Open Journal of Discrete Mathematics, 2011, 1, 127-135
doi:10.4236/ojdm.2011.13016 Published Online October 2011 (http://www.SciRP.org/journal/ojdm)
The Equilibrium Distribution of Counting
Random Variables
Shuanming Li
Centre for Actuarial Studies, Department of Economics, the University of Melbourne, Australia
E-mail: [email protected]
Received July 1, 2011; revised August 3, 2011; accepted August 15, 2011
Abstract
We study the high order equilibrium distributions of a counting random variable. Properties such as moments,
the probability generating function, the stop—loss transform and the mean residual lifetime, are derived. Expressions are obtained for higher order equilibrium distribution functions under mixtures and convolutions of
a counting distribution. Recursive formulas for higher order equilibrium distribution functions of the
 a, b,0  -family of distributions are given.
Keywords: Counting Random Variable, Equilibrium Distribution, Stop-Loss Transform, Mean Residual Life,
 a, b,0  Family, Recursive Formulas, Probability Generating Function
1. Introduction
Recently, there has been much attention given to higher
order equilibrium distributions associated with a given
distribution function (d.f.), see e.g., Fagiuoli and Pellerey
[1,2], Nanda, Jain and Singh [3], Hesselager, Wang and
Willmot [4] and the references therein. Equilibrium distributions arise naturally in ruin theory and play an important in various settings.
The first order equilibrium distribution of a claim size
d.f., in classical risk theory, can be interpreted as the
distribution of the amount of the first drop below the
initial reserve, given there is such a drop (see for instance
Bowers et al. [5], Chapter 12). Many results on the moments of the time to ruin, the surplus before ruin and the
deficit at ruin, heavily depend on the equilibrium distribution of the claim size d.f. [see Lin and Willmot [6,7]
for details].
Some classifications of reliability distributions are
based on properties of higher order equilibrium distributions. Whence, bounds for the right tail of the total claims
distribution and ruin probabilities, can be obtained from
the properties of equilibrium distributions associated
with the single claim size d.f., see [7-9].
Although much attention has been paid to the equilibrium distributions associated with a given d.f., most results are for continuous random variables. Instead, we
discuss higher order equilibrium distributions associated
Copyright © 2011 SciRes.
with a discrete probability function (p.f.). Throughout the
paper,  = 0,1, 2, and   = 1, 2, .
2. Notation and Definitions
Let X be a non-negative r.v. taking integer values, with
probability function (p.f.) p  x  = P  X = x  , survival
function P  x  = P  X > x  =  y  x 1 p  y  , x   and
n -th moment n = E  X n  .
Consider the equilibrium distribution of p, defined as
p1  x  :=
P  x
1
=
1
1
 p  y ,
x .
y  x 1
n
Now, define  n  := E  X    to be the n-th factorial
 n 
moment of X, where x = x  x  1  x  n  1 de0
notes the n-th factorial power of x and x  = 1. It is
well known in summation calculus (see e.g. Hamming
 k = xk
y
 n
n 1
 x
n 1
, for
n 1
n    , x, y  , and x  y. Hence the n-th factorial
moment 1: n  of p1 is given by
[10], p.182) that
1: n  = x n  p1  x  =
x 1
=
1
1
y 1
=
 y  1
1
1
x   p  y 
n
x 1
 p  y  x  ,
y 2
n
y  x 1
n 1,
x =1
OJDM
S. M. LI
128
y
1
=
1
y2
 n 1
=
1  n  1
 n 1
n 1
p  y n 1
=
pn 1  x  =
1  n  1
(1)
, for 1 = 1 .
Similarly, the probability generating function (p.g.f.)
of the equilibrium distribution p1 is given by
1  pˆ  s 
pˆ1  s  = s x p1  x  =
1 1  s 
x0
Pn 1  x  =
, 1  s  1 ,
=
1
1 y  x 1k  y 1
 p  k   k   x  1 .
(2)
1:1
1:1 1

