converges in the neighborhood
of the point z = 0. Therefore, by differentiating both sides of this equality with respect
to z l times and by putting z = 0, we obtain
K̄l =
(
dl
dzl
∞∑
m=0
(z − 1)m
m!
Km
)
z=0
.
Due to the fact that the power series converges uniformly in its convergence region,
all differentiations commute with the summation. As a result, we obtain the following
formula for reduced cumulants:
K̄l =
∞∑
m=0
Km+l
(−1)m
m!
. (28)
The calculation of the reduced cumulants on the basis of formula (28) gives
K̄l =
∞∑
m=0
(−1)m
m!
Km+l =
∞∑
m=0
(−1)m
m!
∞∑
n=1
(m + l − 1)!
|λn|m+l
=
=
∞∑
n=1
|λn|−l
∞∑
m=0
(−1)m
m!
(m + l − 1)!
|λn|m =
=
∞∑
n=1
|λn|−l
[(
d
dy
)l−1 ∞∑
m=0
(−1)mym+l−1
]
y=|λn|−1
=
MULTIPLICATIVE DECOMPOSITION 139
=
∞∑
n=1
(−1)l−1
|λn|l
[(
d
dy
)l−1 ∞∑
m=0
(−1)mym
]
y=|λn|−1
=
=
∞∑
n=1
(−1)l−1
|λn|l
[(
d
dy
)l−1
(1 + y)−1
]
y=|λn|−1
=
= (l − 1)!
∞∑
n=1
|λn|−l
(
1 + |λn|−1
)−l
= (l − 1)!
∞∑
n=1
(1 + |λn|)−l
> 0 .
Then it follows that K̄l ≥ 0 and
K̄l =
∞∑
n=1
(l − 1)!
(1 + |λn|)l
≤ Kl , l ∈ N .
From the proved statement, on the basis of Theorems 1 and 2, we come to the basic
result of the work.
Theorem 3. If the condition ν2/2σ > 1 for the parameters ν, σ determining the proba-
bility distribution of the random variable J̃T is fulfilled, then the Mandel distribution is
infinitely divisible. Thus, it decomposes into the infinite convolution
P = p(1) ◦ p(2) ◦ ... ◦ p(n) ◦ ...
of Poisson distributions p(l), l ∈ N, with steps l and parameters
αl =
1
l
∞∑
n=1
(l − 1)!
(1 + |λn|)l
.
Acknowledgement: We are grateful to RFFI and Belgorod State University for the
financial support of this work
Bibliography
1. M. Lax, Fluctuation and Coherence Phenomena in Classical and Quantum Physics, Gordon
and Breach, New York, 1968.
2. R. Glauber, Optical Coherence and Photon Statistics, in Quantum Optics and Electronics.
eds. C. de Witt, A. Blandin, C. Cohen-Tannoudji. Lectures at Les Houches during 1964 session
of the Summer School of Theoretical Physics at University of Grenoble. Gordon and Breach,
New York (1965), 91–279.
3. L. Mandel, Progress in Optics, Vol.2, North-Holland, Amsterdam, 1963.
4. W. Feller, An Introduction to Probability Theory and its Applications, vol.2, 2d ed., Wiley,
New York, 1971.
5. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sci-
ences. 2d ed, Spriger Series in Synergetics. vol. 13, Springer-Verlag, Berlin, 1983.
6. A.J.F. Ziegert, A systematic approach to a class problems in the theory of noise and other
random phenomena. Part II, examples, Trans. IRE IT (1957), no. 3, 38–44.
7. A.I. Markushevich, Theory of Analytic Functions, vol.2, Nauka, Moscow, 1968. (in Russian)
8. E. Lukacs, Characteristic Functions, Griffin, London, 1970.
9. M.V. Fedoryuk, Saddle–Point Method, Nauka, Moscow, 1977. (in Russian)
E-mail : virch@isc.kharkov.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4434 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T15:35:52Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Virchenko, Yu.P. Vitokhina, N.N. 2009-11-09T15:40:26Z 2009-11-09T15:40:26Z 2005 Multiplicative decomposition and infinite divisibility of the mandel distribution / Yu.P. Virchenko, N.N. Vitokhina // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 131–139. — Бібліогр.: 9 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4434 519.21 Infinite divisibility of the Mandel distribution that arises in quantum optics is proved. On the basis of this fact, the multiplicative decomposition of this distribution into the countable convolution of some Poisson distributions is constructed. This research has been partially supported by RFFI and Belgorod State University. en Інститут математики НАН України Multiplicative decomposition and infinite divisibility of the mandel distribution Article published earlier |
| spellingShingle | Multiplicative decomposition and infinite divisibility of the mandel distribution Virchenko, Yu.P. Vitokhina, N.N. |
| title | Multiplicative decomposition and infinite divisibility of the mandel distribution |
| title_full | Multiplicative decomposition and infinite divisibility of the mandel distribution |
| title_fullStr | Multiplicative decomposition and infinite divisibility of the mandel distribution |
| title_full_unstemmed | Multiplicative decomposition and infinite divisibility of the mandel distribution |
| title_short | Multiplicative decomposition and infinite divisibility of the mandel distribution |
| title_sort | multiplicative decomposition and infinite divisibility of the mandel distribution |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4434 |
| work_keys_str_mv | AT virchenkoyup multiplicativedecompositionandinfinitedivisibilityofthemandeldistribution AT vitokhinann multiplicativedecompositionandinfinitedivisibilityofthemandeldistribution |