On an extension of Lesche stability
Aziz El Kaabouchi
†
Institut Supérieur des Matériaux et Mécaniques Avancés du Mans
44 Avenue Bartholdi, 72000 Le Mans, France
Abstract
In this paper, we give a new method for proving the Lesche stability of several
functionals (Incomplete entropy, Tsallis entropy,
entropy). We prove also that the Incomplete
entropy for 0  q  1 are
Incomplete
  stable
q  expectation
  entropy,
q  expectation
Quantum-Group
value and Rényi
for all 0    q  . Finally, we prove that the new
value is
  stable
1
for all
0   1.
Notations and definitions
We denote by E f 
R
n
. For N  N * and   0 , we define the function d  , N : R N  R N  R  by
nN*
1/ 
 N
 
d  , N ( p, p ' )  
p i  p i'  .


 i 1

We recall that
for   1 , d  , N is a distance on R N

p, p' R N , d  , N ( p, p' )  0  p  p'
p, p' R N d  , N ( p, p' )  d  , N ( p' , p)
p, p' , p' ' R N , d  , N ( p, p' ' )  d  , N ( p, p' )  d  , N ( p' , p' ' )
for   1 , d  , N is a quasi-distance on R N .
p, p' R N , d  , N ( p, p' )  0  p  p'
p, p' R N d  , N ( p, p' )  d  , N ( p' , p)
1
p, p' , p' ' R , d  , N ( p, p' ' ) 
N
2
1
d , N ( p, p' )  d , N ( p' , p' ' )
Let C be a function defined on A a subset of E f . For N  N * and   0 , we denote by C N ,max the number


defined by (when it exists) C N ,max  sup C ( p) , p  A  R N .
Definition 1
A Function C defined on A a subset of E f is said to be   stable if C posses the following property
  0 ,   0 , N  N* , p, p' A  R N , d  , N ( p, p' )   
C ( p)  C ( p' )
 .
C N ,max
Remark.
The Lesche stability correspond to   stability for   1 .
Proposition 2.
Let 0     and C be a function defined on a subset of E f .
If C is   stable then C is   stable.
Proof.
It’s easy to see that for all x ]0,1[ , x   x  and thus if d  , N ( p, p' )   then d  , N ( p, p' )    /  .
Proposition 3.
If C1 ,C 2 are   stable with C1  0 , C2  0 and  ,   R  , then  C1   C 2 is   stable.
Proof.
We can suppose that C1  0 , C2  0 and ,  R * .
We remark that for all N  N * and   0 ,
2
C1  C 2 ( p)  C1  C 2 ( p' )
C1  C 2  N ,max
 C1 ( p)  C1 ( p' )    C 2 ( p)  C 2 ( p' ) 

C1  C 2  N ,max
C1 ( p)  C1 ( p' )
C 2 ( p)  C 2 ( p' )


C1  C 2  N ,max
C1  C 2  N ,max
C1 ( p)  C1 ( p' )
C 2 ( p)  C 2 ( p' ) 


 C1  N ,max
 C 2  N ,max
C1 ( p)  C1 ( p' ) C 2 ( p)  C 2 ( p' )

C1  N ,max
C 2  N ,max



Let   0 , there exists  1  0 ,  2  0 , such that N  N* , p, p' A  R N ,
d  , N ( p, p ' )   1 
C1 ( p)  C1 ( p' ) 
C ( p)  C 2 ( p' ) 
 and d  , N ( p, p ' )   2  2
 .
C1  N ,max
C 2  N ,max
2
2
We put   min 1 ,  2   0 , p, p' A  R N ,
d  , N ( p, p' )    d  , N ( p, p' )   1 and d  , N ( p, p' )   2 
Consequently
C1  C 2 ( p)  C1  C 2 ( p' )
C1  C 2  N ,max


2


2
 .
Definition 4.
Let q  0, \ 
1 . The incomplete entropy S qI is defined on


A
nN*
S qI ( p) 
1

N
i 1
pi
1 q
n
  p  [0,1] ,   p 

n
i
i 1
q


 1 by


for all p [0,1] N .
Theorem 5.
For all q  0, \ 
1 , the incomplete entropy S qI is   stable for all   1 .
Proof.
It’s easy to see that
i)
S 
I
q N , max