y  x 1
 y   x  1  p  y  , x   ,
1: n 1
1:1  n  1
=
1: n 1
1:1  n  1
.
x0
1  pˆ1  s 
1:1 1  s 
1
, 1 < s < 1 ,
n!l :1
 p  y   y   x  1
 n
, x   , (3)
y  x 1
l =0
=
1
 n 
 
 p  y   y   x  1 ,
n
y  x 1
and accordingly
Copyright © 2011 SciRes.
y 1
n
n!l :1
 p  y 
y x2
t = x 1
 y   t  1 
n
y  x  2 
1
n
n!l :1
 p  y 
y x2
k
n
k =0
=
1
n
 n  1!l:1
 p  y   y   x  1
 n 1
,
y  x 1
l =0
verifies (3) also for n  1. Further, since Pn  1 = 1,
we conclude from (3) that
n!l :1 =  n  , n    .
(6)
l =0
Hence Pn  x  is also given by (4). To prove (5), use
P  x
and (6). □
pn 1  x  = n
Example 1: If X is geometrically distributed with
p  x  = 1     x and survival function P  x  =  x 1 ,
for x   and    0,1 then
Define similarly the subsequent equilibrium distributions of p, from the third order p3  x  = 1 2:1 P2  x 
up to the n-th order pn  x  = 1  n 1:1  Pn 1  x  for
x  , where the following theorem gives an expression for Pn  x  and pn 1  x  .
Theorem 1 The survival function Pn  x  , of the n-th
order equilibrium distribution pn can be expressed as
n 1
Pn  t 
l =0
Pm  x  =
with pˆ 2 1 = 1, and the corresponding survival function
P2  x  =  k  x 1 p2  k  .
Pn  x  =
n:1
n:1
Then the p.g.f. of p2  x  is given by
pˆ 2  s  = s x p2  x  =
1
n 1
where 1:1 is the first order moment of p1 . The factorial moments of p2 are obtained as in (1) to be
2: n  =
t  x 1
1
=
1 k  x 1
1
(5)
,
l =0
Now define the equilibrium distribution of p1 , or
equivalently, the second order equilibrium distribution
of p :
P  x
p2  x  = 1
=
n
y  x 1
t  x 1
1
y  x 1
 p  y   y   x  1
 pn 1  t  = 
=
 p1  y  =    p  k 
 n 1
where l :1 is the mean of l -th order equilibrium distribution and 0:1 = 1 is the mean of p (or 0-th order
equilibrium distribution).
Proof: (2) shows that (3) holds for n = 1. By induction, assume that (3) holds for any n, then
with pˆ1 1 = 1 and its survival function is
P1  x  =
 n  1
=
1
 m 
1
 m 

 1    y  y   x  1
m
y  x 1
1    x 1  y y  m  =  x 1 ,
y 0
where the last equality holds true as
 m  =  y  0 1     y y  m  ,
by definition.
This shows that any order equilibrium distribution of
the geometric distribution is identical to the original distribution.
Example 2: Let X be a discrete uniform with
1
p  x =
, x = 0,1, 2, , m.
m 1
(4)
As
 n  =  x =0
m
 n 1
n
 m  1
x 
=
,
1
m

m

   1 n  1
OJDM
S. M. LI
then for n  m,
pn  x  =
1
m
n
 y   x  1 

 n  y = x 1  m  1 
n  n  1
=
m   x 1

 m  1  y =0
n
 n  1 m  x  
=
n 1
 m  1 
n > m, pn  x   0.
while for
n 1
129
for an arbitrary n = k and m  , then for n = k  1
the left-hand-side (LHS) of (10) becomes
 n 1
y
LHS = x m   y  x 
y
n 1
y
= y x m   y  x 
y
  m  k  x
, 0 x  mn,
=
n
n
k
k =1

n
k
S
= S = = S =1, S
n 1
k
=S
n
k 1
 m  k 1
 m  k 1
s = n!

