..
On
the Cumulants of Cumulative Processes
by
Walter L. Smith*
Department of Statistics
University of North Carolina at Chapel Hill
Chapel Hill, North Carolina
October 1979
•
*The work of this author was supported by the Office of Naval Research under
Grant N00014-76-C-0550.
On the Cumu1ants of Cumulative Processes
by Wa1terL. Smith
(University of North Carolina)
§1.
Irttroduct~on.
Cumulative processes were introduced in 1955 as a natural concomitant to
the study of what the author called regenerative processes (Smith, 1955).
In
that first paper some basic results concerning cumulative processes were
obtained; these were also explained, and the notions of cumulative and regenerative processes further detailed, in Smith (1958).
For the purpose of the
present discussion we shall introduce the cumulative process as follows.
Let {Xn}~=t' be a positive renewal, process" that is: an infinite sequence
~
of independent and identically distributed random (iid) variables which are,
with probability one, positive.
Let us set So = 0 and, for n = 1,2, ... ,
Sn = Xl + X2 + ... + Xn · Then suppose W(t), for t > 0, to be a stochastic
process, such that W(o) = 0, satisfying the following requirements:
(C1)
If Yl = W(X 1), and, for n = 2,3, •.. , Yn = W(S n+·1)
. - W(Sn );. then
{Y };=l is a sequence of iid random variables.
n
(C2)
Wet) is, with probability one, of bounded variation in every finite
t-interval.
(C3)
If we set
Wet) = fo_ldW(u) I
...
for the associated "variation process" (de:f:ining Wet) as identically
zer~ should Wet) be of unbounded variation), then Wet) also satisfies
(Cl) .
2
proaess~
Such a process Wet) is called a /?q,nuZative
and since such
processes were introduced they have been found to have wide applicability
especially to problems in operations research.
Typically Wet) represents the
total cost, fuel consumption, lost customers, and so on; the time points
. {Sn } are so-called "regeneration points" of the underlying process.
For r
= 1,2, ••• ,
set \)r = E(yl)r when this mciient exists (Le. when the
relevant integral is absolutely convergent).· Similarly set
the implication of the tilde is obvious.
F(x)
= P{Xl
~ x} andFn(x)
= P{Sn
vr = 86\) r,
Let us also write II
~ xl, for n
= 1,2,3, ....
r
where
= t(X l ) r,
Finally, when the
product moments exist, we shall write
llrs
(r = 0,1,2, ... ; s
= 0,1,2, •.• ).
~r
Thus
= ~ro
and vs
= IIos'
We. similarly write
Prs for product moments of Xl and Y .
l
In Smith (1955), amongst other things, the following was proved:
Theorem 1.
PO'l02
=
llll'
Suppose 112 <
Then~
as t
00
and
-+
00,
"2 <
00.
Let
2
°1 = 112 -
Var Wet) - III-1 t {0'22
-
2
IIp
°22 = v 2 -
2p0102(V1/lll)
+
2
v p and
2 .
2}
0'1 (V 1 /11 l ) •
A very special cumulative process is N(t), the renewaZ aount, which gives
the number of regeneration points to have appeared in the interval (O,t]; thus
N(t) is the maximum k such that Sk
~
t.
In Smith (1959) the cumulants of N(t)
were studied, and it was shown that under a weak condition on F(k) and certain
reasonable moment conditions, the cumulants of N(t) are asymptotically linear.
In more detail, if IIn+p+ 1 <
00
for some n
= 1,2,3, .•.
and some integerp
~
0, then
there exist constants a and b such that the nth cumulant of N(t) is given by
n
n
.
3
ant
(1.1)
•
+ b
n + t n (t)1
where various things can be said about therflpidity of conve.rgence of r (t) to
n
zero, as t + 00, depending on pand the assumption on F(x). We refer to the
original paper for details.
These cumulant results could have been substantially improved if the
results of Smith (1966) on general remainder tems in renewal theory been
available at the time the cumulants were studied.
The obj ect of the present role is to ext'end ·Theorem 1 considerably artd::to
demonstrate that results like -(1.1) can be proved for certain classes of cumulative processes.
However; a difficulty blocking the way of a very general
treatment is that there is such a wide variety of possible behavior for Wet)
between regeneration points even when the distribution of the· {y } is detern
mined. To circumvent this difficulty we shall introduce two special classes
of cumulative process:
Class A.
Let {(Xn'Yn)}:=l bea sequence of independent and identically dis00
tributed random vectors such that {Xn}n=l is a positive renewal process, and
define
N(t 1 I
+.
Wet) =
ClassB.
l
k=l
Yk ·
Let· {(X ,Y )}oo I be as above, but define Wet) = 0 for t ~ Xl and, for
n n n=
..
Nft)
Wet) =.
k=l
.
"'k'
4
The Class A process can be
levied for the first
th~ught of as follows. At zero time a cost is
e
~egenerative
cycle of the process, this cost Y depends
l
on Xl' the duration of that first cycle~ At time Xl a further cost Y2 is
levied for the next cycle, and so on.
the interval [O,t].
ThenW(t) is the total cost incurred in
The Class B only differs from the Class A in that one pays
for a "cycle" at its end, rather than at its beginning.
The Class A and Class B processes pose very similar mathematical problems
with very similar solutions.
processes.
This paper will be devoted exclusively to Class A
However, at the end of this report, in §6, we briefly discuss the
ClassB process and show that the theory and formulae appropriate to that
process are easily obtained from the theory provided for Class A processes.
The work is heavily dependent upon theorems which establish the nature of
remainder terms in renewal theory and related topics.
We shall depend very
much on the results given in Smith (1966), and some words of explanation are in
order at this point.
It is possible that the {Xnl are, almost surely, multi-
ples of some constant
w> O.
If
is chosen to be maximal then we would say
Wet) is a periodiacumulative process, with period til.
