This research was supported in part by the Air Force Office of Scientific
Research under Contract AFOSR 75-2796 and the U.S. Army Research Office
under Grant DAH-C04-74-C-0030.
POWER SERIES DISTRIBUTIONS,
A DUAL CLASS AND SCME EXTENSIONS
Carlos B. Segami
Institute of Statistics Mimeo Series 1060
Department of Statistics
University of North Carolina at Chapel Hill
April, 1976
CARLOS B.
SEGAMI.
Power Series Distributions, a Dual Class
and Some Extensions.
(Under the direction of NORMAN L.
JOHNSON and GORDON D.
SIMONS.)
The class of power series distributions is considered. A
dual class of absolutely continuous distribution functions
is defined using the expression
P(~
as a function of the parameter x.
are discussed,
~
00
k)
=
L a.x
j=k J
j
00
/
L a.x j
j=O J
Properties of this class
in particular, the closure of the class under
weak convergence.
The class is then extended by removing the
condition that the coefficients a. be non-negative.
J
properties of this new class are studied.
Again,
Its closure under
weak convergence is found to be the class of all distribution
functions of non-negative random variables.
Finally, an
extension to the bivariate case is introduced.
ACKNOWLEDGEMENTS
I would like to express my deep feeling of gratitude
toward my advisers, Dr. N.L. Johnson and Dr. G.D. Simons.
Their continuous guidance and encouragement,especially in
the difficult moments, made this work possible.
It has
truly been a privilege to work with them.
I would like to thank the members of my committee,
Dr. W. Hoeffding, Dr. D. Quade, Dr. E. Wegman and,
in
particular, Dr. W.L. Smith whose help made possible the
proof of Theorem 3, Chapter I.
My thanks go to the Department of Statistics which
provided me with financial support throughout my graduate
studies.
To June Maxwell, thank you for a very precise and expeditious typing of this work.
Finally,
I would like to express my gratitude to the
person without whose determination, support and encouragement
I would not have been able to complete my graduate education:
my wife.
ii
e·
to maJUa a.nd ~ a 6.<a
TABLE OF CONTENTS
Acknowledgements----------------------------------------- ii
Dedication-----------------------------------------------iii
INTRODUCTION--------------------------------------------CHAPTER I:
The Class C
1.1
Power Series Distributions---------------------
5
1.2
Some Properties of Power Series Distributions--
7
2.-
P-----------------------------.------ 9
The Class C------------------------------------ 9
Some Simple Properties of C-------------------- 14
3.1
3.2
The Class
3.3
Necessary Conditions for a Distribution
Function F to Belong to C --------------------- 17
3.4
The Closure of the Class
3.5
Relationships Between the Weak Convergence of
C---------------------
25
Sequences in C and P -------------------------- 35
CHAPTER II:
The Class V
1.-
Introduction----------------------------------- 45
2.-
The Class
3.-
Necessary and Sufficient Condition for a Given
Distribution Function to Belong to V ---------- 49
4.-
The Closure of the Class
CHAPTER III:
V------------------------------------ 46
V---------------------
54
The Bivariate Case
1.-
Introduction----------------------------------- 60
2.-
Independence of
3.-
Independence of X and Y -------------------.--- 73
~
and
n-------------.-----.----
70
APPENDIX------------------------------------------------- 76
REFERENCES----------------------------------------------- 79
iv
INTRODUCTION
x2
Let X be a random variable with a
v
with v degrees of freedom.
distribution
Then
If v is even, integrating by parts we obtain
Pr(X s::x)
v
1
- e-
~xf
2
11
\.
+
~x
+
_2_
I!
1
(~x)2
2!
( 1'2X
)~V-l }
(~v-l)!
)~v+l
'2X
(~v+l)
(
+ .' .. +
!
+ •.• }
so that
Pr(X v s::x)
where
~
Pr(~
=
x ~~v)
(1)
is a Poisson variable with expected value
x
~x.
This relationship is discussed by Johnson [5], where
several extensions of it are shown:
Let
(i)
X'
, have a non-central XL distribution with
V,I\
noncentrality parameter A and v degrees of freedom.
Pr(X
I
V,A
s::x)
Then
dt.
Lv+j
22,
r(~v+j)
Again, if v is even we obtain
Pr(X'\)
,
s::x)
A
1
)~V+j+l
'2X
(~\)+j+l)!
(
+ ... ]
so that
00
Pr(X'\1,/\1
where £\ and
::; x)
~x
=
are Poisson variables with expected values
\/2 and x/2, respectively.
Furthermore, if £A and
~x
are
independent, then we can write
Pr(X'
::; x)
\I,A
(i i)
Let F'\1,\1 ,/\1 have a non-central F distribution
0
with \I and \1 degrees of freedom and non-centrality para0
meter A.
If \I is even, then
Pr(F'\I \1 ,/\1 ::; ~)
\I
0
where
~'
=
Pr(~'
-
2,
has a negative binomial distribution with mean
x
2
and variance
x
x
Yz\l --(1 + --)
0\1
\1
=
0
0
Poisson distribution with mean A/2.
x
-(1
2
x
, and £
0
In particular,
+
--)
\1
if F' has a central F distribution, then
Pr(F'
\1,\1
::;
0
~)
\I
=
Pr(~'
(2)
In each of these cases, a continuous distribution is
related to a discrete distribution, or, more precisely, a
family of continuous distributions indexed by a discrete
parameter (e.g. x 2 )
is related to a family of discrete
distributions indexed by a continuous parameter (Poisson).
Other relationships of a similar nature are studied
by Sarkadi [16].
For example, let us consider a lot with
N elements of which some are defective.
2
Let
l;
be the
fraction of defective items and suppose that
form prior distribution.
lot,
~
has a uni-
If we select n elements from this
let X
be the number of defective items found.
nN
Then,
Sarkadi shows that
Pr(X n+ 1 , N+ 1
In this context, if
A and A
~
k+ 11
~
=
M+I)
N+I
has a Poisson distribution with mean
~
has a uniform prior distribution over the half-
line (0,00), then
Pr(A
<
Q,I~
=
k)
=
Pr(~ ~ k+IIA = £)
Similarly, if
This relation is equivalent to (1) above.
~n
has a binomial distribution with parameters n and
~
has a uniform prior distribution, then
Pr(z;; <
p I~n
=
k)
=
Pr(~n+l ~ k+ll z;;
=
~
and
p)
This is equivalent to (2), above.
These relationships suggest the possibility of a
similar connection between larger classes of discrete and
This possibility is explored
continuous distributions.
in this work, where the class of power series distributions
is considered.
As it turns out, a "dual" class of absolutely
continuous distributions can be defined for the class of
power series distributions.
Some properties of this dual
class are studied, in particular, closure under weak convergence.
In Chapter II, a larger class of absolutely
3
continuous distribution functions
is introduced and in
Chapter I I I the bivariate case is studied.
4
e-
CHAPTER I
THE CLASS C
1.1
Power Series Distributions
In 1950, A. Noack [13] introduced a class of discrete
distributions known as "Power Series Distributions."
<Xl
f(x)
=
where a. is real and a.
J
Define a random variable
a
for each j = 0,1 ,2, ...
a
<
:::;
<Xl
j
=~
Pr(~=j)
u
<
with distribution
~
a.x
where x is fixed and
Ix I
a.x
j=O J
~
J
j
I
Let
j = 0,1,2, .. ,
f(x)
x
<
U,
Then
~
is said to have
a power series distribution.
Later, this definition was extended by Khatri
multivariate distributions.
~
n
The random variables
[10] to
~1'~2'"
are said to have a joint power series distribution if:
Pr(~1=jl'~2=j2""
'~n=jn)
jl j2
jn
. xl x
... x
2
n
J l J 2 ·· ,I n
f (x 1 ' x 2 ' . , . , x n )
a..
for jl = 0,1,2, ... ; j2
=
0,1,2, ... ; ... ; j
and a . .
JlJ2
.
"')n
~
=
0,1,2, ... ,
..
Jl J2
.
In
-o···I· -oa..
. Xl x 2 ... x
J J J 2 ... J n
n
I nl
l
<Xl
where f(x l ,x 2 , .. " x ) =
n
n
I·
<Xl
a for all jl,j2" ,·,jn'
Examples
(i)
Poisson:
j
Pr(~
x / j !
j)
e
and x
Negative binomial:
Pr(~
and 0 < x < 1, N
(iii)
=
>
j)
=
0,1,2, ...
j
=
0,1,2, ...
j
=
1,2,3, ...
O.
j)
=
(l +x)
x
where
(iv)
_Pl-p
=
and
N
O<p<l.
Logarithmic series:
Pr(~
(v)
j
Binomial:
Pr(~
and 0
...
O.
>
(ii)
o, 1 ,2,
j
x
<
x
<
j)
=
-£n(l-x)
1.
Multinomial:
6
(vi)
Multivariate negative binomial:
k
(r (N
+ i~ 1 j i) / j 1 ! •.. j k!
j
j
r (N)) XII ... x k k
=
for j.
1
1.2
=
0,1,2, ... , where
0
x.
$
1,
$
1
\~
L1 =
IX.
1
$
1,
Some Properties of Power Series Distributions
~
Let the random variable
have a power series distri-
bution, given by
a.x
P(~
=
j)
j
-L-
=
j
f(x)
o, 1 ,2,
=
. ..
Then, following the notation in Johnson and Kotz
[6], the
probability generating function is:
get)
= E(t~) =
f(tx)/f(x)
The moment generating function is
~~(t)
=
t
=
gee )
t
f(xe )/f(x)
and the factorial generating function is
g(l+t) = f(x(l+t))/f(x)
.
The r-th factorial moment and the r-th factorial cumulant
are given by
).J(r)
7
and
r
r
K(r) = x (d/dx) log f(x)
The factorial cumulants and the cumulants satisfy the following recurrence relations:
and
dK
K
r +l
r
= x(fX
Khatri [10] shows the following property:
"The power series
distribution is uniquely determined from its first two cumulants (or moments)" and this is further extended to the
multivariate case where the means, variances and covariances
determine the distribution.
In the multivariate case, a subclass, the class of sumsymmetric power series distributions, is studied by Patil
[ 14 ] .
This subclass possesses several characteristic pro-
perties, some of which are:
(i)
The regression curve for a bivariate
sum~
symmetric power series distribution determines the distribution uniquely in form, i.e., it determines the power
series f(x l ,x ).
2
(ii)
A bivariate sum-symmetric power series distribu-
tion is uniquely determined in form by its single marginal
distribution alone.
(iii)
The class of sum-symmetric power series distribu-
tions consists of the power series mixtures on the parameter n (sample size) of the multinomial distributions.
8
2.
The Class P.
We will restrict ourselves to a subclass of the class
of power series distributions.
radius of convergence U
>
f(x) --
Let
0, where
a.
I.
oo
Oa.x j have
J=
J
~Oforallj.
J
For
each value of x in (O,U), this power series determines a
power series distribution, that is, a function g of the form
a. x
g(j;x)
We let P
=
j
--Lf(x)
o, I ,2,
j
...
be the class of power series distributions g for
which the corresponding power series f satisfies
00
lim f(x)
xtu
=
j
lim I a.x
xtu j=O J
=
00
We let G(o;x) be the distribution function corresponding to
g(o ;x).
Notation 1
For a power series
radius of convergence U
to denote the fact that
f(x) =
>
I.
oo
Oa.x j
J=
J
with a.
J
>
0 and
0, we will use the notation "fEP"
f(x)
+
00
as xtU.
Then fE P implies
that the family of power series distributions generated by
f is in P.