sn 1  s y
1  s 
x =0

n 1
sn
x  s x = n!
n
x0
1  s 
n 1
y 1
mk 2
n
=
x =0
n! s n  k
k
y   s y , (8)
 k!
n  k 1
k =1
1  s 
n
, s   0,1 .
 m
 y  1
=
 m  1
(9)
.
 1  n  1! 1  pˆ  s 
pˆ n 1  s  =
n 1
 n 1
1  s 
n

pn 1  x  =
 n 1

k =1
nk
k!
(11)
 k 
1  s 
n  k 1
.
n 1
 n 1
 y  x 1p  y   y   x  1
n
,
pˆ n 1  s  = s x pn 1  x 
x0
=
m 1
, then (10) holds
 n  1! n  1
Proof: Since
(10)
when n = 0 and m  . Assuming that it also holds
Copyright © 2011 SciRes.
  n  m  2
The following theorem gives an expression for pˆ n 1  s  .
Theorem 2
n
m!n!
n  m 1
 y  1  .

1
!
m

n


 x =0x
y
  n  1   m  1
Let pˆ n 1  s  =  x  0s x pn 1  x  be the p.g.f. of pn 1 .
1 n y
s
1 sy
I n 1 
y s , while I 0 =
. Then
1 s
1 s
1 s
(8) is verified by mathematical induction. To prove (9),
simply let y   in (8).
Lemma 3 For y , m and n  ,
Proof: Since
m!k!
 y   m  k  
 m  k  1!
= y n  m 1
In = n
x  m   y  x 
m k  2
□
Proof: Let I n =  x =0x  s x . It is easy to show that
y

k! m  1 !
 y  1

 m  k  2 !
y
Proof: See p.160 in [10]. □
Lemma 2 For n   and y    ,
x
m!k!
k! m  1 !
with s0n = 0 and 1  k  n .
x
k
n
m
m  n 1
0 x  y  x  dx = y   m  1, n  1
snn11 = snn =  = s11 = 1 , skn 1 = skn1  kskn ,
n
 y  x 
Remark: (10) is a discrete version of the formula
 nS ,
n
k
with S0n = 0 ,
y 1
x =0

 y  1
 m  k  2 !
m! k  1 !
m  k  2
=
.
 y  1
 m  k  2 !
k =1
1
1
k 
 m  k  1!
m  k 1
 m  k  y  1  k!m!

 m  k  1!

(7)
where S , s , k = 1, 2, , n, called the first and the
second Stirling numbers respectively, are given recursively by
y  y  1
=  y  1
n
k
n
n
 m
y
   x  m  x m   y  x 
x =0
 n  = S k , and n = s  k  , n   ,
n 1
n 1
k 
x =0
In deriving the properties of the higher order equilibrium
distributions of p, the following lemmas will be needed.
Lemma 1 The relationships between raw and factorial
moments are given by
n
k
y  x k
x =0
3. Properties of the Equilibrium Distribution
n
k 
=
n 1
 n 1
n 1
 n 1
s x  p  y   y   x  1
x0
y  x 1
y 1
 p  y  s x  y   x  1
y 1
n
n
x =0
OJDM
S. M. LI
130
=
n 1
y 1
 p  y  s y 1 s  x x n
 n 1
y 1
x =0



1  s y  1  y 1
n

=
 p  y  s  1 n!
n 1  
 n 1 y 1
1  s   s 

nk
n
 1 n! y  k   1  y 1 

 , by Lemma 2 ,
n  k 1  
k!
 s  
k =1
1  s 
n
 1  n  1! 1  pˆ  s 
=
n 1
 n 1
1  s 
n 1

y 1
 n  1!  1
n

 n 1
nk
1  s 
k!
k =1
 k 
n  k 1
n:m =
n!m!  n  m 
, m, n   ,
 m  n  !  n 
n!
m
 n 
k!s m
  n  kk ! n  k  ,
E  X m  x 
(12)
m    , n   , (13)
k =1
E  X m  x  =
=
 n 
n
 n 
y 1
 n 1
m! n  1 !
n!m!  n  m 
=
.
 m  n  !  n 
=
m
k =1 k
n
n
Let X m be a random variable following the probabil n
ity function pm . Define E  X m  x  to be the n -th
n
factorial stop-loss transform of pm and E  X m  x 
Copyright © 2011 SciRes.
 m 1
Pm 1  x 