If Wet) is not periodic
it is aperiodia and this report assumes throughout that Wet) is such an
aperiodic process.
There would be a similar, and in some ways easier, treat-
ment for the periodic case, but we shall not consider it in this report.
If the marginal distribution of the {X l is F(x) "= p{X < xl, let us write
n
nFk(x) = P{X I + X2 + ••• + ~ 2 xl. If, for some k = 1,2, ..• , Fk(x) has an
absolutely continuous component then we say F is in the class S.
this paper we must assume F
E S;
Throughout
this is a technical assumption which is
unlikely to present any difficulty in applying the results of this report to
real problems.
But the theorems of Smith (1966) are all based on the hypothesis
..
5
F
€
S and our statements about the rapidity with which various remainder
functions of t tend to zero, as t ...
= Xl
Let us write Sn
..
so-called
00,
+ ••• + X .
would be false without this hypothesis.
Then we shall see below that the
n
~-moments:
00
~R,(t) =
(1.2)
for R,
= 1,2, •.. ,
pisn-< d,
L
n=O
play a crucial role in our arguments.
In Smith (1966) a useful class
A function M(x), defined for 0
M(x)
M(2x)
~
1 for all x
= O(M(x))
~
0; M(x
~
x <
+ y) ~
for all x > O.
M* of monotone functions Mwas introduced.
00,
Mif: it is non-decreasing;
belongs to
M(x)M(y) for all x
~
0, y
~
0;
A function B(x),of bounded total variation on
(0- ,00) is said to belong to the class B(M;V), for some integer v
f 0OO
xVM(x) IdB(x)
In Smith (1966) it is then shown that if F
~R,(t)
(1. 3)
= PR,(t)
€
I·
<
00
~
0, if
•
B(M,R, + l)n·S then
+ R~(t)
where PR,(t) is a polynomial of degree R, in t and RR,(t) is a remainder term which
tends to zero as t ...
value of R, > 1.
(1. 5)
00
and belongs to B(M;O).
This is true for any integer
Thus the following statements are true, as t ...
00,·
6
(1.6)
It is clear that a Chebyshev argument gives (1.6) from (1,5) and (1,4).
Thus
(1.3) implies'
= PJ/,(t)
(1. 7)
1
+
'.
O( M(t) ) ,
although, of course, (1.7) is much less than is true.
In §2 and §3we shall prove, using these results on
~~moments
that Wet)
does indeed have, asymptotically, cumulants which are linear functions of t.
If Km(t) be the mth cumulant of Wet) then Theorem 3.1 will show
Km(t)
= Amt
+
Bm + Rm(t)
where A and B are constants and R (t) tends to zero at a rate shown to depend
m
m
m
on the tail behavior of G and F.
Ensuing sections are then devoted to the
rather off-putting calculations of the constants {Am} and {B } which are, as
m
in advances, increasingly involved rational functions of the product moments
~ae'
Essentially the {Am} are given for m = 1, •.. ,8 and the {Bm}, for the case
BY. = 0 only, for m = l,.~.,6.
J
In §5 checks are discussed of the detailed formulae given for the {Am} and
{B}.
m
In §6 there is a first discussion of the Class B process.
We close this introductory section with two comments on the reason for being
interested in the cumulants of cumulative processes.
In general, these processes
are difficult to discuss theoretically with exactitude.
But they arise widely
in practice and it is desirable to have, at the least, reliable information
about the
sampl~ng
characteristics of sample means and sample variances.
one can easily be needing information on the fourth cumulants of Wet)
Thus
7
e
and even on the sixth cumulaJit if the skewness of the distribution of Wet)
should interest us.
A second reason for investigating these cumu1ants is that
we may wish to approximate the distribution of Wet) with an Edgeworth,or
similar, expansion based on successive refinements of a dominant normal approximation.
The need for higher cumu1ants of Wet) then quickly appears.
Incidentally, the rather long formulae for An and B , when n grows large,
n
are not as unwie1d1y as a first encounter with them might suggest. We have
found that certain hand-held programmable calculators can be easily programmed
to provide AS and B6 ,for instance, for changing values of the product moments.
Lastly we wish to acknowledge that this work was triggered a few years
ago by an enquiry by the late Mindel C. Sheps.
In preparing her book with
Jane A. Menken (Sheps and Menken, 1973), Dr. Sheps needed detailed information
on the asymptotic behavior of the first few cumulants of a pepiodia cumulative
.
process.
Such results appear in the book in connection with interesting
applications ·of the theory of cumulative processes to mathematical models of
conception and birth.
8
§2.S0me
basicre~ults.
Let us set G(x,y) =p{X l ~x, Yl'~ y} for the joint d.f. of (Xl~Yl)' and
hence of (Xn,Yn ) for n = 2,3, •••• We have already written F(x) = p{X ~ x}
l
for the marginal d. f. of Xl; let us therefore write H(y) = P{Yi ~y} for the
-- --
marginal d.f. of Yl . We introduce the followi~g notation for transforms, in
which sand 8 are taken to be real and s > o.
F(s)
i8Y l
H(8)=Ee-
(2.1)
G(s,8)
= F(s)
l; (t) = 1
n
Thus G(s,O)
Let
,
.
-sX
= Eel
~n(t)
to
~n
and G(O,8)
=E e
-sX l +i8Y
.
l
= H(8).
if 5n -<t and r; n (t) = 0 if Sn >t; we shall abbreviate
where no ambiguity can arise.
Our argument will be much clearer if we deal with a specific case rather
than giving the general ar.gument; the way in which a general argument can be
constructed shOUld be clear from our discussion of the specific case, which we
choose to be an examination of the fourth moment of Wet); this, particular
problem being sufficiently non-trivial to bring out the features of the general
argument.
A
certain notation will prove helpful.