Similarly, "G(O;X)EP" will indicate that the
corresponding g(o;x) belongs to P.
3.1
The Class C.
The type of relation illustrated in the introduction
connects a family of discrete distributions (e.g. Poisson)
with a family of absolutely continuous distributions (chi-
9
square) .
The class P is actually a class of families of
distributions, each family being determined by a power
series f.
The fact that Poisson, binomial and negative
binomial distributions belong to the class P, suggests that
all distributions in P might satisfy similar relations.
This is indeed the case as is shown in the following theorem.
THEOREM 1
f(x) -- roo. Oa.x j
Let the power series
of convergence U
v
~
J=
0, belong to
>
J
, with radius
P. For a fixed integer
1, define the function F as follows:
F(x) = 0
_00
<
x
~
0
00
F(x) =
j
a. x
r
J
j=v
00
r
a.x
e
0 < x < U
j
j=O J
F(x)
If for some i
<
=
otherwise.
I
v and for some j
v, a. and a. are different
~
J
1
from zero, then F is an absolutely continuous distribution
function.
Proof:
Let us take x in the interval (O,U).
for some j
~
Since a.
v, we can write
r a.x j
00
F(x)
=
j =v J
00
j
r
a. x
j 0 J
=
I
=
v-I
I
+
.
r
a. x J
j 0 J
=
00
•
La. xJ
j =v J
10
=
1
I + H(x)
J
>
0
where
v-I
.
La. x J
H(x)
=
j =0 J
=
00
La. x
j=v
j
00
+
•
a v- IX
•.• +
00
La. x J
j =v J
J
La. x
j =v J
a
H (x)
=
+
00
La. x
j=v J
Since a.
J
~
+ ••• +
00
j
'I
a.x
L.
v-I
j
v-I
00
j -1
'I
a.x
L.
j-v+1
j =v J
j=v J
0 for all j, then each term in the last expres-
sion is non-increasing, so that H(x) is non-increasing in
the interval (O,U) and therefore F(x) is non-decreasing in
(O,U).
Hence, F is a non-decreasing function.
Next, let i
<
v be such that a.
a.x
H(x)
~
i
a.
1
=
00
La. x
j =V J
Then
O.
>
1
j
1
00
L a.x
J
j -i
j=v
a.
since i < v . ,
1
00
L a.x
J
-+
as x -+ O.
00
Thus, H(x) -+
j -i
j=v
as x -+ 0
and therefore F(x)-+ 0 as x -+ O.
Now, since fE P, f(x) -+
v-I
H(x)
=
as xtU.
00
L a.J x
j
j=O
00
L a. x
j=v J
Thus
-+ 0
as xtU
j
so that
F(x) =
1
-+ 1 as
1 + H(x)
11
xtU.
00
Hence, F is a continuous distribution function.
Since F
is the ratio of two power series in the interval (O,U), F
is also differentiable, i.e. absolutely continuous.
Remark
The reason for the restriction that f belongs to P
is so that F will be continuous at x = U, if U <
00
For
example, if f is
f(x)
then U
=
1, and lim f(x)
x+l
=
n 2 /6.
In this case we would have
lim F(x) < 1.
xtu
Let fEP and let {g(·;x): XE(O,U)} be the corresponding
family of power series distributions.
g(j ;x)
=
j
We have
= 0,1,2, ...
and, with no loss of generality we can assume that a
=
1.
O
Unless otherwise stated, we will always assume that a = 1.
O
From the previous theorem, we see that there is also
a family of absolutely continuous distribution functions
associated with f, namely
00
•
L a.x]
{F(';v)
=
j
=v
]
00
•
L a.x]
j =0 ]
for all v such that a. > 0
]
for some j
12
~
v}
These two families are connected by the relation
00
F(x;v)
L g(j ;x)
=
j=v
for each XE(O,U) and all v = 1,2, ... such that
~OO=vg(j;X:)
O.
>
In terms of random variables, if X
has distribution
v
function F(-;v) and
~
has distribution g(-;x), then
x
Pr(~
Pr(X$x) =
Definition
x
~v)
I
We will call the family {F(- ;v)}, the "dual" family of
Furthermore, we will let C be the class of all
{g(-;X)}.
distribution functions F(-;v) generated by the power series
in the class P.
The class C is then, the dual of P.
Examples
Let
(i)
f(x) = e
00
AX
L
=
(Ax)j/jl
j=O
Here,
U
00
The corresponding family in C is
hence fE P.
00
LP,x)j/j!
F(x;v)
=
j=v
e
o
AX
. h d ens1ty
.
FI( x;v ) -_ A,vxv-le-Ax/ r (,,)
W1t
v
<
X <
00,
o
X
<
=
v
<
00
1,2, ...
i.e., the
gamma density with v integer.
(ii)
Let f(x) = (l_x)-N =
I
j=O
In this case U = 1 and fEP.
is
13
(N+~~lJxj
, N
>
O.
J
The corresponding family in C
F(x;v)
o
<
o
< x < 1
x < 1, v = 1,2, ...
with density
xV-I(l_x)N-I
F'(x;v)
=
B(v,N)
N
00
(i i i)
Then, U
=
Let
f(x)
I
.
(.)x J
j =0 J
(l+x)N =
=
, N integer.
e
and the corresponding family in
00
.
is
00
I
F(x;v)
=
(~Jxj
j=v J
(l +x)
o
N
<
x
<
v :::; N
00
with density
F'(x;v)
x
=
v-I
B(v,N-v+I) (l+x)
3.2
<
x <
00
•
e
Some Simple Properties of
(i)
o
N+I
Every distribution in
e
is infinitely differ-
entiable in the interval (O,U).
(ii)
function
a
>
O.
If X is a random variable with distribution
FEe,
then the distribution of aX is also in
The distribution of X
+
a is not in
e,
00
L a.
(x-a)j
00
•
I
a. (x-a)J
j =0 J
14
a
<
for
but has the
form
j =v J
e
x < U + a.
e·
00
(i ii)
Two different power series
= L
flex)
a.x
j
j =0 ]
00
and f 2 (x)
in C.
=
L b.x j
may generate the same distribution F
j =0 ]
A simple example is provided by
00
L a.x
]
flex) =
j
1 + x + x
2
...
+
j=O
00
f 2 (x) =
L b.x
]
j
= 1
+
x
2
x
+
4
+
x
6
+
...
<
x
j=O
Then:
00
00
L a.x j
j =2 ]
00
= j=2 ]
•
00
(iv)
=0
=
•
x
2
o
<
1.
L b.x]
L a.x]
j
•
L b.x]
j
]
=0
]
C is not closed under convolutions:
G (x) = G (x) = x
2
l
If
o
<
x < 1
1
<
x
then
2
G * G (x)
l
2
x j2
x2
Gl * G2 (x) = 2x - 2 - 1
~
2 ,
and the second derivative of Gl * G2 does not exist at
x
=
1.
(v)
It will be seen that C is not closed under mixtures.
Remark
With regard to Definition 1, we must note the following.
Given a distribution g(·;x) in P, this determines uniquely
15
the dual family in C: {F(o;v)}.
not true.
Let
FlOC.
However, the converse is
eo
It was shown above that there may be
more than one power series
oo
L]. = Oa.x
]
j
, such that
00
L aox j
=v ]
= j
F(x)
00
•
L a.x]
j=O ]
Thus, suppose
oo
L]. = Ob.x
]
j
is another power series such that
00
L box j
F(x)
j=v ]
00
0
L box]
j =0 ]
Then F generates the two sequences in C: {F(o;k)}
and
{F*(o;k)} where
00
j
aox
r
j=k
]
F(x;k) =
00
L aox
]
k = 1,2"
..
j
j=O
and
00
j
b.x
r
j=k
]
F* (x,k) =
00
L
j=O
k = 1,2, .•.
b.x j
]
Let U and u* be the radius of convergence of Ia.x j and
]
Lb.x],
respectively.
]
Then each of these two sequences has
a dual family in P: {G(o;x), XE(O,U)}
where
16
and {G*(o;X),XE(O,U*)},
e
G(k;x)
=
1 - F(x;k+l)
G*(k;x)
=
1 -
and
3.3
Necessary
F*(x;k+l)
Condition for a Distribution Function F to
Belong to C.
Notation 2
Given a distribution function F, the statement "on the
interval (O,U)" will mean that F(O)
= 0
and lim F(x)
xtU
=
I,
Notation 3
+
Let D F(x) denote the right hand derivative of F at
the point x:
+
=
D F(x)
D~F(X)
Let
+
=
lim F(x+h) - F(x)
hiO
h
F(x)
, D;F(X)
=
+
n
and D F (x)
D+F(x)
=
+
D (D
IF(x)).
1 n-
Definition 2
Let F be a distribution function on (O,U) having
derivatives of all orders in the semi-open interval [O,U).
We say that F is of index v if D:(O)
J
v-I
+
and D (0) 1
v
THEOREM
=
0 for j
=
0,1,2, ... ,
o.
2.
Let F be a distribution function on (O,U) with index
k
~
1.
Then FEC only if there exists a function of a
17
complex variable H(z) analytic on C
= {z:
u
a polynomial
Izi < U} and
\~-olB.zj
with real coefficients, such that
= ]
L]
k -1
H (x)
=
.
2 B.x]j(l - F(x))
for all XE[O,U).
j =0 ]
Proof:
If FEC, then:
00
2 a.x j
F(x)
=
j =v ] __
00
o
•
:0;
x
<
U
,
2 a.x]
j =0 ]
for some power series 2a.xj in P and some integer v.
]
a O = 1, there exists
j
2]. = Oc.x
] .
E
> 0
Since
and a power series p(x) =
00
with Co # 0 such that
v
F(x) = x p(x)
for
o
:0;
x
< E.
But
=
for i
0
= 1,2, ...
and
d
dx
V
v
v x p(x)
= v!p(O) = v!c
x=o
O
#
Hence, v = k.
Now,
k -1
.
2 a. x]
F(x) = 1 _ j=O ]
o
00
2 a. x
j =0 ]
so that
18
j
<
x < U
0
.
,v-l
e-
k -1
.
L a.x J
00
L a.x j =
j =0 J
j =0 J
(l-F(x))
o
x
<
<
U
and the theorem is satisfied with
00
H(z) =
a.z
j=O J
L
j
Iz I
<
u
and
k-l
k-l
lB. z j = L a. zj
j =0 J
j=O J
Remark
An equivalent statement is: "FEC only if there exists
k -1
.
a polynomial Lj=oBjzJ and a function of a complex variable
F (z) such that F ex) = F(x) for all XE[O,U) and
c
c
k -1
.
Lj=oBjzJ/(l-Fc(z)) is analytic in CU'" or equivalently,
"FEC only if there exists a function of a complex variable
Fc(z) such that Fc(x) = F(x) for XE[O,U) and (l-Fc(z))-l
is analytic in C ' except possibly for at most k-l poles."
U
Corollary
00
Let
•
L a.x J
F (x)
= j =k J
00
L a.x j
j =a J
be in C.
Then
oo
l J. = Oa.x
J
j
is the unique power series associated
k- l
- LJ=
. Oa.z j / l . Oa.z j
J
J= J
with F if and only if l-F (z) c
zeros in C = {z: Izi < u}.
oo
has k-l
u
Proof
Suppose 1 - Fc(z) has k-l zeros in CU'
oo
LJ. =Oa.z
J
j
Since
has radius of convergence U, the zeros of l-F c (z)
19
k l
j
I J. =- Oa.z
J
are the zeros of
.