k  x 1
= 1
while
hm  x  =
=
(16)
 m  1  m Pm  x 
1
.
rm 1  x 
= 1
m
E  X  x  =  k =1 k  skn Pk  x  1 .
(15)
k!skn
  m  k ! m  k  Pm  k  x  1
k =1
n
(17)
rm  x  = E  X m  x X m > x  =
, by Lemma 3 ,
To prove (13), use  n:m =  s n: k  , as stated in
Lemma 1, and (12). □
Consider
now
the
stop-loss
transform
π  x  = E  X  x  of the r.v. X (where the notation
 a  = aIn a > 0 ). For n    and x  , denote by
E  X  x  the n-th stop-loss transform of X (with
n
probability function p) and by E  X  x  its n-th factorial stop-loss transform. Theorem 1 and Lemma 1 show
n
that E  X  x  =  n  Pn  x 1 and
k =1
Proof:
x =0
 p  y  y  n  m  n  m !
y 1
 m 
rm  x  = 1 
x 0
y 1
m!
mk 
function of pm . Then the following result holds.
Theorem 5 For x   and m    ,
hm  x  =
 p  y  x m  y   x  1
 m 
k!s n
n
  m  kk ! E  X  x 
Proof: The argument is similar to that in the above
proof. □
N o w d ef in e rm  x  = E  X m  x X m > x  , x  ,
m    , to be the mean residual lifetime (MRL) of
P  X m = x
pm , and hm  x  =
to be the hazard rate
P  X m  x
n:( m ) = x m  pn  x 
n
m!
n
where n:m is the m -th moment of the distribution
pn .
Proof:
=
(14)
m!n!  n m 
=
P  x  1 ,
 m  n  !  m  n  m
=
Theorem 3
 n m 
m!n! E  X  x 
=
 m 
 m  n !
 n
.
□
n: m  =
to be the n-th stop-loss transform of pm . The following
theorem holds.
Theorem 4 For n    and m, x  ,
  k  x  pm  k 
k  x 1
 k   x  1  pm  k  
Pm  x 
E  X m   x  1 
Pm  x 
 pm  k 
k  x 1
1
Pm  x 
 m 1
 m  1  m 
pm  x 
pm  x   Pm  x 
Pm 1  x 
Pm  x 
=
,
by (14).
1
Pm  x 
1
pm  x 
1
1
=
, by (16) .
m m 1 Pm  x 
rm 1  x 
1
 m  Pm 1  x 
This proves the conclusion. □
OJDM
S. M. LI
where X 1 , X 2 , , X n are i.i.d. with common p.f. p,
which can be computed recursively by
4. Equilibrium Distribution and
Convolutions
This section studies the equilibrium distribution of the
n-th fold convolution of a counting distribution.
The following lemma shows that the usual formulas
n
n
 x  y  =  nk =0   x k y n  k
k 
and
n! l1
n
l
x1  xmm ,
 x1  x2    xm  =  l1 lm = n
l1! lm!
for n  , also hold for factorial integer powers.
Lemma 4 For n  
 x  y
n
 n  k nk
=    x  y   ,
k =0  k 
n
131

*kn = E  X 1  X 2    X n 1   X n 
The following Theorem gives an expression for
pm*n  x  .
Theorem 6 For m    , n  2,3, and x  ,
 x =
*n
m
p
n!
l  l 
 l  (19)
x1 1 x2 2  xmm .
l1  l2  lm = n l1!l2! lm!
Proof: Clearly (18) holds for n = 1. Assume it holds
for an arbitrary n, then
 x  y
=  x  y
 n
 x  y  n
n
 n  k nk
=    x   y    x  k    y  n  k  
k =0  k 
 n  1  k   n 1 k 
= 
x y
k =0  k 
and (18) holds by induction. A similar argument proves
(19). □
Next we will discuss the high order equilibrium distributions of the convolution of p with itself.
x
Let p*n  x  =  k =0 p  k  p*n 1  x  k  be the n-th fold
convolution of p, with p*1 = p, and pm*n be the
m-th order equilibrium distribution of p*n . Consider
*kn , the k-th factorial moment of p*n , then
pm*n  x  =
=
=
=
=
=
k!
 l    l    l  ,
1
2
n
l1  l2  ln = k l1!l2! ln!
Copyright © 2011 SciRes.
1
 x
(20)
m
* n 1
  l  l   m l  pl  x  .
l =1 