If Pl,P2"" ,P,t are integers, we
shall set
(2.2)
Thus, for example,
...
9
Now it should be apparent from the" definition of a Class A cumulative
process that
00
(2.3)
Wet)
If we note that ~
rl
= r~o
~r Yr +l .
~r = ~r when r 2 < r l and that ~r2 = ~r' then a careful
2
1
.
calculation based on (2.3) yields the conclusion:
(2.4)
{W(t)}4
= I t (4)
+6
I t (2,2)
+ 12 It(1,1,2)
+ 12 I t (2,1,1) + 12 It(1,2~1)
+ 24 It(l,l,l,l) + 4 It(3,1)
+ 4
I t (1,3)
.
We wish to learn about E{W(t)}4; thus we must be able to deal with individual expressions such that E I t (1,2,lL and so on.
expectations like
This leads us to consider
(2.5)
But it is to be noted that Yr +1 is independent of
l
,
~.
r
l
only on S' • Thus (2.5) simr
(2.6)
E It (1,2,1)
= ViE
t
I·
r I
>" >i:>O
11:2
3-
~
rl
l
10
vari~
In notation already introduced, if 62 and 63 represent real dUDDIly
ables,
= {F(s)}
(2.7)
r -2
1
G(s,e ) G(s,e ) .
2
3
Also let us note that
(2.8)
Thus (2.8) and (2.6) yield
E{y
.
2
st
r 2+l y r +l ~o. e- .~ r (t)dt}
l
3
-s5
T
1
}
.
The manipulations performed here are easily justified if we assume v2 exists
(absolutely) for we can first replace the Y-terms by their absolute values, thus
making all terms non-negative, and we shall see below that the resulting triple
series must be (absolutely) convergent.
Let us set, for k
= 1,2, .•• ,
k
(2.10)
i
Ck(s)
= (i~e)~(S,e)
I ;: ~ Y~. e-sX-, n,
n
6=0
= 1,2,....
~s5
Then we see that, for r 1 > r 2 > r 3
This result, used in (2.9) yields,
. {2 .
~ 0, E Y + Y +
e
r2 1 r3 1
r 1 .}
=C
.
2 (s)C 1 (s){F(s)}
r 1 -2
11
(2.11 )
1:Oe-st
E
VI'
Lt (I,2,I)dt =5-
t· t·· t
r l >r 2>r?0.
r ~2
CI(s) C2 (s) {F(s)} 1
00
VI
=s- Cl (s) C2 (s) l
r=2
r(r~l)
2
{f(s) }r-2
VI ,C l (s)C 2 (s)
=
3
•
2s[1-F(s)] .
Let us set
S(I,2,1)
= Jt»O e-st
E £t(1,2,1)dt,
t
with an obvious extension of this notation to terms like S(3,1), S(4), and so on.
Thus (2.11) yields
(2.12)
S(I,2,1) =
Incidentally, since F(s) < 1 for s > 0, the convergence of the triple
series needed earlier to justify certain maneuvers is any easy consequence of
the obvious finiteness of S(1,2,1).
It should be clear now how one can obtain
a variety of similar results which we shall need; for some examples:
V
S(4)
=s[l
4
_ f(s)] ,
C
V
S(2,2)
(s)
2 2
=-----.:;-
s[l - f(s)]2 '
Ii
'V
SCl,l,l,l)
=.
1
[C (s)]4
.1
3! s[l - fCs)]
4 .
12
The relevance of all these
. .-
Laplace Transform of'E{W(t)}.
(2.13)
S~terms
is that, together, they give us the
--'-4~
For (2.4) shows that
E~ e- st {W(t)}4 dt
=S(4)
+
6 S(2,Z) + lZ $(1,1,2)
+
lZ S(2,l,l)
+
Z4 S(l,l,l,l)
+ 12
S(l,2,l)
+
4 S(3,l)
-+
~ of {W(t)}4 we
+ 4 S(l,3).
Thus to gain information about the asymptotic behavior as t
must look upon each S-term as the Laplace Transform of a function whose
asymptotic behavior must be separately determined.
use of the results of Smith (1966).
..
At this point we will make
But the latter paper, essentially dealing
with renewal theoretic
problems in which the lifetimes {Xn } are not restricted
,
to being positive, is couched in terms of Fourier Steiltjes transforms; we must
therefore make certain necessary adjustments to those results.
Consider S(l,2,1) as a specific example.
We may regard s S(l,Z,l) as the
Laplace-Stie1tjes transform of some non-decreasing function H(t; l,Z,l), say.
Thus, from (2.1 ),
(2.14)
"I Cl(s)CZ(s)
Z[1-F(s)]3
= 10
00
e
-st
H(dt; l,Z,l) .
Equation (Z.lO) shows that Ck(s) is the Laplace-Stieltjes Transform of
,II
(2.15)
13
If M(x) is a moment function of the kind introduced in
~
some
= 0,1,2 1 " "
§I
and we ~ssume, for
\
e
that
(2.16)
then it follows that Ck(x) e
B(M;~).
Moreover one can write
(2.17)
where Vk(s) e B(M;O) and is the Lap1ace-Stie1tjes Transform of a function
O~k(x),
say, such that
Theorem
3
O~k(x)
=0
for x < 0, and
of Smith (1966) and Lemma
4
O~k(oo)
= 1.
This follows from
of Smith (1959), with suitable, but fairly
obvious, modifications.
With reference to (2.14), if we assume temporarily that (2.16) holds with
~ = 3
and k
2, we have that C (5) C2 (s) e B*(M;3) and by the same argument
l
behind (2.16) we can write
=
(2.18)
say, where VO(s) e B*(M;O) is the Lap1ace-Stie1tjes
0o(x) .: 0 for x < 0 and 00 (00) = 1.
