Proceedings 2nd ISI Regional Statistics Conference, 20-24 March 2017, Indonesia (Session CPS09)
The Characteristic Function Property of Convoluted Random Variable from a
Variational Cauchy Distribution
Dodi Devianto
Department of Mathematics, Andalas University,
Padang 25163, West Sumatra Province, INDONESIA
E-mail address: [email protected]
Abstract
The variational Cauchy distribution is constructed by setting the shape parameter of Cauchy
distribution multiplied with part of random variable in standard Cauchy distribution. The convolution
of a variational Cauchy distribution is obtained by using properties of inversion theorem of
characteristic function. Then it is given some potensial peoperties of characteristic function of
convoluted random variable from a variational Cauchy distribution. The properties of uniform
continuity and complex conjugate of characteristic function is determined by mathematical analysis
methods, and it is confirmed that the characteristic function has never vanish on the complex plane
and infinitely divisible.
Keywords: variational Cauchy distribution; convolution; characteristic function, infinitely divisible.
1. Introduction
The Cauchy distribution is due to Augustin Louis Cauchy in 1853 who introduced a continuous
distribution as standard Cauchy density. The Cauchy distribution is a case of continuous stable
distribution for which does not have mean, variance and other moments. Furtermore, the infinitely
divisible property of Cauchy distribution is the closed invariant property under convolution, then it is
on class of infinitely divisible because all convolutions belonging to this type again yield the same
distribution after convolution stability of this distribution. The specific case of inifinite divisibility of
this distribution has shown by Bondesson [2] for the half Cauchy densities, while Takano [8] has
exhibited the infinite divisibility of normed product of Cauchy densities. However, Dwass [6] has
developed the convolution of Cauchy distribution by using induction from definition of convolution of
two random variables. However, the convolution theory also very widely use to express properties of
some distribution, such as Devianto et. al [4, 5] are also using convolution to determine the sum of
exponential distribution with stabilizer constant and its properties. These previous result of
convolution is to confirm that convolution of random variables now very interesting to establish its
mathematical properties.
The study of convolution and infinitely divisible Cauchy distribution has widely developed to the
property of its characteristic function, since the characteristic function is always exist as the most
general tools to determine convolution of random variables. The characteristic function from a random
variable X is defined as Fourier-Stieltjes transform as the following
 X (t )  E [exp (itX )]
where exp (itX )  cos tX  i sin tX and i as imaginary unit. The characteristic function has inversion
theorem as the uniqueness property that is for every probability distribution function f (x) from
random variable X has a unique characteristic function  X (t ) such that the probability distribution
function can be obtained by using inversion of characteristic function as follows
f ( x) 
1
2

  exp[itx]  X (t ) dt .
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Proceedings 2nd ISI Regional Statistics Conference, 20-24 March 2017, Indonesia (Session CPS09)
The refinement of this characteristic function property also can be used to determine convolution of a
distribution. The convolution of Chauchy distribution can be obtained from its characteristic function
by using linerity property of expectation.
The establishment theory of characteristic function and its uses to determine convolution and infinite
divisibility have developed the stability properties of Cauchy distribution and its variation such as half
Cauchy distribution by Bondesson [2] and normed normed product of Cauchy densities by Takano [8],
while Devianto [3] also defined a variational Chaucy distribution and its concolution. The probability
density function of a variational Cauchy distribution is constructed by setting parameter γ multiplied
with part of random variable in standard Cauchy distribution, so that it can be defined in the following
term
f ( x) 
12
for   0 and x  (, ) .
 (1   x 2 )
The cummulative probability distribution function of a variational Cauchy distribution is obtained as
follows

1 2 arctan 
F ( x)  
2

12
x

for   0 and x  (, ) .
The direction of this paper is to contruct the characteristic function of convoluted of a variational
Chaucy distribution and its property of characteristic function. The section 2 is devoted to establish the
property of characteristic function of convoluted of a variational Chaucy distribution, while in Section
3 is discussed its infinite divisibility.
2. The Characteristic Function properties of Convoluted Random Variable from a Variational
Cauchy Distribution
This section is started by using definition of non-negative function and necessary and sufficient
conditions for a function to be a characteristic function in Bochner’s Theorem. We use definition of
non-negative function as necessary and sufficient conditions to be a characteristic function from
Lukacs [7].
Definition 2.1. A complex-valued function  (t ) with real variable t is said to be non-negative if the
following two conditions are satisfied
(i)  (t ) is continuous;
(ii) For any positively defined function with quadratic form
 c
j
cl  X (t j  tl )  0
1 j  n 1l  n
for any complex number c1 , c2 , ..., cn and real number t1 , t 2 , ..., t n .
Theorem 2.2. (Bochner’s Theorem). A complex-valued function  (t ) of a real variable t is a
characteristic function if, and only if,
(i)  (0)  1 ;
(ii)  (t ) is non-negative definite.
The characteristic function of a variational Cauchy distribution has introduced by Devianto [3] by
using concept of complex integration, that is for X as andom variable from a variational Cauchy
distribution has characteristic function