Suppose now, there exists a power series
oo
I J. = Ob.x
J
j
e·
such that
F (x)
=
00
•
I
a.x J
00
=
j =k J
j =k J
00
j
~ a.x
=0 J
•
L b.x J
for XE[O,U)
00
j
I
b.x
j
j =0 J
Then
00
l/(l-F(x))
=
j
L a.x
=0
k-l
I
j
00
=0
~ b.x j
j
J
=
.
a.x J
j=O J
k-l
.
I b.x J
for XE [O,U).
j =0 J
J
As we noted before, all k-l zeros of
k l
j
I J. =- Oa.z
J
are in
CU'
On the other hand, the only singularities of
oo
k I
.
k l
. ob.z j / I . - Ob.z j in Cu are the zeros of \'-Ob.z J in
I J=
LJ=
J
J
J= J
oo
k
l
j
I
j
Thus, the two functions I . Oa.z / . Oa.z and
J= J
J= J
oo
j Ik- l
j
. Ob.z / . Ob.z are analytic in CU' except for a finite
I J=
J
J= J
number of poles.
But they are equal on the set [O,U).
Hence, they must be equal at all points of C
u
where they
are analytic.
This forces the two functions to have the
l
j
k-l
j
same poles so that I k. - Oa.z
and I . ob. z must have the same
J=
zeros in CU'
'"
Since a
O
J
J=
= b
J
O = 1, then a.J = b.J for j = 1,2,
,k-l, which implies a. = b. for all j and the theorem
J
J
holds.
\00
Conversely, suppose Lj=Oajx
j
is the unique power series
j , 0
associated with F(x) = L\~ = ka.xj/\~
oa.x
J
LJ =
J
J
k l'
$
.
X <
U, and let
us assume that 1 - F (z) = \. -oa.zJ/\~ oa.z J has N zeros in
c
LJ=
J
LJ=
J
Cu with N < k-l.
20
..
Let p(z) be a polynomial of degree N whose zeros are
the zeros of l-Fc(z) in CU'
k -1
Then we can write
.
L a.z J
= p(z)q(z)
j =a J
for some polynomial q(z).
Since a j is real for all j and
since the roots of p(z) are in the circle C ' the coefU
ficients of p(z) are real and so are the coefficients of
q (z) .
Thus, p(z)/(l-F (z)) is analytic in C and there
c
u
L~=objzj
exists a power series
such that
00
.E.(z)
=
(l-F (z))
L b.z j
j =0 J
c
for z in CU'
The coefficients b. are real for each j since
J
all derivatives of p(z)/(l-F (z)) are real at z = O.
Thus,
c
N
.
zJ , then
if P (z) =
OI3·
J= J
r.
00
r
(l-F (z))
b.z
c
. a J
J=
j
N
.
L l3.z J
=
u .
for z in C
j =0 J
Setting z = 0, we obtain b
O
= 13 ,
0
Taking the first deriva-
tive:
N
.
j l
j
(l-F (z) L jb.z - + (-F'(z))
b.z = L jl3.z J
c
j=l J
c
j=O J
j =1
J
00
00
r
Since F has index k, F(i) (0) = a for i = 1,2, ... ,k-l and
c
F(k)(O). 1- O.
Thus, setting z = a in the last expression,
c
we obtain b = 13 1 . Similarly, from the second derivative
l
we obtain b
= 13 2 , and, in general, b. = 13. for
2
J
J
j = 0,1,2, ... ,N, b. = a for j = N+l,N+2, ... ,k.!.
J
21
Thus
N
k~l
0
L boz J
so that
00
e·
0
I boz J
j =0 J
= j=O J
I-F (z)
I-F (z)
c
c
00
I boz
=
j
j =0 J
•
L
b.z J
J
F (z) = j=k
00
c
L b.z j
for z in Cu '
j =0 J
oo
I J = Oboz
J
and the power series
0
j
is associated with F.
But,
for any real a, p(z) (z-a)/(I-Fc(Z)) is also analytic in CU.
Hence, we can write p(z) (z-a) /(l-Fc(Z)) = Ij=ocjz
j
and
we also have
00
F
c
(z)
=
I co zj
j=k J
00
I coz
j
=0
.
for z in C
u
0
J
J
Therefore, the two distinct power series Ibozj and Icoz
J
j
J
are associated with F, which is a contradiction.
Hence
I-Fc(z) must have all its k-l zeros in C
u and the theorem
holds.
Applications
(i)
Let F I and F 2 be two distribution functions in
C, given by
00
j
x /j !
I
j=v
F I (x) =
I
e
00
j=v
o
~
x <
00
o
~
x
00
j
x /j !
I
F 2 (x) =
x
2
e
x
22
<
where vI
< v
2
These are gamma distributions.
•
Let us
consider the mixture
o
< a.
<
1 .
Then, the index of H is vI and
l-H (z)
c
But U =
00
=
v -1
(
1
.
la.
zJ/ j ! +
I
v -1
(l-a.)
I
z./j!]/e
j =0
j=O
z
J
Then all v 2 -l roots of l-Hc(z) lie in C and
u
there does not exist any polynomial p(z) of degree vl-l
such that p(z)/(l-Hc(Z)) is analytic on CU'
Hence, the
mixture H does not belong to C, by Theorem 2.
(ii)
Let F be a gamma distribution function given by
00
F(x) = (
L xj/jl)/e x
j=v
Then FEC and U =
00
Thus, l-Fc(z) =
(I~:~zj/j!)/ez has its
j
By the corollary, eX = I;=ox / j ! is the
v-I zeros in CU'
unique power series associated with F.
(iii)
Let
~
have a negative binomial distribution
with parameters x and N, 0 < x
integer.
<
1, N > 0, not necessarily
Then
P(~=j) = (
N + ' - l 'J
~
)x (l-x)
J
N
j
= 0,1,2, ...
This defines a family of power series distributions and the
dual family in C is given by
23
F(x;M) =
(I-x)
o
-N
<
x
I
(N+~-l)jxj-l
1. M = 1.2""
<
The density of F is
f(x;M)
=
d
dx F(x;M)
= (I-X)N-l{(I_X)
j=M
J
r (N+~-l)xj}
-N
j =M
~L (N+,j)x j
= N(l-x) N-l{(N+M-l)
M-l x M-l +
j=M
=
J
J
N+M-l) M-l
N-l
N ( M-l x
(I-x)
,
thus:
f(x;M)
xM-l(I_X)N-l
B(M,N)
and the family of beta distributions with parameters M,N,
M = 1,2,3"
" , N > 0 is the dual family of the negative
binomial family with parameters x and N, 0
<
x
<
1, N > O.
Now, the power series (l_x)-N is the unique series
associated with F(o;M) if and
Q(x)
24
only if the polynomial
,
e·
has all its zeros inside the unit circle C
For N
1, Q(x)
=
=
l+x+x 2 + ... +X M-I
=
I
=
{z~
Izi
n.
<
(l-xM)/I-x and this
polynomial has all its zeros on the unit circle.
Hence,
(l_x)-l is not the only series associated with F(o;M).
Suppose now N is an integer greater than 1.
Q(x) has
the same zeros as
R(x)
=
M-I
r
(N+~~l)! x j .
=
(N-I)!Q(x)
=
d N- I
2
M+N-2
N_I(l+x+x + ... +X
)
dx
J.
j =0
But
R (x)
Then, by Theorem A in the Appendix, the roots of R(x) are
inside the convex polygon determined by the roots of
l+x+ ... +X
M+N-2
, so that all roots of Q(x) are in C I and the
power series expansion of (l_x)-N is the unique power series
associated with F(o;M), for N > 1, N integer and M = 1,2, ...
3.4
The Closure of the Class C
Lemma 1
Let us consider the sequence of power series
p (x)
=
n
where a~n) ~ 0 for all j
J
bounded.
and all n.
Suppose there exists
Then, there exists a number U such that Xo
a subsequence {n.} and a power series
1
for all j, such that
25
rJ.
oo
U
:0;
j
a.x, with a.
=1 J
J
~
:0;
00,
0
00
p
n.
j
1
and p
a.
-+
(x)
ni
=1
a.x J
as i -+
00
as i -+
00
for XE[O,U)
J
00
(n.)
1 im a.
i-+oo J
=
J
I
-+
(x)
, for x
U (if U
>
<
Furthermore,
00).
for all j.
1
Proof
Since {Pn(xO)}:=l is bounded and since Pn(y) ~ Pn(x)
for y ~ x, the sequence {p } is uniformly bounded at least
n
in the interval [O,x ].
O
Let
If U is finite there exists a subsequence {Pn.}~=l such that
1
p n. (x) -+
00
as i -+
00
for all x
U, and we can assume with no
>
1
loss of generality that p (x) -+
n
a
Let 0 >
be arbitrary.
00
as n -+
for all x > U.
00
Then there exists M
a such
>
that p n (x) < M for all XE[O,U-O] and all n = 1,2,...
By
Theorem B in the Appendix, there exists a subsequence
{Pn.}:=l and a non-decreasing right continuous function p
1
such that p n. (x) -+ p(x) at all continuity points of p in
1
[O,U-o] .
We now define the following sequence of functions of a
complex variable:
en.)
ex>
L a.
f. (z)
1
j
=1
.
zJ
1
i
Each of these power series converges for
(n.) .
a. 1 zJ
j =1 J
00
/f.(z)1
1
=
I I
00
~
=
1, 2 , . . .
J
(n.)
I a.
j=l J
26
1
.
Izl J
Izl
~
<
U-O
(n.)
r a.
00
j =1 J
1
and
.
(U_o)J
e
u-o.
for all z such that Izi ~
But
en.)
00
L a.
j =I
1
.
(U-o)J
= p
J
Hence If.(z)1
1
n.
(U - 0)
for all i = I ,2 , ...
< M
1
< M
for all z such that
Izi
< U-o
and all i.
Since the discontinuities of p are at most countable,
we can find a sequence {x } in (O,U-o) such that lim x = 0
k
k-+oo k
and such that
~im
1-+ 00
Pn. (x k )
1
= p(X
for each k .
k)
Thus, we can apply Theorem C in the Appendix to the sequence
(n. )
{fi(z)} and conclude that lim a.
i
1
exists for each j, say
.
\'00
J
the series Lj=lajz converges for
(n. )
I im a. 1
= aj'
i-+oo J
and for each positive R
I
<
J
-+00
I zl
< U-o,
U-o we have
00
lim f. (z) =
j
i-+ oo 1
La. zj
=I
J
uniformly with respect to z in {z:
00
lim p
(x) = L a.x
i-+oo n i
j=l J
Since 0
>
j
Izi ~ R }.
I
Therefore
for XE[O,U-O]
0 was arbitrary, the convergence holds in the
interval [O,U) and if U is finite, then p
n.
(x) -+
00
as
1
i -+
00
for x
>
U.
Having proved this lemma, we now consider the problem
of determining the class of all distribution functions that
are weak limits of distributions in C.
As it turns out,
the limiting distributions can have at most two discontin27
uities and are restricted to intervals [L,U], or [L,U)
if U
= ~,
where L is not necessarily O.
uities can
The discontin-
only occur at the points L or U.
THEOREM 3
Let Fn be a sequence of distribution functions in C
given by
o
F (x)
n
x < 0
o
=
x < U
~
n
~
~
otherwise .
I
Suppose the sequence Fn converges weakly to the distribution
function F.