Transfo~
of 00(x) such that
In (2.18) the constants YO'Y 1 'Y 2 'Y3 are
rational functions of 1101,1111,1121,1131,1102,1112,1122,1132'
On the other hand, the same a.rgwnents al10w us to write
\0..
(2.19)
14
e
where fc3Y(s) is the transform of the distribution function .of some positive
random variable, andbel0,ngs to 8* (M; 0) •
In a similar way, since' {2 - F (s)}
..
~
,
(2.20)
say, where Vl(s)
x < 0 and 01(00)
€
..
{I - F (s)}
2
2 2
= llis
€
8,* (Mj3), we can show
. 3
...
- l-'211ls 1\ (s),
B(MjO), is the transform of 01 (x) such that 0l(x) - 0 for
= 1.
If we combine (2.19) and (2.20) appropriately, we find from (2.18) that we
can write
(2.21)
,,,
say, where gO,gl,g2,g3 are rational
functions~of 1l1,1l2,1l3'
and the product
with r = 0,1,2,3 and s = 1,2. Furthermore V2 (s) € B*(M;O) is the
rs
transform of 02(x), say, where D (X) = 0 for x < O. A careful calculation shows
2
that 02(00) must ,be a rational function of the same moments and product moments
moments II
as are go, ..• ,g3'
In Smith (1959) certain convenient
(2.22)
~k(t) =
It was shown that, for k
(2.23)
were introduced:
E[N(t) + l][N(t) + 2] ... [N(t) + k],
=0
.
~-moments
t.< O.
= 1,2, ••• ,
and s > 0,
t
~
0,
15
But
(~.14)
and (2.21) shoW' that
flO
(2.24)
JO e
-st
.
H(dt; 1,2,1)
.80
---'="3 +
[l-F (s)]
Consider first
~2(s).
. 2
[l-F(s) ]
+
[l ... F(s)]
We can write
~2(S) =
(2.25)
.81
K(s) 3
'
[l-F(s)]
•
and, by (2.Zl), claim that
Evidently
K(s)
€
B*(M;3) and those will be a K(x) such that
K(5)
flO
= JO-
e
-sx
K(dx).
Thus K(-i6) is the Fourier-Stieltjes transform of K(x) and, in view of the fact
that
K(s)
= s3
tilZ (s)
16
we can easily show K( -;i.e)
For real
e we
= ocl e13 )
in a ne,ighborhood of the origin
e = o.
have
(2.26)
This expression is in a form suitable for the direct application of Theorem 1
of· Smith (1966).
The conclusion of that theorem is that there exists a func-
tion P2(x) in B(M;O) such that
~
However, as we shall show a little later, the variation of P2 (x) is confined
to [0-,00). If we assume this fact, temporarily without proof, then
(2.27)
But P2 (x) is of bounded variation, so the right-side of (2.27) is continuous in
16 ~ 0 and analytic in 16 > 0, if we now allow 6 complex values.
But F(-i6) and K(-i6) are also continuous in the closed upper half-plane
16
~
0 and analytic in the open upper half-plane 16 > O.
Recall that
IF(-ie)1 < 1 for all real 6 ~ 0, in view of oup assumption that F
€
If fl(z) and f 2 (z) are functions of complex z, continuous in Iz ~ 0 and
analytic in Iz > 0, and if f l (z) :: f 2 (z) on the real axis, then f (z) :: f (z)
1
2
throughout the upper half-plane. This is a fairly easy result in complex
analysis, best proved by a conformal mappi?g of the half-plane onto the unit
~
disc and appeali?g to the Maximum Modulus Theorem.
is played. by
Thus, if the role of fl(z)
17
and that of f (z) by
2
.... 'K(;;.iZ) .
[1 - F(_z)]3
since (2.26) and 2.27) show f 1 (z) - f 2 (z) on the real axis, we may conclude that
for real s > 0,
K(s)
=
[1 _ F(s)]3
Thus we have shown that If 2 (s),
e: B* (M; 0),
variation is confined to [0-,00).
from. lim If 2 (s) .
roo
-sx
J O_ e
P2(dx)
is the transform of P2 (x) whose
We can take P2 (0-)
= 0 and
determine P (00)
2
Thus
s+O
P (00)
2
= lim
s+P
=
s
3
'0 2 (s)
[l - F(s)]3
'0 2 (0)
Ji1
Therefore, in view ·of our earlier consideration of '0 (0), we may c1aimP (cXl)is
2
2
a rational function ofJ.l1'J.l 2 ,J.l 3 , and the product moments llrs' with r = 0,1,2,3,
and s
= 1,.2.
We must now cover a point. glossed over earlier, that P2 (x) has all its
variation in [0"',00).
To do this we refer to Theorems 4 and· 5 and the ensuing
discussion, in Smith (1966).
It is shown there that there exist constants
18
4
.A.(4)
J
I
(2.28)
+ . 1\ (e) ,
j=l
where A(e) is the Fourier-Stieltjes Transform of a function L(x) , say, which is
of bounded variation, identically zero for x < O.
A. (4)K(-ie)
4
= I
'i'2(-ie)
J.. . 4-j
(-l-li i8 )
j=l
But (2.26) and (2,28) give
~ K(-i8) n(8) .
It is immediate that K(-ie)A(8) is the Fourier-Stieltjes Transform of a function
identically zero in (-00,0).
We therefore need only deal with terms like
K(-i6)
for r
C(_i8)r
e
= 1,2,3.
For r = 3 this term is simply V (-i8}, and there is nothing to prove.
2
we must turn to (2.21).
Suppose, for
instance~
r
= 2.
If r < 3
Theorem 3 of Smith
(1966), especially the proof of this result and the accompanying Lemma 1, shows
that we can write
(2.29)
where
V3 (8)
in (-00,0).
is the Fourier-Stieltjes Transform of a function, identically zero
This is obtained by expressing the left-side of (2,29) as a linear
combination of characteristic functions in B~(M;l), and by noting that it is
0(leI 3) near e
= o.