m
*mn 
 p*n  y   y   x  1
y
m

 p k  p 
x

t  x  k 1
*n
m
p* n 1  t  t   x  k  1 
 p k  p 
* n 1
*n
 m  k  x 1
* n 1
 m
 m 1
yk
p k  
*n
 m k 0

 m 1
 p  k   p* n 1  y  k   y   x  1
m

 m 1
y  x 1
*n
 m  k  x 1
m

 y  k   y   x  1 
 p  k   p* n 1  y  k   y   x  1
*n
 m  k =0
m

* n 1
*n
 m  y  x 1k =0
m

 m 1
*n
 m  y  x 1
y 0
 m 1
 y   y   k   x  1 
 m 1
x
m
 p  k  pm* n 1  x  k    *n  p  k 
 m
k =0
k  x 1
m 1 m  1


 l   m 1 l 
  p* n 1  y   
  k   x  1  y
l 
y 0
l =0 
=
*m n1

*n
m
p * pm
* n 1
 x 
m
*mn 
 p k 
k  x 1
m 1 m  1


 l  * n 1

  k   x  1   m 1l 
l 
l =0 




k 
*kn = E  X 1    X n  

k!
l 
l  
X1 1  X n n 
= E 
l1  ln = k l1! ln!

m

n 1

* n 1
Proof:
n
n
 n  k 1 n  k
 n  k n 1 k 
=    x  y       x  y 
k =0  k 
k =0  k 
n 1
 n   k   n 1 k  n  n   k   n 1 k 
= 
   x y
x y
k =1  k  1 
k =0  k 
p * pm
*mn 

n
 n 1
*m n1
(18)


k
k 
l 
k l
=    E  X 1  X 2    X n 1  E  X n  


l
l =0  
k
 k  * n 1
=     l   k l  .
l =0  l 
 x1  x2    xm  
=
k 
=
*m n1


*n
m
p * pm
* n 1
 x 
m

 p  k  k   x  1
m 1
 m  1 * n 1
  m 1 l 
 l 

*n
 m  l =0
l 
k  x 1
OJDM
S. M. LI
132
=
*m n1

p * pm
* n 1
*n
m
  m 1l 
* n 1
=

* n 1
m

 l 1
l 1

*n
 m
 m  1
 l 
l =0 

1

= z g * g m
*  1
m 1
pl 1  x 
p * pm* n 1  x  
*n
m
m
 x 
m
m
 l 
*n
 m  l =1
 
  l   m l  pl  x  .
where z =
 x   1  z  g  x  ,
 1
. This shows that the m-th order
m   1
equilibrium distribution of a negative binomial is a mix*  1
ture of the distributions g * g m  and g , where the
 1
.
mixing factor z =
m   1
* n 1
5. Equilibrium Distribution of a Mixture
This completes the proof. □
Remark: Theorem 6 gives a recursive formula for the
high order equilibrium distributions of the convolution
pm*n . First obtain pl  x  and *lk , for l = 0,1, , m
and k = 1, 2, , n. Then compute the starting p.f.
pm  x  , followed by the convolutions pm*2  x  up to
pm*n  x  .
Example 3 Consider X  negative binomial  ,   ,
for   2 and    0,1 , that is
 x    1
 x
p  x = 
 1     , for x  .
x


Since the NB  ,   distribution can be viewed as the
 -th convolution of a geometric distribution
g  x  = 1     x , then p  x  = g *  x  and the above
theorem can be used to compute pm  x  = g m*  x  recursively.
Here the k-th factorial moment of g  x  is
  
 ,
1   
 k  = 1     x 0x  k  x = k! 
k
by Lemma 1. Now Pk  x  =  x 1 and
*k =
k!
 l   l    l 
1
2

l1  l2  l = k l1!l2!l !

  
= k!

1 
k

1
 1
    k    1
= k!
.