Then there exist numbers Land U, 0 ~ L ~ U ~ ~
and a Laurent series expansion I~
J=
j,
-~
b.x
with b. ~ 0 for all
j
J
J
such that
x
0
~
I
F(x) =
L
b.x j
j=O J
L
~
I
<
b.x
j=-~
<
x < U
j
J
x > U if U <
I
00
Proof
The limit distribution function can be degenerate in
which case the theorem holds with L
that F is not degenerate.
=
U.
We will assume
Then, with no loss of generality
28
e·
we can further assume that 1 is a point of continuity of
F and that 0
This implies that lim inf U > 1.
n-+ co
n
Hence, we can assume that U > T > 1 for all n and some
<
F(l)
1.
<
n
number T > 1.
Let
co
(n)
r.
J
=
.
L a~n)
.
l=J
1
Then
r(n)
\!
Fn(l) =
n
----cnr
-+ F(l)
as n -+
co
(1)
•
r
O
In the interval (O,U n ) we can write
F
n
(x)
=
=
Let
p
n
o ::;
(x)
x
<
U
n
and
q
Now, since
n = 1,2' ...
n
o ::;
J
(x)
x > 0 .
=
a (n+) . / r (n )::; 1 for all j
\!
n
J
\!
= 0, 1 , 2 , ... and all
n
then the sequence {Pn(x)}~=l is bounded for
any fixed xE(O,l).
29
By lemma 1, there exists U ~ 1, a subsequence {n.}
1
oo
j
and a power series p(x) = l . Ob.x such that
] =
]
00
= I
p n. (x) -+ p(x)
b.x
j=O ]
1
for all
XE
[O,U)
and p
n.
(x) -+
00
j
as i -+
00
for x > U if U is finite.
1
(n.)
Here, b.
]
=
(n.)
1
1
lim a \).+]./r \)
i-+ oo
1
n.
With no loss of generality, we
1
can assume that n.1
o
s;
x
<
nand p n (x) -+ p(x) for x such that
=
U.
Let x < 1 be fixed and consider the sequence {qn(x)}~=l'
If this is an unbounded sequence, then
•
since Pn(x) -+ p(x)
<
00
Hence, since we have assumed that 1 is a point of continuity of F, there must be a number X
o
1 such that
<
Replacing x by l/y and using lemma
1, we conclude that there exists a number L s; X
subsequence {n.} and a power series q(x)
1
=
l~l
] =
o
_00
<
1, a
b.x j
]
such
that
-1
q n. (x) -+ q(x)
=
j =
1
for x
>
Land q
n.
(x) -+
L
00
-00
.
b.x]
as i -+
00
]
for x such that 0
<
x
<
L, if L
=
-1,.2,-3, . . . .
>
O.
1
(n.)
As before, we have b.
]
(n.)
lim a \) 1 +]. Ir \) 1
i-+oo
n.
n.
1
1
for j
Again, with no loss of generality, we can assume that
n l· = nand q n (x) -+ q(x) for x
30
>
Land q n (x) -+
00
for x such
~
that 0
x
<
<
L, if L > O.
Thus, unless p(x)
=
0 and q(x)
=
0, we conclude that
the limit distribution function F must be of the form
o
x < L
00
I
F (x)
box
j
j =0 J
=
L
x < U
<
otherwise.
1
Suppose now, for purposes of contradiction, that p and q
This means that b.
are identically zero.
o, ± 1 , ± 2 , . . .,
J
lim a(n) .fr(n)
v +J
v
n
for j
0
=
n
Consider the expression
v +j
a
H
0
n ,J
(x)
=
(n)
n
oX
v +J
n
00
•
\'
(n) 1
La.
x
. 0
1=
<
x
<
0 for j
=
i. e .
n+ oo
For 0
=
1
1 we have
v +j
a (n) oX n
v .+J
H
0
n, J
(x)
n
$
- I v n -1
v
I
x n
i=O
a
(n)
.x
a~n)
1
j+l
v +J
n
=
(r
(n)
o
-r en))
v
n
31
=
O,±1,±2,...
(2)
e·
or
H
. (x)
n, J
$
Hence, using (1) and (2), we conclude that
limH
.(x) = 0 , forxE(O,l),
n-+ oo n,J
and all fixed j
= O,±1,±2,"".
"
Now, again for 0 < x < 1, we have
v +k
n
00
a~n)xj
J
j=v
n
I
F
n
(x) =
00
.
I
=
a~n)xj
j=O J
00
a~n)xj
J
j =v
n
a~n)xj
j=v +k+l J
n
+
00
2
00
a~n)xj
j=O J
I
I
a~n)xj
I
j=O J
But
v +k+l
n
x
(n)
r v +k+l
n
v -1
v -1 n
n
x
\
(n)
a.
L
J
j =0
x
k+2
n
=
r
x
=
(n)
r v +k+l
(n)
o
k+2
-
r
(n)
v
n
(n)
/ (n)
r v +k+l r v
n
n
(r6n)/r~n)) - 1
n
Thus, for 0
<
x
<
1
/ r (n)
x k+2 r (n)
v +k+1
v
n
F (x)
n
$
+
n
(r6n)/r~n)) - 1
n
32
(3)
Let the integer k
~
0 be fixed,
As n
~
oo~
the first
term of the last sum goes to zero, because of (3).
I
(n)
TV
n
as n
~
(n)
+k+l r v
00,
Also
=
n
because of (2).
Finally,
r6n)/r~n) ~ I/F(I)
n
Therefore, for 0
<
x
<
1
lim sup F (x) ~ x
k 2
+
n
n~oo
I ((I/F(l)) -
1).
Since k is arbitrary, lim F (x) = 0 for all x such that
n~oo
o
<
x < 1.
n
This is a contradiction, since F(l) > 0 and 1
is a point of continuity of F.
Hence, p and q cannot be
both identically zero and the theorem holds.
Let I~
b.x
J=_OO J
for j
by L
j
be a Laurent series expansion with b.~ 0
J
= O,±I,±2, ... and with region of convergence given
<
Ixl
<
U.
Define the function F as follows:
x < L
0
00
F(x) =
J
L b.x
J
j=O
00
r
b. x
j=_oo J
L
x < U
x > U,
1
and F(L) = lim F(x).
x+L
<
j
if U
<
00
,
Then, F is a distribution function.
33
e·
On the other hand, all distribution functions of this
form can be obtained as limits of distributions in C, since
F(x)
=
lim Fn (x), where
n+ oo
00
L box]
F
n
(x)
= j =a ]
00
I
a
0
<
x < U •
box]
j = -n ]
Thus, the class of distribution functions of the form given
above is the closure of C.
Examples
(i)
Let
=
n n n+1 n+2
2 x +x
+x
+ ...
n n-1 n n n+1 n+2
1+2 x
+2 x +x
+x
+ ...
=
I
F (x)
n
+
a
x < 1/2
Fn (x)
+
x/(l+x)
1/2 < x < 1
F (x)
n
+
1
x
Fn(x)
=
n
x /(l+x+x 2 + ... +x n )
F (x)
n
+
a
F
n
(x)
, a
$;
x
<
x > 1 .
Then
(ii)
>
1
Let
Then
a
$;
x < 1
00
Fn (x)
+
1/
L
(l/x)j
j=O
34
=
1 - l/x
x > I
.
I
(iii)
Let
F
(x)
n.
=
n
2
n
nIx jC1+x+2!x + ... +n!x )
Then
o
5
X <
o
<
x < 1
x
>
1
o
<
x
00
•
1
F
(iv)
(x)
-+
(x)
=
n
x
e
o
x <
<
00
•
Let
00
F
n
L nx j
00
j=3
=
j
l
j(l+x+
nx )
j=2
1
Then
00
F (x)
n
xjj
j=1
-+
3.5
l
-+
00
l
x
j
=
x
1
<
j=O
1
x > 1.
Relationships Between the Weak Convergence of Sequences
in C and P.
In Theorem 3 we
. considered sequences {F n } in C with
index v
n
There were no restrictions on the sequence {v}.
n
We now consider the case where v
n
=
v for all n; that is,
we consider a sequence of distribution functions in C with
common index v.
As we have seen before, given a distribution function F
in C there may be more than one associated power series
oo
. Oa.x
LJ=
J
j
, with a
O
=
1.
Hence, F may generate several families
of distribution functions in C.
All elements in one family
are associated with the same power series and they differ
only in their index.
If La.x
J
j
is one power series assoc-
iated with F, with radius of convergence U, then {F(o;k)}
35
00
is one such family, where FCx;k)
<
x
<
l
a.x /
j=k J
o
00
j
l
=
a.x
j
j=O J
U, for all values of k for which there exists j
such that a. > O.
J
Thus, if La.x
j
2
k
reduces to a polynomial
J
of degree r, FCo;k) is defined for k
1,2, ... ,r.
=
Other.,.
wise, FCo;k) is defined for all integers k = 1,2,3, ...
Corresponding to each of these families generated by F,
there is a dual family {GCo;x): XECO,U)} in P.
Let us now consider a sequence {F } of distribution
n
functions in C.
We will let f
series associated with F , U
n
f
n
and I
n
sup{j: a~n)
=
J
>
n
a}.
n
Cx)
=
\~ = oa~n)xj
be a power
J
L
J
the radius of convergence of
Then, each element F
sequence generates at least one family,
say {F Co;k)
n
with corresponding dual {G Co;x): O<x<U }.
n
v,
of the
n
If F
n
:l~k~I
n
has index
n
then FnCo) = FnCo;v).
As we know, the following relation holds for each n:
1 - G Ck-l;x)
n
=
1
F Cx;k)
n
k
~
~
I
n
, 0
<
x
<
U .
n
Notation 4
Following the notation in Gnedenko and Kolmogorov [1],
F
n
=>
F will denote weak convergence of the sequence {F }
n
to the distribution function F.
THEOREM
4.
Let {F } be a sequence of distribution functions in C
n
with common index v.
For each n, let {F CO ;k):
n
a sequence in C generated by F
corresponding dual family.
n
l~k~I
n
} be
and {G Co;x): O<x<U } the
n
n
If {F } is relatively compact
n
36
}
and lim sup F (E) < 1 for some E
n-+ oo
n
>
0, then there exists
V > 0 such that the sequence {Gn(o;x)}:=l is relatively
compact for each XE(O,V).
Proof
Let us define V as follows:
V = inf{sup{x: F(x)<l}: F is a limit point of {F }}
n
Then, V
0, and in fact, V
>
>
E.
Let {G ,(o;x)} be an
n
arbitrary subsequence of {G (o;x)}. We must show that there
n
exists a subsequence {G (o;x)} of {G ,(o;x)} that converges
m
n
weakly to a distribution function.
{F
n
,(o;v)} has a subsequence,
say
But the sequence
{F
n
II(o;V)}
that converges
weakly to a distribution function, say F.
For XE(O,U n ) we have
( n)
F (x;v)
1
n
-
(1+a l
X+ ...
(n)
+av~lx
v-I
)
00
La~n)xj
j =0 J
Let a~n) = a~n)/(l+al(n)+ ... +a(n)l) for each j and each n.
J
vJ
Then
(0.
=
1 •
(nil)
(n l ') v-I
+
...
+av_ljc
)
0
00
'i'
(nil) j
La.
x
j =0 J
.
Slnce
(nil)
(nil)
, ... ,a _
are less than or equal to one for all
0
v l
nil, the r e ex i s t s a sub seq u e n c e {nil'} 0 f {n "} s u c h t hat e a c h
(nil' )
(nil')
one of the sequences {a O
}, ... ,{a v _ l } converges.
Let
0.
us call these limits
0.
0
,0.
1
" , .,a _ ' respectively.
v l
37
Since
lim sup Fn(e;v)
1 and Fn,,(e;v)
<
~--;>
F(e), then F(e)
<
l.
n+ oo
Let V
~
V
by
=
V.
sup{x: F(x)
<
I}.