Thus
19
and the desired result is obtained.
We must now hark back to (2.24) and discuss the significance of the term
'1'1(s) .
But it is plain from (2.23) that '1'1 (s) is a linear combination of
transforms of the
~-moments ~1'~2j
the asymptotic behavior of
~3'
and
~-moments;
Thus we merely need to know about
this matter was investigated in Smith
(1959) but, as we have already remarked, more general conclusions are possible
if we use the results of Smith (1966).
Computation shows that, for k
= 1,2, ... ,
00
Thus
~k(t)
is precisely the function covered by Theoremi5 of Smith (1966), in
which the substitution
The cases k
= k.+
~
= 1,2,3
L is to be made.
are of immediate interest.
Consider k
Here we must take ~ = 4 in Theorem 5 of Smith (1966).
4
.
E X M(X) < 00, and can then conclude that, for t > 0,
say, where A(t)
~
0 as t
~oo
are rational functions of
and A(t)
~
V(M;O).
= 3,
for example.
We must assume
The constants A33,A32, ••• ,A30
~1'~2'~3'~4.Simi1arresults
hold for
~2
and
~1'
Thus we see that '¥l(s) is the Lap1ace..Stieltjes Transform of a function Pl(t),
say, such that for t > 0:
20
where pet)
-+
0 as t
-+
00,
pet)
E:
B(M;O), and the constants II , ..• ,II are
O
4
rational functions of ~1'~2""'~4'
If we combine our findings on
~l(s) and~2(s)
we obtain the following
result. ,
If E X4M(X) and E X3y 2M(X) are both finite, then for t > 0,
Lemma 2.1.
e
where kO,kl'k 2 ,k 3 are rational. functions of ~l"" ,]J4' and th.e product moments
]J rs for r = 0,1,2,3 and s = 1,2. The remainder function pet) -+ 0 as t -+ 00,
and pet)
E:
B(M;O).
It should not be too difficult to see that the treatment we have just
afforded S(1,2,1) can be given any of the S terms.
letPl,P2, ..• ,Pk be, k integers.
S(P l ,P 2 "",Pk)
=
k
e
C
Pj
(s)
gene~al
example,
Then one can show
"'p C (s)C (s) ...
1 P2
P3
(k-l)! [1 -F(s)]
Letq be the largest of P2,P 3"",Pk'
can expand the C-product as follows:
nj=2
For a more
R
We must assume E X~ 1Yl,q M(X 1 ) <
00
and
=
In this expansion, the coefficients gO,gl,.·.,gk will be functions of
and the product moments
~rs
for r
= O,l, ••• ,k
and s
=P2,P 3"",Pk'
~1,]J2, ••.
,Uk
21
The argument then goes much a.s before., making use of the <P-moments, an'd " _
leads to
thefol1owi~g.
Lemma 2.2.
Assume E
t
k integer-8 and 1J1!lite q = max(P 2,P3, .. .,Pk}'
EIYl.lPl < 00, and E X~IYl.lq M(Xl)<oo. Then,Ifor
Let Pl'P2, ... ,Pk be
x~+l
M(X l ) <
00,
> 0,
(2.30)
where the aoeffiaients cO,cl, .•• ,ck arae r-ationat funationsof lJ ,1J2 , ... ,lJ +1
l
k
and of the pl'Oduat momentSlJrs 7JJith r
remainder funation
pet)
~
Oas t
~
00
= O,l, ..• ,Rand
and pet)
~
s
= P2 ,P3"",Pk'
The
B(MiO).
We can use this lemma to deduce the asymptotic behavior of E{W(t)}4.
From
(2.12) we can infer that
EIW(t)}4
= H(ti4)
+ 6 H(ti2,2) +12.H(til,1,2)
+ 12 H(ti2,1,1}+ 12 B(til,2,1)
+ 24 H(til,l,l,l) + 4H(ti3,1)
+ 4 H(t;1,3) .
If it is supposed that EI Yl l4 < oo,E X~ M(X l ) <00, and that E X~IYlls M(X ) < 00
1
for all integer r ~ 4, s ~ 3, such that r + s ~ 5, then Lemma 2.2 can be applied
to each of the above H-functions in turn and the results establish that, for
t > 0,
22
where 'P4 (t) is a polynomial in t and R4 (t) is a remainder term, tending to zero
as t ~ co, and belo.ngi,ng to B (M; 0). The coefficients of the polynomial Pet)
will be rational functions of '1-11''1..1 2 ,. • .,'1..1 5 , "1'''2'''''''4' and the product
such that r ~ 4, s ~ 3, and r + s < 5.
rs
We can evidently anticipate at this stage what will be true in the general
moments '\-I
case; we shall have, for integer
m~
1,
where the I * means a sum over 'all partitions PI + P2+ ... + PR
order of the parts is significant.
= m,
where the
When we apply Lemma 2.2 we find the following
moment conditions arise:
(i)
Since one partition of m is into the sum of m units, it is possible
..m+l
to have k = m. Thus we must have E Xl M(X l ) < co.
(ii)
Since the H-function H(t;m) can arise, we must have
(iii)
If q
= max(P2,P3, ••• ,Pk)
is given, then the maximum possible value for
k is when all the pIS except the maximum one equal unity.
(k - 1) + q
every q
~
= m,
i.e. k + q
m - 1, k
~
Elyl,m <~.
= m + 1.
Thus we need E X~
m, such that k + q
~
This makes
Iy Iq M(X) <co for
1
1
m + 1.
We have therefore deduced the following.
FoT' integer- m ~ 1, assume E X~+ M(Xl) <
1
Theorem 2.3.
E X~IY1Is M(X ) <:'co foT' r ~ Jil;
l
S
< m - 1, and r +
5
co,
EI Y11 m < co,
< m + 1.