1
 S qI (1 / N )1 / q ,  , (1 / N )1 / q 
ii) For all q  0, \ 
1 , the function x 
1 N
1 q
1
1 x
Indeed if q  0,1 , then lim
1
x 
1 x
1
1
q
1
.
is bounded on 2,
 1 and if q  1 , then lim
1
x  
1 x
Let M  0 , such that for all N  N * \ 
1 ,
1
1 N
  
For all   0 , there exists    
M 
implies
1
q
1
q
1
1
q
1
1
q
 0.
M .
1/ 
 0 , such that for all N  N * , p, p' A  R N , d  , N ( p, p ' )  
3
S qI ( p )  S qI ( p ' )
S 
I
q N , max
 p
N
i 1

1 N
 M


N
i 1
'
i
 pi
1
1
q
 

N
p i'  p i
i 1
1 N
p i'  p i  M

N
i 1
1
1
q
p i'  p i

M  d  , N ( p, p ' )   M   
Definition 6.
Let
q  0, \ 
1 .
S q ( p) 
1

N
i 1
The
 p i q
Tsallis
entropy
Sq
is
defined
A
on


n
 p  [0,1] ,
* 
nN 

n
p
i 1
i


 1


by


 1


by
for all p [0,1] N .
q 1
Theorem 7.
For all q  0, \ 
1 , the Tsallis entropy S q is   stable for all   1 .
Corollary 8
a) For all   1,1 , the   entropy is   stable for all   1 .
b) For all q  0, \ 
1 , the Quantum-Group entropy is   stable for all   1 .
Proof.
a) Let
  1,1 .

S  ( p)  
N
i 1
  entropy
The

pi  pi    pi 

2
S
is
defined
A
on


n
  p  [0,1] ,  p
nN*
n

i 1
i
for all p [0,1] N .
1
S  1 ( p)  S1 ( p) and we conclude by proposition 3.
2
n




n
p

[
0
,
1
]
,
p i  1 is defined by
b) Let q  0, \ 
1 . The Quantum-Group entropy is defined on A 



i 1

nN* 
It’s easy to see that S  ( p) 

  p 

N
S qQG ( p)
i 1
q
i
  p i 1 / q
1
q
q


for all p [0,1] N .
1
q
q
q
1
It’s easy to see that S qQG ( p) 
S q ( p) 
S 1 ( p) 
S q ( p) 
S 1 ( p) and we conclude by
1
q 1
q 1
q 1 q
1 q
q
proposition 3.
For the proof of the theorem 7, we have need of the following lemmas.
Lemma 9.
x 1 q
is bounded on 2, by M 1  0 ;
1  x 1 q
1
b) For all q  1 , the function x 
is bounded on 2, by M 2  0 .
1  x 1 q
a) For all q  0,1 , the function x 
Indeed if q  0,1 , then lim
x 1 q
x   1 
x
1 q
 1 and if q  1 , then lim
1
x  1  x 1 q
4
0.
Lemma 10.
 

If   1 , then  p   p   d 
Let   0 , N  N * ,  pi , pi'  0,1N .
N
a)
'
i
i
( p, p' )  ;
,N
i 1
b) If   1 , then
 p   p 
 N 1 d1, N ( p, p' )  ;
 p   p 
  d1, N ( p, p' )  ;
N
' 
i
i
i 1
N
c) If   1 , then
' 
i
i
i 1
 
N

d) If   1 , then
p i  p i'
 N
1
1

d  , N ( p, p' ).
i 1
Proof.
a) We put I1  i  1,, N, pi'  pi and I 2  i  1,, N, pi  pi' .







 p'  

 p' 
p'   p' 
p' 
p'
For all i  I1 , we have: 1   i   1  i    i   1  i    i   1  i
 pi  

 pi 
pi   pi 
pi 
pi
  

 
 pi   pi' 


 p i  p i'
And
by
We conclude that for i  1, , N ,
 pi 
p    p   p
'
i
i
i
 pi'