Then V > 0, and by definition,
For simplicity, let us denote the sequence {n"'}
{nd.
At all points of continuity of F in [O,V), we have
1 -
F (x)
(m)
oo
l ] =Oa.]
so that the sequence of power series
o
[O,V'], for each U'
bounded in the interval
<
x
j
is uniformly
U.
Applying
Theorem C in the Appendix, we conclude that the sequences
{a~m)}
converge for all
]
= 0,1,2,...
j
be aO,a l , ... , respectively.
l
oo
l ]= Oa.](m) x j
+
o
F (x)
=
oo
0
]=
1
Oa.] x j
-
Let these limits
Then, from Theorem C,
, and
00
L a.x j
j =0 ]
00
I
0
aox]
= j=v ]
00
I
o
0
x
~
<
U
.
a.x]
j =0 ]
For k
0,1,2, ... , define the functions
o
0
x
<
x
< V
00
F(x;k)
=
I
j
00
•
a.x
j =k ]
I
o
~
a x]
j =0 ]
0
otherwise
1
38
(1)
Let k be arbitrary but fixed and consider the sequence
{F (o;k)}.
We can write
m
o
F (x;k)
m
As m
+
00
x < U
:'>
m
this sequence converges to
00
•
I
a.x J
F(x;k)
j =0 J
for XE[O,U).
But this is true for any k.
Thus for any
fixed XE[O,U), hence for any XE(O,V), we have
=
for all k
=
1 - Fm (x;k+1)
+
1 - F(x;k+1)
Hence, the subsequence {G m( ;x)}
0,1,2, ...
of {G ( ;x)} converges weakly and the theorem holds.
n
Remarks
(i)
U
=
In the previous proof, U is defined as
sup{x: F(x) < 1} and F was found to be
o
F (x)
=
00
•
I
a.x J
00
•
I
a.x J
j = \! J
o
:'>
x
<
0
x
<
U
j =0 J
1
otherwise .
Note that F is not necessarily in C since it may be dis39
u.
continuous at
be O.
Also, a
O
is not necessarily 1 and it can
Note also that U is not necessarily the radius of
convergence of
flOj=Oajx j .
e
For example, let
00
x
L xj/n
+
j=2
00
I
1 +x +
o :s:
x
x
1.
1
<
xj/n
j=2
1
=
Then, F (x;l)
n
+
x/l+x
for 0 :s: x
+
1
for x
~
~
1
<
1
so that U = 1, while the radius of convergence of
OO
. Oa.x j -l J=
J
(ii)
l+x is infinite.
The fact that {F (o;v)} converges for a fixed
n
value of v does not imply convergence for other values of
v.
For example, let
F (x;2) =
n
2 3
x +x + .
= x
2
2
0 :s: x < 1
l+x+x + .
for n even, and let
F (x;2) =
n
for n odd.
2 4
x +x + ...
2
= x
4
2
0 :s: x < 1
l+x +x + ..
Clearly, F (x;2)
n
+
x
2
for 0 :s: x
<
1,
but
{F (o;3)} does not converge, since
n
F
F
n
n
(x;3)
(x;3)
=
x
x
3
4
for n even, and
for n odd.
However, it is clear from the proof of the previous theorem
that if {FnCo;v)}:=l is relatively compact, then
40
e
-
{Fn(o;k)}:=l is relatively compact for each k ~ v.
In
fact,
00
L Ct·(n'"
Fn" , (x, k) =
00
) xj
L
j=k J
00
-+
(1.
j=k J
x
00
j
Ct.(n '" ) x
L
j=O J
L Ct.x
j=O J
j
= F(x;k)
j
and F(x;k) is a distribution function for k
it may be degenerate.
For values of k
some n, then F (0 ;k) _
n
a
>
~
v, although
v, if k
so that the sequence {F
n
>
I
CO
'k)}OO
'
n=l
n
for
is not necessarily a sequence of distribution functions.
For example, if
v
<
k
<
F (x) = x/l+x, then F (x;2) :: O.
n
n
If
I , for all n, then the sequence {Fn(o;k)}:=l
n
is not necessarily relatively compact.
x+
F
n
(x)
=
l+x+
1
n
1
n
x
2
x
2
For example, let
Then, {Fn(o;l)}:=l is relatively compact, In = 2 for all n,
2
but F (x;2) = (! x 2 )/(1+x+ ! x ) -+ a for all x.
n
n
n
We now consider a sequence {Gn(o;x): xE(O,Un)}:=l in
P and assume weak convergence for some fixed x to a distribution function G.
This implies weak convergence of the
sequence {Fn(o;k)} for each k such that G(k-l) < 1.
THEOREM 5.
Let {Gn(o;x): xE(O,Un)}:=l be a sequence of families
in P and {Fn(o;k): 1 ~ k ~ In}~=l the corresponding sequence
of the dual families.
If for some fixed x, G (0 ;x)
n
41
=>
G(o)
e·
for some distribution function G, then for each k such
that G(k-l)
F(o;k)
1 there exists a distribution function
<
such that
Fn(o;k) => F(o;k).
Proof
The fact that G (o;x)
G(o)
+
n
finitely many values of n.
forces U
> x
n
for all but
Let
k = 0,1,2, ...
o
<
x
U
<
n
Let v be the smallest value of k such that G(k)
>
O.
+
00
•
00
,
Then
G (v-l;x) + 0 so that
n
G(v)
+
as n
0,
>
Therefore
x
v
+
G(v)
>
0
as n +
l~
which means that
J=O
,
(a~n)/a(n))xj remains bounded as n
J
v
Then, by Lemma 1, there exists a number
U
~
+
00.
x, a subsequence
, such that
(n.)
a.
J
1
(n.)
/a
1
v
+
8.
J
as i +
00
for each j, and
en.)
en.)
.
(a. 1 / a 1 ) yJ
j=O J
v
00
co
l
for y E: [ 0 , u)
j =0 J
+
42
L 8. yj
+
00
for y > U,
if U <
00
Thus
X
\I
X
\I
<lO
L
as n
•
8·
+
<lO
x]
j =0 ]
and indeed we have a~n)/a(n)
]
+
8 ..
\I
]
Now, for an arbitrary k:
k
.
L 8.x]
+
G (k;x)
n
j=O ]
<lO
•
L 8.X]
j =0 ]
Therefore,
k
.
L 8.X]
G(k)
= j =0 ]
<lO
•
k
=
0, 1 ,2, . "
.
L 8.x]
j =0 ]
But,
<lO
L
8. yj
for all YE[O,U)
.
j =0 ]
Hence,
for each such y we define
k
.
L 8.y]
G(k;y)
=
j =0 ]
<lO
•
L
8·
k = 0,1,2, ...
y]
j =0 ]
and for each k = 0,1,2, ... , such that G(k-l;x)
o
F(y;k) =
1, we define
<
y
<
y.
< U
0
1 - G(k-l;y)
o
1
otherwise
43
~
Thus,
in the interval
Fn(y;k)
=
and for y > U,
°
$
e·
Y < U
1 - Gn(k-l;y)
+
1 - G(k-l;y)
=
F(y;k)
if U is finite,
+
which implies that
1 - Fn (y;k) = Gn (k-l;y) + 0, as n +
Fn(y;k)
+
1 = F(y;k)
44
00,
, for y > U.
or
00
CHAPTER II
THE CLASS V
1.
Introduction
In Chapter I, we considered the class of distribution
functions C which consists of all distribution functions
that can be written as
x < a
a
'"
r
F
a.x
j
j :: k J
ex) ::
'"
r
a.x
a : :;
x < U
j
j :: a J
I
r'"J. :: Oa.x
j is
J
lim r'". Oa.x j
J ::
J
where
and
::
xtU
otherwise ,
a power series with a.
J
The restriction a.
'"
J
~
a
for all j
~
a
for all
j,
allowed us to establish a duality between the classes C
and
P,
since every power series with non-negative coeffic-
ients defines a family of power series distributions.
How-
ever, when considering the weak limits of sequences of
distribution functions in C, we found that C was not significantly enlarged.
The weak limits could still be
expressed in terms of a Laurent series expansion with all
coefficients non-negative.
Clearly, some expressions of
oo
j
the form r . k d . x I
J=
J
roo. Od.x j
with the coefficients d. not
J=
J
J
restricted to be non-negative, will be distribution functions.
For example
o
F(x)
X
x
2
2
o
l(l-x+x)
1
<
0
x < 1
~
otherwise ,
is a distribution function on the interval
(0,1).
A dis-
tribution of this form, however, does not have a dual
discrete distribution.
The Class V
2.
We define the class V as the set of all continuous
distribution functions that can be written as
o
x
r
d.x J
j=k J
00
F(x)
r
<
0
•
00
•
o
x < U
~
d.x J
j =0 J
1
for some power series
R
~
j
rooJ. = Od.x
J
U and some integer k.
that dO
=
otherwise ,
with radius of convergence
Again, we will always assume
1.
Examples
00
(i)
Let
H(x)
=(
r (-l)jxj/jl)lej=k
46
X
,
x > 0, k even.
e-
Then
H'(x)
Thus, if we let
k -1
=
x
k -1 x
e
f(k)
B be the only positive root of the po1yk
..
nomia1 Lj=O(-l)JxJ/ j ! , then F defined as
o
F(Xi k )
x
<
0
H(x)
=
1
otherwise ,
is a distribution function on the interval (O,B ).
k
(ii)
Let us consider the power series
(l+x)-N
Let
k
~
Ix I
=
<
1.
1 be fixed and
r(x;N,k)
Then
r(x;N,k) = 1 Suppose k is even.
( N)
1 x + (N+l)
2 x 2 - (N+2)
3 x 3 + ... + (N+k-2)
k-l
(-x) k-1 .
Then
+ [(N+k-3) k-2
k-2 x
•
But, for j even we have:
47
j
( N+j-1)
.
x
J
If N
>
_ (N+j) j+1
. 1 x
J+
e·
(N+~ -l)x j (1 _ N+j x)
=
j +1
J
1, then
1 - N+j x
j+1
<
1 _ l+j
j+1 x
1 - N+j x
j+1
<
0
1 _ x .
=
Hence
so that r(x;N,k)
<
0 for x
for x
>
1, which implies that r has
no roots larger than 1 if N > 1.
enough, we see that r(x;N,k)
1,
>
However, by taking x small
0 for some x
>
has at least one root in the interval (0,1).
>
O.
Hence r
Similarly,
if N < 1 then r has no roots in the interval [0,1] and if
N
=
1, then x
Let
=
1 is a root of r.
DN, k be the smallest root of r in the interval
(0,1], when N ~ 1.
Define, for k even
o
Q(x;N,k)
x < 0
=
(l +x)
-N
1
otherwise .
The derivative of Q in the interval (O,D
Q' (x;N,k)
=
~-1 (1+x)N-1
B(k,N)
48
k) is given by
(k even)
Thus, Q is strictly increasing in (O,D
if and only if r(x;N,k) = O.
N,
k)'
But Q(x;N,k) = 1
N,
Hence Q(D k;N,k) = 1 and Q is
N,
e
V, for N
a distribution function in the class
~
1 and k
even.
Note that ON , k is actually the only root of r in (0,1],
when N
~
1.
(iii)
In a similar way, let us define, for k even
o
H(x;N,k)
x
=
(I-x)
N
otherwise ,
1
where N is an integer, N
\~-Ol(~)
(-l)jx
=
J
L..
j
J
~
0
<
k, and eN k is the only root of
,
in the interval (0,1).