Then~ foT'
E WCt) m = 'P (1:) + R (t).
m
m'
7JJherae 'Pm(t) is a poZynomiaZ ofdegpee m in t" Rm(t)
~
0 as t
~ co,
and
and
t > 0,
23
..
..
of the momentslll'lJ2"" '~m+l'
that r
~
m, s < m-
I, and r +
'V1''V 2P
s < m+
'' ,'Vm" and the produat moments llrs suah
1~
24
moments and product moments which arise as the coefficients in the polynomials
"
#
If g(x ,x , ... ,xm) be a rational function of the real variables xl ,x 2 , ... ,x
m
1 2
in Euclidean space of m dimensions then it is uniquely determined by its values
in any sphere, however small.
This idea motivates the argument of this
section.
For real a, define
(3.1)
as the characteristic function of Wet).
Thus
(3.2)
If we calculate the Laplace Transform, for real s > 0,
o
Ma(s)
flO
=
)0 e
-st
Ma(t)dt,
we find from (3.2) that
00
(3.3)
MaO = E L"
f
s
y•
-st+l·e~"f+l
t:.o
J dt
~+ I e
p.,=o SR,
=E
"
00
ieE~+1 Y.J" (""-SSR,"
e
-
Ie"
~o
"-SSR,+l")"
s
e ""
"
2S
r
00
= s-l
{[G(s,6)]JI,H(6) - [G(s,8)]R.+l}
JI,=O
H(6) .. G(s6)
=S[1-6(sJ)f'
This formula is essentially that derived by a much less transparent method
in Smith (1955,
equa~ion
(5.2.11)).
Let us fix an integer k > 1 and choose another integer
than k.
m,~
much
larger
For j = 1,2, •.. ,m, let X.J be a negative exponential random variable
.
with mean value A., say.
.
J
variables Y. = A~ X..
J
J
J
We assume X ' .•. ,X are independent. Now define new
1
m
Let a random selection of theX 1 ,X 2 ""'Xm be made, each
X. being equally likely, and let the resulting selection be calledX.
J
then set Y. = Y.
J
= .X
Thus we have defined the probability distribution of the random
vector (X,Y), and we see that for integer r,s: m E Xry s =
m
A~sE X:+ s
r
J
j=l J
m
= (r + s) 1
r
j=l
A~s+r+s
J
It will be convenient to say (X,Y) is generated by a A-model.
Hence,·i$'we set 1.I = E Xry s , then
rs
m
(r + s) 1 l.Irs
For simplicity, let us
(3.4)
If X.J
r+(k+1)s
= Al
r+(k+1)s
+ A2
+ •.• (m terms) •
set
l;
rs
.:, ..... Jil.
1.I
~ (r + s)! rs
26
The argument we wish to pursue is clearer at this stage if we look at a
specific case. .If we choose k
:.
ie
!~
I
=~
and m = 15
1;10
= Al
+A +
2
·..
+A
1;20
= A1
2
+ A2 +
2
·....
+ A2
1;30
= Al
3
3
+ A2 +
....
3
+ ~15
1;01
= Ai
4
+ A2 +
·..
+ A4
1;11
= A~
5 + AS +
2
·..
+ A5
1;21
= Al6
...
+ A6
·..
8
8
1;02 = A1 + A2 + ...
+ 7
1;31
1;33
= Al7
15
= A1
6
+ A2 +
+ A27 +
15
+ A2 +
15
15
15
15
15
\5
+ A8
••• +
15
A15
15
These equations establishes the vector
i=
(1;10 ' 1;20'
... , 1;33)
in Euclidean Space of 15 dimensions as a function of the vector
also in Euclidean space of 15 dimensions.
If we compute the Jacobian
I
e
U~J = J,
'I~t'
say,
/'
27
we
are led to a determinant of the familiar Vandermond type, and by a
well~
known representation of that determinant we find that
J • (151)
Thus if
~
I
II ,(A r
r>s
I.
~,As)
be restricted to a region D, say, in which none of its coordinates
are equal, then J will be non-zero throughout D.
It follows from the theory of
implicit functions (see, e.g. Goursat (1904), pp. 45-51) that there will be
sphere E, say, in t'hel;-space and a vector-valued function
~ = ~(Z;)
which maps
E into some part of D.
What we have seen is that if we are given the product moments
rs, for
f = 0,1,2,3 and s = 0,1,2,3, and if these lead via (3.4) to a point ~ in E then
there exists a random vector (X,Y), generated by a A-model.
~
Plainly there is
= 3.
nothing special about our choice of k
The above argument will work for
arbitrary integers k ~ 1; we evidently must take m = k2 - 1. Thus, given k,
there is a sphere in the space of vectors with coordinates:
such that these product moments are all generated by an appropriate A-model.
For such a model
m
1
G(s,6) = -m
j=l
r
(3.5)
If we use the equation
= G(0,8),
H(8)
m
1
1 + SA.
. J
=
•
then (3,3) gives
.}; (1 ~ i8Ak)~1 - .
(3.6)
- ieA~
' J
m
I
(1 + SA. _ i8A k)-1
j=l,j"
j=l'
'J
" .. j
¥._------------------m
k -1
m
~
}; (1
j=1
+
SA. - ieA.)
'J
J,
28
Thus ~(s) is a rationabfunction,of s, it maybe expanded in partial
fractions and then easily inverted to yield Me(t) as a linear combination of
M~(S) in the comple~ s-plane are all distinct,
.
.
'.
.
z,,(e)t
for example, we would have an expression of the form Me(t) = L'I'"e
in
exponentials.
If the poles of
which the critical numbersz,,(8) are roots of the characteristic equation
~
(3.7)
l.
j=l
(1 +
Z"A. 'v
J
ieA~)-l
'J
:: m •
From this it is clear that there will be exactly m such roots.