  1!
 1 
After some simplifications, we have
g m*  x  =
*m 1


=
*
m
1
*m
g * g m
*  1
 x
m
*  1
  l  l   m l   pl  x 
l =1 

m
 1
*  1
g * g m   x 
m   1

m
1    x
m   1
Copyright © 2011 SciRes.
p  x  =  p  x   dU   , is given by
0

pn  x  = 0 pn  x   dU n   ,

(21)
E  X     dU  

where dU n   = 
.
n
E X   


Proof
( n 1)
n
pn  x  =
p  y   y   x  1 


n
 n
y  x 1
 n 1
=n
=
l1  l2  l = k
k
This section discusses the equilibrium distribution of a
mixed p.f.. For x  . let p  x   , be the conditional
distribution of X, given  =  .
First assume that  has a continuous distribution
function U with density u over  0,   . Then the p.f. of
X, given by p  x  = 0 p  x   dU   , is a U-mixture
of distributions. The high order equilibrium distributions
of p are given in the following theorem.
Theorem 7 The n-th order equilibrium distribution of
the mixed p.f.
E  X   x  1 
E  X n 


n
E X


= 0 n
0 E  X   x  1 
 n 1

n 


 n 1
E  X   x  1  
E X

n
 

 dU  

 E  X  n    dU  


n
E X   


= 0 pn  x   dU n   .

□
This theorem shows that the n-th order equilibrium
distribution of a mixed p.f. is the mixture of the n-th order equilibrium distribution of the conditional p.f., mixed
by a new distribution U n .
An important special case
is the mixture of geometric
1
p.f.’s, where p  x  =  1     x dU   . Geometric
0
mixtures benefit from an important property, is that they
OJDM
S. M. LI
are completely monotone distributions, in the sense that
n
 1  n p  x   0, for all x, n  . Then pn  x   =
1    x and
E  X     dU  

dU n   = 
n
E X   


n
n
=
=
=
x 1   
x
x0
E X

discrete p.f., that is p  x  =  j =1 f  j  1   j   jx , for
k
x  , where 0 <  j < 1,
dU  
k
x

(n)
x
 j

 1 j
n! n
1   
n 1
, by()
showing that the n-th order equilibrium distribution of a
geometric U-mixture is still a geometric, with same parameter. Here the new mixing density is proportional to
  
the original one, un    
 u   .
1 
Example 4 (Waring Distribution)
If p  x   = 1    x , for x  , and
n
 a 1 1   
b 1
,
 1    
p  x = 0
x  a 1
d
6. Equilibrium Distribution of the (a,b)
Family
  a, b 
=
  x  a, b  1
  a, b 
b  a  b 
 a
  x  a
  x  a  b  1
We consider here the equilibrium distributions of the
 a,b  class of discrete distributions, or more precisely,
of the important subclass of the  a, b  family called the
 a, b, 0  class, see [11,12]. This class of counting distributions has support on the non-negative integers on
which the recurrence relation p  x  =  a   b x   p  x  1
,
which is called a Waring distribution.
It follows that
pn  x   = p  x  
= 1    x , un     a  n 1 1   
b  n 1
,
which is a   a  n, b  n  distribution when b > n.
Then
pn  x  = 0 1     x dU n  
1
=
  x  a  n,b  n  1
  a  n, b  n 
Copyright © 2011 SciRes.
The following theorem gives the aging properties of
the higher order equilibrium distributions of geometric
mixtures.
Theorem 8 The n-th order equilibrium distribution of
a geometric mixture is DFRd and IMRLd . Also Pn  x 
 Pn1  x  , for x  0, or equivalently, pn < st pn 1 .
Proof: Since pn is also a geometric mixture, it is
completely monotone. Then pn is DFRd , see [9].
Further, rn  x  = 1  hn 1  x   , by (17), hence pn is
IMRLd . Lastly,
Pn  x 
r  0 1
= n
 1,
Pn 1  x  rn  x   1
by (16). □
for   (0,1), then
=
n