Then, Q is a distribution function in the class
V.
Its
density is
h(x;N,k)
x
=
k-l
B(k,N-k+l) (I-x)
=
N+l
o
<
x
<
CN, k
elsewhere .
0
Finally, we note that if X has a distribution function
H(o;N,k) then Y
ON , k
3.
=
=
XCI-X) has distribution Q(o;N-k+l,k) and
CN, k / (l-C N, k)·
Necessary and Sufficient Condition for a Given Distribution Function to Belong to V.
The necessary condition for a distribution function to
belong to the class C that was discussed in Theorem 2 of
Chapter I, turns out to be a necessary and sufficient
49
condition for a distribution function to belong to the class
V.
Theorem 1.
Let F be a distribution function on the interval
with index k
~
1.
Then FEV if and only if there exists a
function of a complex variable H(z) analytic on C
{z:
Iz I
<
(O,U)
U} and a polynomial
r~:~Bj zj
u
=
with real coefficients,
such that
H (x)
=
F(x)) for all XE[O,U)
Proof
The proof of the necessity of this condition is the same
as the proof of Theorem 2 of Chapter I.
Suppose now that there exist H(z) and
j
l
,
rk. - OB.z
] =
]
with B.
]
real, such that
k -1
H (x)
I
=
.
j =a
B.x] /1 - F(x)
for XE[O,U)
]
Since H is analytic on CU' we can expand it in a power series
H(z)
r
oo
= ]. =Ob.z
]
00
I
j
, for uC '
Thus
U
k-l
.
B.x]
b.x
j=O ]
j
r
j =a
]
1
F (x)
-
for XE[O,U)
r.
oo
This implies that all derivatives of
are real.
] =
Ob.x
]
j
at the origin
Hence, the coefficients b. are real for all j.
]
As in the proof of the corollary to Theorem 2 in Chapter
I, by taking the first k-l right-hand derivatives at x
50
=
a
~
k -1
00
(l-F(x))
I
b.x
I
j
j =0 ]
.
b.x]
, for XE[O,U)
j =0 ]
which implies
00
F(x)
=
I
j
00
•
b. x
j =k ]
,
0
x < U
$
I b.x]
j=O ]
and FEV.
Remark
As in Theorem 2, Chapter I, an equivalent statement is
"FEV if and only if there exists a function of a complex
variable F (z) such that F (x) = F(x) for XE [O,U) and
c
c
(l-F (Z))-l is analytic in C except possibly for at most
c
u
k-1 poles."
Applications
(i)
o
$
x
$
n
We note that the distribution functions F(x) = x ,
1 belong to V for all n.
Then, as a consequence of
Theorem 1, all mixtures of the form
o
$
x
1,
$
In fact, they belong to C.
(ii)
If the distribution function F is a polynomial of
1 t- 0, then
the index of F is one and F will belong to V if and only if
the form F(x)
=
alx+a2x2+ ... +akxk , 0'5 x
is analytic on C
u
is, if and only if 1 -
(a
k
1
=
$
{z:
U,
0.
Izl
<
U}, that
z+ ... +akz ) has no roots in
51
e U.
(iii)
e·
Let
2
f(x)
12/rr e -x /2
=
o
<
x
<
00
•
This is the density of the half-normal distribution.
The
distribution function is given by
x
F(x)
o
=f
o
<
x <
00
•
Let us consider the function of a complex variable
G (z)
f
=
r
(z)
where fez) is any continuous curve that joins the points 0
and z.
Then, for all real x, G(x)
F(x)
=
The function G is analytic everywhere.
- 1.
The number of
zeros of G inside the circle C
= {z: Izi < R} is given by
R
1/2rr times the change ln the argument of G(z) as z goes
around the circumference {z:
Izi
=
R},
Now, if z
Re iT ,
=
then
R
G(z) =
2
f 12/rr
e
-x /2
dx
o
where K(O,T)
=
f
+
12/rr e-
w2 2
/ dW
~
I
K(O,T)
{z
=
Re i8 : 0
$
8
$
T},
We then have
R
G(R) =
I
12/rr
o
and
Argument (G(R))
=
Argument (
=
7T
S2
..
f
R
o
2
12/rr e- x /2 dx - 1)
e
Also
R
J 12/7T
=
e-
X2 2
/ dX
o
+
2
J
12/7T e
.. w /2
dw -
1
.
K(O,27T)
But
J
K(O,2n)
so that
Argument
(G (Re
i27T
=
))
Argument
(G (R))
and there is no change in the argument of G(z).
G(z) has no zeros in CR.
Since R is arbitrary, we conclude
that G(z) has no zeros in the complex plane.
the function H(z)
and H(x)
=
l/(F(x)
l/G(z)
=
- 1)
belongs to the class
Hence,
Therefore,
is analytic in the complex plane
By Theorem 1, F
for all x > O.
V.
Remark
The corollary obtained after Theorem 2 of Chapter I can
also be obtained in this case.
applies now.
The same proof we had then
Thus, we have
Corollary
Let
F(x)
00
•
I
d.x J
= j =k J
00
•
I
d.x J
o
~
x
<
U
j =0 J
be in V.
oo
Then
l J. = Od.J x
j
is the unique series associated with
F if and only if
53
k -1
I
1
-
F
c
j =0 J
=
(z)
e-
0
doz J
00
0
I
doz J
j=O J
has k-l zeros in C
u = {z: Izl
4.
UJ.
<
The Closure of the Class V
Lemma 1
Let {Fn}~=l be a sequence of distribution functions
that converges weakly to the distribution function F.
For
each n, let {Gn,k}~=l be a sequence of distribution functions such that Gn, k
=>
Fn as k
+
Then, there exists a
00.
subsequence {G n. ,k. }ooi=l of the double sequence {G n, k} such
1
1
that Gn . ,k.
1
=>
F as i
+
~
00
1
Proof
By Theorem D in the Appendix, L(F ,F)
L(G n, k,F)
n
+
be arbitrary.
0 as k
+
00
as n
0
+
n
for all n = 1 , 2 , 3 , . . .
Then there exists nO such that L(F
+
00
and
Let
E
>
0
,F)
<
E/2,
nO
and there exists k O such that L(G
k ,F ) < E/2.
TherenO' 0 nO
fore, L(G
k ,F) < E.
This implies that F is a limit point
nO' 0
of the sequence {G n, k}oo
n, k=l and there exists a subsequence
{G n . ,k. }:=l such that L(G n . ,k. ,F)
1
1
1
1
+
0 as
i
+
00
Again, by
Theorem D, the sequence {Gn.,k.}:=l converges weakly to F.
1
1
Lemma 2
Let F be a distribution function on the interval (0,1)
of the form
F(x)
S4
Then, there exists a sequence in the class V that converges
weakly to F.
Proof
Let
G(x)
=
l/(l-F(x))
G'(x)
=
F'(x)/(l-F(x))
O$x<l.
,
Then
2
and G' is continuous on the interval [0,1).
Let 0
0
<
1 be arbitrary but fixed.
<
By the Stone-
Weierstrass theorem (E in Appendix) there exists a sequence
of real polynomials, say p', that converges uniformly to
n
G' on [0,1-0].
By Theorem F in the Appendix,
x
J
f
p'(t)dt -+
n
o
x
F' (t)/(l-F(t))2 dt
o
for all xE[O,l-o], that is, p (x) -+ l/(l-F(x)) for all
n
XE [0,1- 0], where
x
p
and dxd Pn ()
x
=
n
(x)
=
f p'(t)dt+l
n
o
Pn' (x) .
For each n, let a.
n
=
min{p'(x): 0
n
$
X
$
l-o} and let
Since p'(x) converges to F'(x)/(l-F(x))2,
n
uniformly in the interval [0,1-0], and since FI(X)
Also, pI (x) + 8
all xE[O,l-o], then lim 8 = O.
n
n
n-+ oo n
all xE[O,l-o] and all n = 1,2, ...
55
?
?
0 for
0 for
e·
lim q (x) = lim (p (x) + S x) = lim p (x)
n
n
n-+ co n
n-+ co
n-+ co n
= l/(l-F(x))
Thus, q n converges
to 1/(I.F), for all X in [0, 1-0].
.
more, q'(x) = p'(x)
n
n
+
Sn
a
~
for all xE[O,I-o], i . e . , q
Furthern
is
non-decreasing in the interval [0,1-0] for all n = 1,2, ...
For x in the semi-open interval [0,1-0), let us define
co
s
n
(x)
=
q
n
(x)
I
+
(x/(l-o))j.
j=n+l
For each n, s (x) is a power series in x with radius of
n
convergence 1-0.
Since both q
n
and \~
l.J=n+
non-decreasing in the interval [0,1-0), s
l(x/(l-O))j are
n
is also non-
decreasing in [0,1-0) for all n = 1,2,3, ...
Now, for any xE[O,I-o), we have
00
lim s (x) = lim q (x) + lim
2 (x/(l-o))j
n
n-+ oo n
n-+ oo
n-+ co j=n+l
so that
s n (x) -+ l/(l-F(x))
as n -+
co
for xE[O,l-o).
Writing
00
s
n
(x)
=
2 a~n)xj
j =a J
a
~
x
<
1-0 ,
and letting x = 0, we see that lim a(n) = 1. Now, the fact
n-+ co a
that sn(x) -+ l/(l-F(x)) implies that 1 - l/Sn(x) -+ F(x).
Let us define Fa and Gn as follows:
56
Fo(x) =
{
F(x)
_ 00
1
x
<
~
X
<
1-0
1-0
and
x
0
<
0
00
G (x) =
n
L a~n)xj
J
j =1
= 1
00
a~n)xj
j=O J
-
L
a (n)
O
sex)
n
0 s x < 1-0
x
1
~
1-0
Also, since s
Then, F o is a distribution function.
n
is non-
de cr e a sin gin [0, 1 - 0), Gis
non - dec rea sin gin [0, 1 - 0) .
n
Furthermore, G (0) = a and lim G (x) = 1, so that G is a
n
x+l-o n
n
continuous distribution function for each n.
But, the fact that a(n) + 1 as n
o
+
00
implies that
(n)
lim G (x) = lim(l n+ oo n
n+ oo
:0 (X))
n
= lim(l
n+ oo
so that, lim G (x) = F(x) for all x in [0,1-0), which means
n+ oo n
that the sequence {G }, in the class V, converges weakly
n
to the distribution function F o .
We now let {o.} be a
1
sequence of positive real numbers such that lim o. = O.
i+ oo 1
Then, it is clear that the sequence {F .} converges weakly
o1
to F.
Since for each i we can find a sequence {G i , n}~=l
in the class V that converges weakly to F o . by lemma 1 there
1
exists a sequence of distribution functions in the class V
that converges weakly to F.
S7
e.
Remark
Note that the sequence in V that converges to F can
be taken to be of the form
Definition (taken from G.G. Lorentz [12])
For a function f(x) defined on the interval [0,1], the
expression
B (x)
n
= Bf(x) =
n
n
.
.
L f(j/n)(~)xJ(l_x)n-J
j=O
J
is called the Bernstein polynomial of order n of the func~
tion f(x).
THEOREM
2.
Let F be any distribution function on the interval (0,1).
Then, there exists a sequence in V that converges weakly to
F.
Proof
Let
n
.
.
B (x) = I F(j/n) (~)xJ (l_x)n- J
n
j=O
J
n
=
1,2, ... ,
be the sequence of Bernstein polynomials of the distribution
function F.
Then
n-l
B' (x)
n
n
so that B'(x)
n
.