It might be
noted als.o that '(3.7) is exactly the same as
(3.8)
,e
G[z,8)
= 1,
a fact that will be important later.
We may, with no loss of generality,' suppose
•.. >
A-
n
Consider the function
m
1
G(z,O) = -m
l
j=l
for
Z
real.
-1
towards ,-AI
1
(1 +
zA.)
J
Evidently G(z,O) increases without bound as z decreases from 0
In the open interval (_A;l, ,-Ail) it .again increases continu-
ously as z decreases, from arbitrarily
l~rge
negative values when z is near
_A~l to arbitrarily large positive values when z is near-A;l. This behavior
-1
-1
-1
-1
is repeated for the intervals C.-A3 ,-A 2 ), (-A 4 ' .-A 3 ), and so on.
Thus
29
e~~h
= 1,
of these intervals contains one real root .of the equation G(z,O)
. giving m - 1 distinct
~egative
real roots.
But there is obviously a root at
the origin, and so all possible roots .of the equation G(z,O)
for (there
Can
= 1 are
accounted
be no complex roots).
Thus, by the theory of algebraic functions there are mdistinct functions,
roots of (3.7),
which are analytic functions of the complex S in some neighborhood of S
Let zl(S) be that one of these functions which satisfies Zl(O)
= O.
= O.
Then, in
view of the preceding discussion, we may claim that, for S in a sufficiently
small neighborhood of 0,
(3.9)
Rzj(S) <
_,-1
11.
2 '
j=2,3, ... ,m.
Thus we can write, with a nod to the partial fraction discussion above,
(3.10)
where the O-term can be shown to be uniform for all sufficiently small
IS I.
The function Zl(S), by the usual logic, can be calculated from the relation
i' (3.11)
It follows from (3.10) that
30
(3.12)
But the left-hand member of this equation is the cumulant generating function,
r~th
and we see frOm (3.12) that, if Kr(t) be the
cumulant ofW(t), then
,,-1
K (t)
(3.19)
r
= Ar t
-tl\
+ O(e· 2
+ B
r
)
In (3.19) the constants Ar are determined by the expansion
00
(3.20)
zl(6)
= r=l
I
.
(ie) r
!
r
A
r
and the constants Br are determined by the expansion
00
(3.21)
log Zl(6) =
I
r=l
(i6) B
r!
r
We have thus established the following.
Lemma 3. 1.
If a aZass A aumuZative proaess Wet) is based on a sequenae {(Xn ,Yn )}
generated by a
A-modeZ~
then the aumuZants of Wet) are all asymptotiaaZly
Zinear~
in the sense of (3.19).
Let us use this result in conjunction with Theorem 2.3.
For clarity we
shall be specific and consider the third cumulant of Wet); the way the general
argument would go will then be apparent.
Suppose, then, that Wet) is based on a. general sequence {(Xn ,Y)},
Le. one
n
not necessarily produced by ~ A-model. In order to use Thoerem 2.3we shall
assume;
31
(a)
(b)
(c)
E X1 M(X1) < 00;
EI Y
l3 < 00;
l
E Xr
1Y11s
= 0,1,2,3,;
M(X 1) < 00 I for r
s
= 0,1,2,r+s<4~
Let us denote moments of Wet) as follows:
(3.22)
Mk(t)
= E {wet)}k
,
k
= 1,2, ....
Then Theorem 2.3 gives
M2 (t)
Ml(t)
where R2 (t) € B(M;l) and R1 (t)
K (t), we obtain
3
(3.21)
The function R(t) is given by
€
= P2 (t)
= Pl(t)
B(M;2).
+
+
R (t)
2
Rl(t)
Thus, if we compute the third curnu1ant
32
= R3 (t)
R(t)
(3.22)
~
~
3R2 (t)P (t)
l
3P2 (t)R l (t)
+ 6[P (t)]2 Rl (t) +6Pl(t) [R (t)]2 + 2[R (t)]3
l
l
l
Note that if f(t)
B(M;r) and f(t)
€
+
0 as t
+
00, then
t +00,
irnpliesthat M(t)t r f(t)
+ 0
as t
+
This fact makes it a quick task to show·
00.
that all the terms on the right-hand side of (3.22) belong to B(M;O).
Thus K (t) is shown by (3.21) to equal a polynomial of degree 3 plus a
3
remainder tending to zero and belonging to B(M;O). The coefficients of this
-e
polynomial must all.
be rational
. functions of the moments lJ rs with r -< 3, s -< 2,
r + s
~
4 and lJ l ,lJ 2 ,lJ 3 ,lJ 4 , "1'''2'''3' These rational functions will be the same
if the {(Xn,Y )} were generated by a A-model with the same marginal and productn
moments. But we have seen in Lemma 3.1 that in the latter case all the coefficients must be identically zero except fot the coefficients of the first and
zeroth.
p~wers
of t.
Thus we have the following result.
Assume the conditions of Theorem 2.Z.
Theorem 3.1.
Then Km(t), the mth.
aumulant of Wet), is given by
K (t)
m
where
~
R (t)
m
+
0 as t
+
00
= Amt
and R (t)
m
€
+
B + R (t)
m
m
B(M;O).
The constants A and Bare
m
functions of the moments lJ l , lJ 2, ••• ,lJm+l' vl ,"2""'''m and aZZ the
lJ
with t ~ m, s < m - 1, r + 5 < m+ 1.
rs
m
p~duct~moments
33
§4 •. On the
ca1cu1~tion
of the constants An and Bn
We have seen that the desired constants A and B are rational functions
n
n
of the product"moments
and it follows from the discussion of §3 that, in
rs
order to determine these rational functions, it will be sufficient if we assume
~
For low values of n, the coefficients {A } are reasonably simple expressions;
n
.
functions of the probut as n increases the {An } become increasingly cumbersome
.