 f  j 

where f n  j  =
, for j = 1, 2, , k .
n

k  i
 i =1  1    f  i 
i 

n
b
k
pn  x  =  f n  j  1   j  jx , x   ,
  

 dU   ,
 1 
1
 j =1 f  j 
= 1, then
x0
n
E X   


f  j  > 0 and
j =1
1    dU  
1
when b  n,
If instead, the geometric p.f. is mixed over another

n
E X   


  a, b 
b  n   a  b   x  a  n
 a  n
  x  a  b  1
then pn  x  does not exist.
=
n 
1    dU  
u   =
133
holds for x = 1, 2, . The members of the this class are
binomial, Poisson and negative binomial distributions
(with their corresponding special cases). It is easily seen
that
1 =
 ak  b   k 1
ab
and  k  =
, for k  2 .
1 a
1 a
Then we have the following recursive formula for pn .
Theorem 9 The n-th order equilibrium distribution
pn in the  a, b, 0  class of distributions satisfies the
following recursion for n    :
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S. M. LI
134
pn 1  x  1
= apn 1  x  

which in turns implies
 n  1 1  a  x  n  1
n
an  a  b
 n  1 1  a   an  a  b  ax 
an  a  b
n
pn  x  1
(22)
pn 1  x  1 = apn 1  x  
pn  x  , x   .

The starting points of the recursion are
1
p1  x  =
P  x  , p1  0  =
1
1
1
1 p  0  
n! 
n 1
n 1  1
 1 1  p  0     k =1
 n  
k!

n 1 k
 k   ,
pn  0  =

=
for n  2.
b

Pro of: p  x  =  a   p  x  1 , or equivalently,
x

 a  b  p  x  =  x  1 p  x  1  axp  x  , for x = 0,1, 2,.
Then
 a  b  pn  x 
n
 n 1
=
 a  b   p  y   y   x  1 , n  1

y  x 1
n
=
n 


  y  1 p  y  1  y   x  1
 yp  y   y   x  1
 n 1 
y  x 1
=
n 


 n   y  x  2
yp  y   y   x  2  

a
y  x 1
n
 n 

a
=
 n 1
 p  y   y   x  1
y  x 1


n
y x2
 n 
n
 n 
 p  y   y   x  2 
 n 1
y  x 1
 p  y   y   x  1
n
y  x 1
n  x  n
 n 
 p  y   y   x  1
 n 1
y  x 1
n  n 1
p  x  1   x  n  1 pn  x  1
 n  n  1 n 1
a
n  n 1
p  x   a  x  n  pn  x  ,
 n  n  1 n 1
Copyright © 2011 SciRes.
=
 n 1
n
 n 
n
 n 
n!
 n 
 p  y   y  1
y 1
 n  1
 n 1 k   k 
y
  1
k =0  k 
n 1
p  y   
y 1
n 1

 1
k =0
n 1 k
k!
 p  y  yk 
y 1
n 1 k
n 1 1

 
n! 
n 1
 k   .
=
 1 1  p  0    I  n  2  
k!
 n  
k =1

This completes the proof. □
For another subclass of the  a, b  family, the
 a, b,1 class of distributions, the relation p  x  =
 a   b x   p  x  1 holds for x  2, where p  0  is
an arbitrarily selected value in  0,1 . In this case, it is
ak  b
 , for k  2. The
easy to see that  k  =
1  a  k 1
above recursive formula (22) and those for the starting
points still hold true here, the only change being that
 n 1
1 =
 n 1 
 
 p  y   y   x  2 
n  x  n  1
a


 y   x  n   p  y   y   x  1 
a  x  n 
=
n 1
 n   y  x 1
a
 n 1
1 a
, we get (22).
an  a  b
=
Finally, we have
and
pn  0  =
 n  n  1
 an  a  b  ax  pn  x  , n  1
 n 1 n
 n 
Since
 n  n  1
p  x  1
 n 1 n n
p 1   a  b  1  p  0  
1 a
.
7. Conclusions
This paper investigates the higher order equilibrium distributions of counting random variables. The above results can be used in Risk Theory to derive bounds of ruin
probabilities in the discrete time risk model. They also
lead to the factorial moments of three related random
variables: the surplus before ruin, the deficit at ruin and
the time of ruin (see Li and Garrido [13] for details).
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