|t | 
12 
  
 X (t )  exp 
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Proceedings 2nd ISI Regional Statistics Conference, 20-24 March 2017, Indonesia (Session CPS09)
for   0 and t  (, ). The convolution of random variables from a variational Cauchy
distribution is determined by using property of characteristic function. Let Random variable Xi has a
variational Cauchy probability distribution with parameter γ, then random variable
n
Sn   X i
i 1
as convolution of random variables Xi has probability distribution function
f (s n ) 
n 1 2
 ( n 2   sn 2 )
for   0 and s n  (, ) . The characteristic function of random variable S n is obtained by using
linearity of expectation for characteristic function as follows
 n |t |
.
12 
  
 Sn (t )  E [e itSn ]  exp 
The characteristic function has some refinement properties, this section gives basic properties of
characteristic function from convoluted random variable of a variational Cauchy distribution in the
propositions.
Figure 1. Curve of characteristic function ϕSn(t)
as simetric distribution from random variable Sn
with parameter γ = 2 and various value of n-fold
convolution.
Figure 2. Parametric curves of characteristic
function ϕSn(t) with parameter γ = 2 and
various value of n-fold convolution.
The simetric shape of distribution for n-fold convolution a variational Cauchy distribution causes the
characteristic function only has real part. Figure 1 has plotted the function ϕSn(t) on the cartesian
coordinate with t in x-axis, the curve comes as smooth lines with extreme values on  Sn (0)  1 .
Furtehermore, the characterization of characteristic function can be seen from the shape of parametric
curves. The parametric curves are governed by using parametric plot at cartesian coordinate system for
x-axis as real part of characteristic function and y-axis as imaginary part of characteristic function.
Figure 2 has shown that characteristic function ϕSn(t) lies only in the real part in the interval
0  Re [ Sn (t )]  1 . These properties imply that characteristic function ϕSn(t) is positively defined
function and never vanish on the complex plane.
Proposition 2.3. Let Sn be a random variable from convolution of a variational Cauchy distribution
with characteristic function
 n |t |
.
12 
  
 Sn (t )  exp 
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Proceedings 2nd ISI Regional Statistics Conference, 20-24 March 2017, Indonesia (Session CPS09)
then it is satisfied  Sn (0)  1 ;
Proof. (i) It is easily obtained that for t  0 then  Sn (0)  1 .
Proposition 2.4. Let Sn be a random variable from convolution of a variational Cauchy distribution
with characteristic function
 n |t | 
 Sn (t )  exp  1 2  .
  
then characteristic function  S n (t ) is uniformly continuous.
Proof. The property of uniformly continuous of characteristic function from convoluted random
variable of a variational Cauchy distribution is explained by setting for every   0 there exists   0
such that S (t1 )  S (t 2 )   for t1  t 2   where  depends only on  . Let us set the the
following equation
 n | t1 | 
 n |t |
 exp  1 22 
12 
  
 

 Sn (t1 )   Sn (t 2 )  exp 
Then let us define h  t1  t 2 , so that we have
 n | h  t2 |
 n |t2 |
  exp  1 2 
12



 

 Sn (t1 )   Sn (t 2 )  exp 
By taking limit for h  0 , then it is obtained
 n | h  t2 |
 n |t2 |
lim exp 
  exp  1 2   0
12
h 0



 

This result of limiting process hold for every    where  Sn (h  t 2 )   Sn (t 2 )   for t1  t 2   .
Then the caharacteristic function of  Sn (t ) is uniformly continuous.
Proposition 2.5. Let Sn be a random variable from convolution of a variational Cauchy distribution
with characteristic function
 n |t |
.
12 
  
 Sn (t )  exp 
then characteristic function  S n (t ) is positively defined function with quadratic form
 