L {F(L!..L)
. 0
n
J=
~
0 for all xE(O,l).
58
Furthermore,
e
Bn(O) = F(O) = 1 and Bn(l) = F(l) = I,
B
Thus, the sequence
is a sequence of distribution functions on (0,1) that
n
converges weakly to F, by Theorem G in the Appendix.
By lemma 2, for each n there exists a sequence in V,
B
as k
+
By lemma 1,
say {Fn,k};=l such that Fn,k
=>
from the double sequence {F
k} we can obtain a subsequence
{F
n. ,k.
1
} such that F
1
n. ,k.
1
n,
n
F, as i +
=>
co
This proves the
co
1
theorem.
Clearly, this result can be extended to distribution
functions on intervals of the form (O,U), for any real
U
>
o.
Therefore, using lemma 1 again, the result is true
for distribution functions of non-negative random variables.
THEOREM
3.
Let F be the distribution function of a non-negative
random variable.
Then, there exists a sequence of distribu-
tion functions in V that converges weakly to F.
The weak closure of the class V is then the class of
distribution functions of non-negative random variables.
Each one of these distribution functions can be approximated
by a sequence of distribution functions of the form
co
1 - 1/
I
a.x
j
j =0 J
where a
O
=
1 and a. is real for each j
J
59
o
~
x
<
U
CHAPTER III
THE BIVARIATE CASE
1.
Introduction
In Chapter I we dealt with discrete random variables
~,
whose distributions belonged to the class of power series
distributions.
The distribution of
~
depends on a parameter
x, so that we actually have a family of random variables
{~
x
P(~
:
x
As it turned out, the function F(x;k)
XE[O,U)}.
0
~k),
$
x
<
tion function.
U,
was an absolutely continuous distribu-
This was a consequence of the fact that the
distribution function of
F(x;k) =
o
$
X
<
=
P(~
U, with a.
x
>
J
~k)
~x
belonged to P, i.e.,
=
0 for all j.
Thus, for the class P we
were able to define a dual class of absolutely continuous
distribution functions: the class C.
It seems natural to consider bivariate power series distributions and to determine whether a dual class of bivariate
absolutely continuous distributions can be defined.
Let
~
and n be two discrete random variables with joint
distribution function given by
P(~=k,n=R.)
=
00
\
L
00
\
L
i
a .. x y
i=O j=O 1J
j
where
~
a ..
1J
0 for all i,j
and x
~
0, y
~
for which the double series is convergent.
0
are values
The first
problem is to find a suitable way to define a function of
x and y, using the expression above.
ate case we used F(x;k) =
P(~
x
~k),
Since in the univari-
it seems reasonable now
to define a function F as follows:
= P(~~k,
F(x,y;k,£)
n~£)
We are only interested in positive values of x and y so we
will always assume that F(x,y; k,£)
such that x
~
~
0 or y
=
0 at all points
(x,y)
o.
In contrast to the univariate case,
in the bivariate
case it is not true anymore that F as defined above will be
a bivariate continuous distribution function for all bivariate power series distributions.
with examples.
c2 '
This will be illustrated
However, we can still define a class, say
of bivariate absolutely continuous distribution func-
tions that can be written as
F(x,y;k,£)
co
\
L
i=O
Clearly, if ;
o
co
\
L a .. x
j=O 1J
i
y
~
x < U,
they are independent, then
= P(~~k,n ~ £)
is a bivariate distribution since
61
~
y
<
V
j
and n have distribution functions
F(x,y;k,£)
0
in P, and
and each of F
and F
I
are in C.
2
In general, for a bivariate distribution function
F(x,y) of non-negative random variables, there are values
u s
00
and V S
00
such that
lim
F(x,y) = 1.
(x,y)-+-(U,V)
This means
that the point (U,V) must be on the boundary of the region
of convergence of the power series I Ia .. xiyj.
IJ
Definition 1
Given a bivariate continuous distribution function F
with marginals F I and F 2 , let
V = sup{y:
F (y)
2
<
U
= sup{x: FI(x)
I}
<
and
I} and let us denote by C 2 the class of
all bivariate continuous distribution functions F such that
00
00
00
00
i i
.. x y
I
I a IJ
F(x,y) = i=k j=R.
i
a .. X yj
I
I
IJ
i=O j=O
for all (x,y) in the set R = {(x,y): 0
x
<
< U,
0 <
y
< V},
for some power series IIa .. xiyj with a .. ~ 0 and some inteIJ
IJ
gers k, L
If F belongs to C , then the second partial derivative
2
of F with respect to x and y exists on the set R so that F
is absolutely continuous.
The density, say f(x,y), can be
written as the ratio of two power series:
f(x,y)
=
00
00
I
Lb . . x Y
00
00
i=k j
I
\'
i
IJ
=R.
I
i=O j = 0
j
"
c .. x l y J
1 J
62
(x,Y)ER .
e-
Thus, there cannot be an open set contained in R on which
f vanishes, for then the numerator would be identically zero
on R; hence f would be identically zero on R.
This rules
out, for example, the bivariate Dirichlet distribution,
since its density is given by
f(x,y)
for (x,y) such that 0
<
x+y
<
1, x ~ 0,
y~
f(x,y)
0,
0
otherwise.
Let us consider a few examples.
Let
P(~
~
and n have a joint multinomial
=
distribution~
N
k £
N
k, n = £) = (k £Jx y /(l+x+y)
where N is a positive integer and
N
(k £)= N!/k!£!(N-k-£)!
=
if k
0
+
Q,
s;
N
if k +
Q,
>
N
This is a bivariate power series distribution with a .. =
1J
The series converges in the set R = {(x,y): x:::: 0 I
restricting ourselves to non-negative values of x and y.
If we define
00
N
00
..
L L (..J )X 1 y 1
. k·
1
F(x,y) = 1= J=£
N
(l+x+y)
on
~,
then letting N = 2, k =
F(x,y) =
Q,
2xy
(l+x+y)
and the limit
63
2
= 1, we obtain
y~O},
1
lim F(x,y) = lim
X
y
x-+a>
x-+a> + 2X
y-+a> 2y
y-+a>
does not exist.
+
1
In fact,
lim F(x,y) = 0 for all y and
x-+a>
lim F(x,y) = 0 for all x. Thus, there is no distribution
y-+a>
function in C2 corresponding to the multinomial distribution.
Let now
~
and n have a bivariate negative binomial
distribution:
P(~=k,
n=R.)
=
r(N+k+R.)
k!R.!r(N)
k R.
x Y
-N
'
x + y
<
(I-x-y)
k,R. = 0,1,2, ... , N > 0 and 0
<
1.
This is a bi-
variate power series distribution with
a>
a>
L
L
\
\
i=O j=O
a .. x
i
Y
j
=
(I-x-y) -N
1)
a>
a>
\
\ r(N+i+j)
i j
= L
L
o"'r(N)x y
i=O j=O 1 . ) .
where the region of convergence in the first quadrant is
s
= {(x, y): x
~
0, y
Define
~
a>
L
G(x,y) =
0, x+y < I}.
a>
I
r(N+i+j)
xiyj
i=k j=R. i!j!r(N)
(I-x-y) -N
on the set S.
Taking N = k = R. = 1, we obtain
G(x,y)
= ~+ ~
I-x
l-y
and
aG
y + y
ax = (I-x) 2 (I-y)
~
0
aG
x +
ay = I-x
~
0
x
(I-y)
64
2
1
=
1
+
(l_X)2
on S.
x+y
=
(l-y)
2 ~ 0
As (x,y) approaches any point on the boundary of S:
1, G approaches one.
(0,1).
Let R
=
{(x,y): 0
1- a
Let a be any real number in
<
x
0 < y < I-a}
a,
<
I-----.,.---..,.-~
o
1
a
Define
1
f(x,y)
(l_x)2
=
I
+
(l-y)
0
2
for (x,Y)ER
otherwise,
and let
F(x,y) =
x rY
J
_ 00
for all x,y real.
J
_
f(u,v)dudv ,
co
Then, F is a bivariate distribution
function with the property that
F(x,y)
on R
65
so that F belongs to C2 "
Other values of k and t
also
provide us with continuous distribution functions in C ,
2
Let us consider functions of the simple form
F(x,y)
=
x 2 0, Y
= 0
otherwise,
2
0,
In this case, the power series has a finite number of terms
and the corresponding discrete random variables can take
We
on only two values.
have
aF
ax
=
2
0
=
2
O.
and
aF
ay
Also
2
=
aOOall+aOOaIOallx+aOOaOlally+all(2aIOaOI-aOOall)xy
s
3
(x,y)
Thus, a necessary and sufficient condition for F to be a
bivariate distribution function is
If this is the case, the marginals are given by
allx
=
(a01+a11x)
allY
66
x
~
0
y
~
0 ,
e·
both of which belong to C.
The conditional distribution of y given x is
r
Y
j
F(ylx)
y
f(t I x)dt
J
_00
f(x,t)
dt
f I (x)
_00
y
J
f(x,t)dt
ax3
_ 00
=
---::-~-=:---
d
f I (x)
dX
F(x,y)
F I (x)
Thus
F(ylx)
F(ylx) =
for y
in
~
o.
This distribution function is not necessarily
V.
For functions of the more general form:
MN
F(x,y)
aMNx y
=
M
L
N
\'
La . . x
i=O j = 0
1
..
J
1
,
0
~
x
<
00
o
y
~
<
00
y
J
a necessary and Sllfficient condition for F to be a bivariate
distribution function can he given in terms of the dual
hivariate power series distribution.
joint distribution given by
67
Let
E;.
and
II
have
P(~=k,
11=£) =
M
L
i=O
N
..
I
a .. x
j=O 1J
1
yJ
k = 0,1,2, ... ,M, £ = 0,1,2, ... ,N, x
~
0, y
x
~
~
O.
Then
F(x,y) =
P(~~k,
11~£)
,
0, y
~
o.
.
1
Letting
s(x,y)
M
N
L
L
i j
a .. x y
1J
i=O j =0
we have
aMNMx
M -1
N
MN M
N
- a~1Nx y . L
y s(x,y)
\'.
1 1a . . X
L
1=0 J. =0
aF
ax
1J
.
y
J
2
s (x,y)
and
aF
ax
s
for all x
~
0 and y
2
~
~
0 ,
(x,y)
O.
aF
ay
Similarly,
~
0 for all x,y
Now let us write
F(x,y)
MN
aMNx y
=
s(x,y)
1
1
MN
s(x,y)jaMNx y
G(x,y)
Then
2
2
-G (x,y) (a Gjaxay)
+
2G(x,y) (aGjax) (aGjay)
4
G (x,y)
and we have
~
0 for all x,y
68
~
o
if and only if
~
O.
2
2
aG aG
a G
- G
ax dy
dXdY
20
for all x,y
2
O.
Replacing, we obtain
M
- s (x , y)
for all x,y
2
O.
N
L
l
La. . (M - i) (N - j ) X yJ
i=O j=O 1J
0
2
2
Dividing through by s (x,y), we obtain
2
d F
dXdY 2 0 for all x,y 2 0 if and only if
2E(M- s )E(N-n)
-
E(M-s) (N-n)
2
0
for all x,y
2
0,
where E denotes expected value.
A necessary condition is obtained by noting that the
coefficient of the largest power in the previous expression
must be non-negative, i.e., F is a bivariate distribution
function only if
For functions of the more general form
00
I
F(x,y)
i=k
00
L
00
a . . x i yj
L
j=,Il, 1J
00
L~ai'xy
i j
i=O j =0
.1
it seems very difficult to find a simple necessary and
sufficient condition for the second partial derivative
with respect to x and y to be non~negative for all x,y
2
O.