.
we
deduce
that
all
, we deduce that
i  I2 ,
we
have:
.
  pi 
 
 pi'
    p
N
b) By a) we have,
symmetry

N
 pi'
i 1
i

 pi  pi'
N

 pi'
, and thus a) is proved.
p

i 1
i
 pi'

 1 , and by Holder’s Inequality,
i 1
 1

1


 1 , we obtain
 1 /  1 /(1   )

N
p
i

pi'

i 1
 N 
 1    pi  pi'
 
 i 1


 1 /    N



  11 /(1 ) 
 

  i 1


1

 N


pi  pi'  N 1 .


 i 1


c) The function x  x  is continuous on [0,1] , differentiable on ]0,1[ , by the Mean-Value Theorem, we have


    p
i  1, , N  , ci  pi , pi' , such that   pi   pi'

We deduce that
 
i  1, , N  ,  pi   pi'
 p   p 
N
Consequently,
i
' 
i
i 1
N
d) By c) we have

 
p i  p i'
i 1
N

i 1
N
Consequently,

 N
 
p i  p i'

 i 1



 
p i  p i'
N
1





 pi  pi'  .
1/ 
N
p
i
 p i' 1 , and by Holder’s inequality we obtain
i 1
 N  /( 1) 
 1



 i 1


1


 pi'  ci  1 ,

  d 1, N ( p, p' ) .


 N

 
p i  p i'   


 i 1

 N
p i  p i' 1  
p i  p i'

 i 1
i
d  , N ( p, p ' ) .
i 1
Proof of theorem 7.
5
1
1

N
1
1

d  , N ( p, p ' ) .
 max
It’s easy to see that S q
 S q 1 / N ),  , (1 / N  
  

For q  0,1 , let   0 , there exists   
 M1 
d 1, N ( p, p ' )   implies
S q ( p)  S q ( p' )
S q max


1 /(q )
 
N
 pq  p'
 i
i

i 1
1  N 1q
.
1 q



q
1  N 1 q
1 , p, p' A  R N ,
 0 , such that for all N  N * \ 
N


p iq

 N
N 1 q 
p i  p i'

 i 1
 

q
p i'
i 1

1  N 1 q

N 1 q d  , N ( p, p ' ) 





q
1  N 1 q

q
1  N 1 q
N 1 q d  , N ( p, p' ) q

1 N
  

For q  1 , let   0 , there exists   
 qM 2 
d 1, N ( p, p ' )   implies
S q max

 M 1 q  
1/ 
 0 , such that for all
 
N
S q ( p)  S q ( p' )
1 q

i 1

1  N 1 q
 N
q
p i  p i'

 i 1

q
p iq  p i'
N  N* ,
p, p' A  R N ,




1  N 1 q
q d  , N ( p, p ' ) 

1  N 1 q
qM 2    

Thus, the theorem 6 is proved.
The   stability of the Incomplete q  expectation
Definition 11.
Let q  0, \ 
1 et C a element of E f
The
q  expectation
Incomplete

p   p  [0,1] N ,

N
 p 
i
i 1
q
of
C
is
defined
by
 C  q ( p) 

N
i 1
 p i q C i

 1 where N  N * .

Theorem 12
a) For q  1 , the Incomplete q  expectation of C is   stable for all 0    1 .
b) For q  0,1 , the Incomplete q  expectation of C is   stable for all 0    q .
c) For q  0,1 , the Incomplete q  expectation of C is 1  unstable.
Proof
We have
N


 

 C  q N , max 
max   C  q ( p ) , p   p  [0,1] N ,
p i q  1 


 
i 1

N
 N




max 
p i q C i , where p satisfies
p i  q  1


i 1
 i 1

q
 max C i 1  max C i




i 1,, N

i 1,, N
6
for
all
 
a) For q  1 , let   0 , there exists    
q
d  , N ( p, p ' )   implies
1/ 
 
N
 C  q ( p)  C  q ( p' )
 C  q N ,max

p iq  p i'
 p  p 
N
q
' q
i
q
i
Ci
i 1

N  N* ,
 0 , such that for all

 C  q N ,max
p, p' A  R N ,
Ci
i 1
max C i 1q
i 1,, N
N


 q
p i  p i' 


i 1
 i 1


 qd 1, N ( p, p' )   qd  , N ( p, p' )   q   
 p  p 
N


' q
i
q
i
b) For q  0,1 , let   0 , there exists    1 /   0 , such that for all N  N * ,
d  , N ( p, p ' )   implies
 p  p 
N
 C  q ( p)  C  q ( p' )
 C  q N ,max