1 j  n 1l  n
c j cl  Sn (t j  t l )  0
for any complex number c1 , c2 , ..., cn and real number t1 , t 2 , ..., t n .
Proof. It will show that characteristic function  S n (t ) from convoluted random variable of a
variational Cauchy distribution satisfies the quadratic form
 
1 j  n 1l  n
c j cl  Sn (t j  t l )  0
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Proceedings 2nd ISI Regional Statistics Conference, 20-24 March 2017, Indonesia (Session CPS09)
for any complex numbers c1 , c2 , ..., cn and real numbers t1 , t 2 , ..., t n . Let us use definition of
characteristic function from convoluted random variable of a variational Cauchy distribution, hence
we have
 
1 j ,l  n 1l  n
c j cl  Sn (t j  t l ) 
 
1 j  n 1l  n
 n | t j  tl | 
c j cl exp 

12 

It is used the modulus property of inequality | t j  tl |  | t j |  | tl | , then we have the following form
 
1 j , l  n 1 l  n
c j cl S n (t j  tl ) 

 
1 j  n 1 l  n

1 j  n


1 j  n
 n | t j |  n | tl | 
c j cl exp 

1 2


 n |t j | 
 n | t |
c j exp  1 2   cl exp  1 2l 
  
   1 l  n
 n |t j | 
c j exp  1 2 
  
2
The last part of equation above has been in the quadratic form then we have
  c j cl S (t j  tl )  0
1 j  n 1l  n
n
It is proved that  S n (t ) as positively defined function where the quadratic form has nonnegative
values.■
3. The Infinitely Divisible Characteristic Function of Convoluted Random Variable from a
Variational Cauchy Distribution
This section will explain the infinitely divisible characteristic function of convoluted random variable
from a variational Cauchy distribution. The definition of infinitely divisible characteristic function is
referred to Artikis [1] that confirmed for distribution function of F with characteristic function  (t ) is
infinitely divisible if for every positive integer m there exists a characteristic function m (t ) such that
 (t )  (m (t ))m .
Proposition 3.1. Let us define a function
 n | t |
 S (t )  exp  1 2 
  
1
m

for every n, m  Z ,   0 and    t   , then it is satisfied
(i) S (0)  1 ;
(ii)  S (t ) is uniformly continuous;
(iii)  S (t ) is positively defined function with quadratic form
  c j cl S (t j  tl )  0
1 j  n 1l  n
for any complex number c1 , c2 , ..., cn and real number t1 , t 2 , ..., t n .
Proof. The outline proof of this proposition is similar way with Proposition 2.3, Proposition 2.4 and
Proposition 2.5, then Proposition 3.1 is obviously proven.
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Proceedings 2nd ISI Regional Statistics Conference, 20-24 March 2017, Indonesia (Session CPS09)
Theorem 3.2. The function
 n | t |
12 
  
 S (t )  exp 
1
m
is a characteristic function.
Proof. Proposition 3.1 has established the necessary and sufficient condition for  S (t ) to be a
characteristic function that required by Theorem 2.2. Then the function  S (t ) is a characteristic
function.
Theorem 3.3. The convolution of a variational Cauchy distribution is infinitely divisible.
Proof. The infinite divisibility is shown by using the property of characteristic function S n (t ) such
that satisfies necessary and sufficient condition in term of characteristic function, that is
Sn (t )  (S (t )) m . Based on Theorem 3.2, there is exist a characteristic function  S (t ) such that

 n | t |
 S (t )   exp  1 2 

  

m
1
m
n

  exp  n | t |    (t )
Sn
12

  

for any positive integer number m. The function Sn (t ) is a characteristic function from convolution of
a variational Cauchy distribution, so that the characteristic function of convolution of a variational
Cauchy distribution is an infinitely divisible.
4. Conclusion
The characteristic function of convoluted random variable from a variational Cauchy distribution is
obtained as  Sn (t ) with some basic properties are uniformly continuous and its complex conjugate of
characteristic function is determined by mathematical analysis methods. It is confirmed by graphically
that characteristic function of convolution a variational Cauchy distribution has lied only at the real
part and never vanish on the complex plane. The most important property of convolution a variational
Cauchy distribution is the infinite divisibility of its characteristic function.
References
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[3] Devianto D. 2016. On the convolution of a variational Cauchy distribution. Proceeding
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