Given a double power series \~ o\~ Oa .. xiyj, if we can
L1
69
=
LJ
=
1J
define a bivariate continuous distribution function in C :
2
F(x,y)
00
00
'
},
La .. x y
\'
i
j
i
j
= i=k j=Q. 1J
00
00
\'
\'
L
a .. x y
L
i=O j=O IJ
then the region of convergence of Ila .. xiyj must contain
1J
a set of the form R = {(x,y):
for some U
>
0, V
>
O.
a
x
$
<
a
U,
y
$
V},
<
That this is the case for every
power series is a consequence of
Abel's Lemma (Herve [2])
Suppose the series s =
k
in m complex variables zl""
the point
(b l ,b 2 , ... ,bm)EC
m
k
I
a
Z
k
ok , ... ,k
1
l ' ... , m2:
1
m
l
k
.Z
m
m
,zm' converges absolutely at
(space of order m-tuples of
complex members) and let b . ., a for j = 1,2, ... ,m.
J
Then
s converges at every point of the open polydisc
I z.J I
Ib·1
J
<
,
j
=
1,2, ... ,m}
and s converges normally on every compact subset of P.
•
\'00
\'00
i
Thus, If s(x,y) = Li=OLj=Oaijx y
(xo'YO),
X
that
<
2.
Ixl
j
converges at the point
o and yO real, then s converges for all
Ixol and
jyl
<
(x,y) such
IYol.
Independence of C and n
Let C and n have a joint power series distribution given
by
70
P(E;=k,
n=Q.)
a , 1 ,2,
k,Q.
00
00
L
L a .. xiyj
. . .,
i=O j=O 1J
and
oo
(x, y)
in the region
f convergence,
0
say T,
of s (x, y)
i j
. a LooJ. = Oa 1J
.. x y .
l 1=
Let
(xO'YO)ET be fixed and suppose that
independent
P(E;=i,
for these values of x and y.
n=j)
= P(E;=i)P(n=j)
E; and n are
Then
for all
i,j
0,1,2, ...
<=>
00
for all
i,j
for all
i,j
00
ffi
a. yn ) ( L a .x )
O
n=O In
ffi=O ffiJ O
( L
a ..
1J
Let
00
(n~oainY~)
A. =
1
(S(xo'Yo))
k
2
00
B. =
(m=O
La.
x~)
ffiJ
J
Then a 1..J
(S(xo'Yo))
= A.B.
1
J
for all
00
s(x,y)
'\
L
k
2
i,j
and for any
00
'\
L
a .. x
i=O j=O 1.1
71
i
y
j
00
'\
L
(x,y)ET, we have
00
'\
L
A.B.x i Yj
i=O j=O 1 J
00
=
L
(
00
i
A.X ) (
i=O
I
L B.yj)
j=O J
If we let s' and n' be the random variables corresponding
to (x,y), then
k £
P(s'=k, n'=£) = ak£x y /s(x,y)
Akx
k
=
00
•
L A.x
I
i=O
00
•
LB. yJ
j =0 J
I
But
00
•
( L a k . yJ ) x
j =0
P(s'=k) =
J
s(x,y)
=
k
=
00
L A.x i
i=O
I
Similarly,
P(n'=£) =
00
LB. yj
j
=0
J
so that
P(s'=k, n'=£) = P(s'=k)P(n'=£)
Thus, independence for one fixed value of x and y implies
independence for all values of x and y in T, and; and n
are independent if and only if a .. = A.B.
I
J
1 00
I
J
0,1,2, ... , for some sequences { AiJi=O and
72
for all i ,j
Clearly, if
~
and n are independent for all values
of x and y in T, then there exists a dual bivariate continuous distribution function in C ' whose marginals are
2
independent:
F(x,y)
=
P(~~k,
n~£)
00
=
00
I A.x
i=k 1
P(~~k)P(n~£)
i
00
I
A.x
i=O 1
i
I B.yj
j=£ _J_
00
lB. yj
j =0 J
Independence of X and y
3.
Let
00
00
00
00
i
.. x yj
I
I a 1J
i=k j=£
F(x,y)
(x,y)E:R ,
i
I
I a .. x yJ
1J
j=O i=O
where R = {(x,y): 0 s:: x
V
>
0,
<
U,
0 s:: y
<
V}, for some U > 0,
be the joint distribution function
of the random
variables X and Y.
Let us suppose that lim s(x,y) =
x-+U
for each yE:(O,V) and lim s(x,y) =
for each XE:(O,U), where
Y-+V
s(x,y) = I:=oI~=oaijxiyj.
This is the case, for example,
00
00
when U
V =
F1
(x)
00
Then, the marginal of X is
lim Fex,y)
y-+V
lim
y-+V
But,
73
00
00
L
L
00
00
i
a . . x yj
j =k j = £ 1J
i
I
I a 1. J. x yj
i=O j=O
£-1
00
L ( L a .. x
lim
y+V
••
1
)yJ
j=O i=k 1J
==
0 ,
for each XE[O,U), since lim s(x,y)
y+V
lim
y+V
00
00
L (
La.
0
Thus, we can write
00
yj ) xi
i=O j=O 1J
For each fixed y, the expression
00
00
L(
i=k
00
Lao. yj ) xi
j=O 1J
00
••
L ( L
a .. yJ)x
i=O j=O 1J
1
is a distribution function in C, so that F
l
is the limit
Then, by Theorem
of a sequence in C with common index k.
i
3 of Chapter I, there exists a power series Lbox , with
1
b.
1
~
0 for all i, such that
00
L b.x i
F 1 (x)
i=k
1
00
L b.x
i=O
o
s x
<
U .
i
1
Similarly, we can show that the marginal of Y is given by
o s
74
y
<
V ,
L]. = Oc.y
]
00
for some series
J
~
with c.
]
Hence, both marginals F
I
0 for all j.
and F
Z
belong to C.
Further_
more, if X and Yare independent, then
F(x,y)
for all
(x,y)ER ,
that is,
00
L b.x
F(x,y) =
i=k
00
i
1
00
L b.x
i=O
(x,y)ER ,
00
i
L c.y]
j =0 ]
1
or, equivalently, there exist random variables ~' and n'
with power series distributions, such that
F(x,y)
(x,Y)ER .
Note that the converse of the result in Section 2 above is
not true.
That is, independence of X and Y does not imply
independence of
~
and n.
F(x,y)
Here, FEC Z '
In fact,
r(~
X
For example, let
xy ,
o
s x s 1, 0 s Y s 1.
and Yare independent but ~ and n are not.
1 n)
o.
7S
APPENDIX
The following are statements of theorems that have
been used in the proof of some results in this dissertation.
Theorem A
Gauss
(Hille [3], page 84)
Let P(z) be a polynomial of degree n having the zeros
zl,z2""
,zn (multiple roots repeated in this sequence
according to their multiplicity), and let IT be the least
convex polygon containing the zeros.
Then P' (z) cannot
vanish anywhere in the exterior of IT.
Theorem B
Helly (Leadbetter [11], page 42a)
Let F ,F , ... be a sequence of distribution functions.
2
l
Then, there is a subsequence {F
n.
},
j=1,2, ... , and a non-
J
~
decreasing right continuous function G with 0
for all x, such that lim F
.
J -+co
n.
G(x)
~
1
(x) = G(x), whenever x is a
J
continuity point of G.
Theorem C
(Hille [4], page 252)
Given a sequence of power series
n = 1,2,3, ... , which converge for
fy
Ifn (z) I
<
Izi
M for such values of z.
sequence {zk} with
<
R
f
n
<
(z)
=
a
,co
Lp=O np
zP
,
R and which satis-
Suppose there is a
and lim zk = 0 such that
k-+ co
lim fn(zk)
exists for each k.
Then,
n-+ co
each p, the series
co
2
a zP - fez)
p=o p
lim a
n-+ co
np
-
a
p
exists for
converges for
we have lim f
Theorem 0
Izi
n
(z)
<
R, and for each positive R < R
1
fez) uniformly with respect to z in
(Gnedenko and Kolmogorov [1], page 33)
For the weak convergence F
n
=>
F each of the follo~ing
three conditions is necessary and sufficient:
(1)
Fn (x) ~ F(x) at every point x which is a continuity point of the distribution function F(x).
(I1)
Fn (x) ~ F(x) on some set C which is everywhere
dense on the real line.
(I I I )
L(F n ,F) ~ 0, where the distance L(F,G) between
two distribution functions F and G is defined as
4It
the infimum of all h such that for all x
F(x-h)
Theorem E
- h ~ G(x)
Stone-Weierstrass
~
F(x+h) + h
(Rudin [15], page 146)
If f is a continuous complex function on [a,b], there
exists a sequence of polynomials P
fex),
uniformly on [a,b].
n
such that lim P ex)
n~oo
If f is real, the P
n
n
may be taken
rea 1.
Theorem F
(Rudin [15], page 137)
Let a be monotonically increasing on [a,b].
Suppose
,R(t).) on ra,bl, for n = 1,2, ... , and suppose f
~ f
n
n
uniformly on [a,b].
Then f(R(a) on [a,h] and
f
b
J
a
b
fda
=
lim J f da
n~oo
a
n
77
=
Theorem G
Bernstein (Lorentz [12], page 5)
For a fun c t ion f ( x ) h 0 u n d e d
lim B (x)
n-+ oo
n
=
.0 n
[0, 1 ], the reI at ion
f(x)
holds at each point x of continuity of f,
holds uniformly on [0,1] if f(x)
interval.
Here, B
n
and the relation
is continuous on this
is the Bernstein polynomial of degree
n of the function f.
78
REFERENCES
[1]
Gnedenko, B.V. and Kolmogorov, A.N. Limit Distributions
for Sums of Independent Random Variables, AddisonWesley Publishing Co., Massachusetts (1954).
[2]
Herve, M.
Several Complex Variables,
Press, London (1963).
[3]
Hille, E.
Analytic Function Theory,
Company, New York (1959).
[4]
Oxford University
Vol.
Analytic Function Theory, Vol.
Company, New York (1962).
1, Ginn and
2,
Ginn and
[5]
Johnson, N.L. (1959)
On an extension of the connexion
between Poisson and X2-distributions, Biometrika,
46, 352-363.
[6]
Johnson, N.L. and Kotz, S.
Discrete Distributions,
Houghton Mifflin Co., Boston (1969).
[7]
Continuous Univariate Distributions I,
Mifflin Co., Boston (1970).
[8]
Continuous Univariate Distributions II,
ton Mifflin Co., Boston (1970).
[9]
Continuous MUltivariate Distributions, John
Wiley and Sons, New York (1972).
Houghton ~
_
Hough-
[ 10]
Khatri, C.G. (1959)
On certain properties of power
series distributions, Biometrika, 46, 486-490.
[ 11 ]
Leadbetter, M.R.
(class notes)
[ 12]
Lorentz, G.G. Bernstein Polynomials, University of
Toronto Press, Toronto (1953).
J
Noack, A. (1950)
A class of random variables with
discrete distributions, Annals of Mathematical
Statistics, 21, 127-132.
[ 13
Probability and Stochastic Processes,
Statistics Department, U.N.C. (1972)
[ 14 ]
Pati1, G.P. (1968)
On sampling with replacement from
populations with mUltiple characters, SankhYa, 33,
Series B, 355-366.
[ 15 ]
Rudin, W.
Principles of Mathematical Analysis,
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Sarkadi, K. (1960)
A rule of dualism in mathematical
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80