 C  q N ,max
max C i 1q

q

p i  p i' 


 i 1





 d  , N ( p, p ' )    

' q
i
q
i
i 1
d  , N ( p, p' )

q

  2  1q
1
c) Let q  0,1 , and    0 , for all   0 , there is N  2W  2 E  
2
  

1
p   pi i 1,, N , p i   
W 


*
  2  N , let p, p' defined by


1/ q
if i is odd and p i  0 if i is even
 
p '  p i'
Ci
i 1
i 1,, N
N
 p  p 
N

' q
i
q
i
Ci
i 1

 p  p 
N
' q
i
q
i
p, p' A  R N ,
1
, p i'   
W 
i 1,, N
1/ q
 pi
We have
N
q
  1 1 / q 
    W  1  1 ,

 W 
W
i 1  

W
 p  
q
i
i 1
N
d1, N ( p, p' ) 

i 1
q
  1 1 / q 
    W  1  1

 W 
W
i 1  

  
N
W
q
p i'
i 1
1
pi  pi'  N  
W 
1/ q
 2W 11 / q  
C  Ci i 1,, N , Ci  1 if i is odd and C i  0 if i is even
W

i 1

 C  q N ,max     W1 
 
1

W   0
 C  q ( p)  C  q ( p' )
W


W
And
1

1
/
q
 C  q N ,max
W


1
1
q
7

  W 11 / q


1  .
Consequently, the Incomplete q  expectation is 1  unstable.
The   stability of the Rényi entropy
Definition 13
1/ q
Let
q  0, \ 
1 .
The
Rényi
Rq
entropy
is
defined


nN
Rq ( p) 
ln 


N
i 1
 pi 
q
1 q



n
  p  [0,1] ,  p
A
on
*
n

i 1
i


 1


by
for all p [0,1] N .
Theorem 14
For q  0,1 , the Rényi entropy R q is   stable for all 0    q .
Proof.
 max
It’s easy to see that Rq
 Rq 1 / N ),  , (1 / N  


ln N 1q
 ln N .
1 q
For q  0,1 , let   0 , there exists    1 /   0 , such that for all N  N * \ 1,2, p, p' A  R N ,
d  , N ( p, p ' )   implies
 N q
 N
ln 
p i   ln 
p i'



R q ( p)  R q ( p' )
 i 1

 i 1

Rq max
ln N
 

q
 
 N q
pi ,
By the Mean-Value Theorem, there exists c q , N  

 i 1

 N q
 N
ln 
p i   ln 
p i'



 i 1

 i 1
 

N
Because

i 1
 p 
N
p iq  1 and
' q
i
 1 , then
i 1
 N q
 N
ln 
p i   ln 
p i'



 i 1

 i 1
ln N

By a) of lemma 10, we obtain
1
c q, N
 
q




Rq ( p)  Rq ( p' )
Rq max
q
 p 
N
' q
i
i 1





 such that


N





 p 
N
p iq 
i 1
' q
i
i 1
1
c q, N
 1 , consequently
N


 p 
N
p iq 
i 1
i 1
ln N

 d q , N ( p, p ' )
' q
i
1
c q, N
 p  p 
N

q
i
' q
i
i 1
q  d  , N ( p, p' )      .
References
[1] S. Abe, Instability of q  expectation value;
[2] S. Abe, Stability of Tsallis entrpy and instabilities of Rényi and normalized Tsallis entropies: A basis for
q  exponential distributions
[3] S. Abe, G. Kaniadakis, A. M. Scarfone; Stability of generalized entropies
[4] G. Kaniadakis, A. M. Scrafone, Lesche Stability of   Entropy. arXiv:cond-mat/0310728v2
[5] B Lesche, Instabilities of Rényi Entropies. Journal of Statistical Physics, Vol. 27, No. 2, 1982
[6] M F. Curado, F. D. Nobre, On the stability of analytic entropic form. Physica A 335 (2004) 94-106
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