Thus any ruse which will diminish the complication is desirrs •
able; unfortunately, few such ruses seem available. However, one that is avai1duct moments
~
able involves replacing the {y } by
n
-Yn = Yn - v 1Xn
(4.1)
The use of these
{yn } will
.
substantially reduce the complexity of our formulae.
But it must be borne in mind that these formulae will now be expressed in terms
of the' product moments
~rs
= E Xnr ySn
Thus a price must be paid for some degree of simplification achieved in the
formulae for the' {An}'
We know that zl(8) is the unique root of (3.8) which is analytic in a
neighborhood of
e =0
and vanishes at 8
= O.
But, in an obvious extension of our notation~
34
=E e
(4.2)
-zX+ieY.. iv 1 ex
= G(z
+ iV
1
0 1 8) •
Thus we see
(4.3)
which implies that for all n > 2
(4.4)
while
(4.5)
Thus, apart from the tedious necessity of calculating the product moments
' there is no difficulty in obtaining the desired {A } from the {A}. To
rs
.
n
n
avOid the frequent appearance of the tilde we may now assume v = 0; we comment
lJ
1
again at the end of this section on the appropriateness of our formulae to the
general case.
It is well known, and easy to deduce, that when v
E Wet) = 0 for all t > O.
r
k=2
A
k~ (i8)k
Equation (3.8) then yields the identity
00
(4.7)
= 0 we must have
Thus we have at once that A = 0 and may set
1
00
(4.6)
1
00
r 1- E(iOY .- X·k=2r
n=l nl
.~
kl
35
Let 6r/r! be the coefficient of (i6)r in the expression on the 1eft~hand side
of (4.7).
Then one can deduce that
A.
s
(4.8)
.
where the summation
(4.9)
Ps
(ST)
L*
is over integers P1 ,P 2 , ..• ,P s ' such that
P1 + 2P 2 + ••• + sP s
=r
and we have set
P1 + P2 + ••• + Ps
=n
.
By examining all partitions of the integers 1, •.. ,8 in turn it fs a fairly
straightforward matter to obtain the following results.
(4.10)
We also obtain 6 ,8 ,8 ,6 , but will not quote them in the interest of
5 6 7 8
saving space. At this point it is also convenient to adjust the X and Y scales
so that 1110 = 1 and 1102 = 1 (bear in mind that we are supposing 1l0l = 0).
every 6r must vanish, by (4.7), we are led in particular from the equations
(4.10) to the result A2 = 1, by setting 62 = O.
Since
4It
36
If we set the remaining e's to zero and use the values
'I-l
10
=11 02 ;:
A
2
we obtain the foUowi,ng equations l from which the A. can be successively
J
calculated.
2
+
105'1-l 40 - 210A4'1-l25 - 280A3'1-l30 + 28A6'1-lZ0
+
28A61-l20 + 56A3A51-l20 + 35A41-l20 •
2'~
=1
37
We have fotirid that the numerical calculation of these. {An }-coefficients
is in no war facilitated by solving the preceding equations for explicit representations
of the· {An } in terms of the product moments.
.
A computer program is
easily prepared to make systematic Use of the successive equations for
A ,A , ... ,A if they are needed.
8
3 4
The coefficients {Bn } of Theorem 3.1 are
far less important than the {An }
. .
since they merely provide a second-order improvement in the formulae for the
cumulants of Wet).
Unfortunately, they are much more troublesome to calculate
than the {A }, and no obvious simplifying tricks seem available.
n
00
<p (S)
(4.11)
=
If we write
B
l -.!fn.
(is)n
n= I
then it follows from (3.11) and (3.3) that
e
(4.12)
<p (S)
=1
- G(O,S) , say,
Zl(S)K(S)
where
(4.13)
K(S) =
r[aas G(S'S)] s=zl (S)
.
Let us write
(4.14)
llor(S) = [(-
a r
as)
00
G(s,S)]s=O =
llrn
I
n=O TiT
(is)n
Then
(4.15)
1
- G(O,S)
-zl(8)
G(zl (8),8) ., G(0,8)
=
-zl(8)
r l
[-z 1 (S) l llor(S)
= I
r!
r=l
00
~
,.,
38
In order to avoid too much
~Ol
= O.
comp1ex~tywe
shaUl at this point l suppose
This represents a genuine loss of generality, not trivially over-
come as a similar maneuver
was for the case of the {Anl.
of attack also becomes clearer to follow.
However I our method
We shall a1s0 1 with no further loss
of generality, assume the X and Y scales have been adjusted to make
If we were only interested in terms up to those in (i8)4, the
side of (4.14) equals, to terms of that degree,
(i8)2
21
{1 + i8~11 +
1
+
31
(4.16)
From the expansion
(i8)2
{
21
2
}
~12 +
(i8)3
(i8) 4
31 ~13 +
41
~14}
~30'
Co
=1
C2
= ~12
1
- ! ~20
~10
=
~02
right~hand
=l.
39
2, ""
we can discover the coefficients CI, C
log 1 - 0(0,8)
-zl(8)
= Ci(i8)
+
2.
C
2T (18)
2
say, in the logarithmic expansion
+ ••• +
Cl.· 4
4! (18)
+
5
0(8 )
The easiest procedure for the calculation of the C! is to make use of the
.
J
tables provided by Kendall and Stuart for the calculation ofcumu1ants from
moments (Kendall and Stuart, 1958, p. 70).
4= ~14
C
-
1
2
In the present instance we find:
~20A4 - 2~21A3 + 2~11~20A3 - 3~22 +V 30 - 4~11~13
+ 6~11~21
2
- 3~12 + 3~12~20 -
3
2
4 ~20
2
+ 12~ll~12
A glance at (4.12) shows that we also need an expansion of log {-K(6)} as
a power series in 8, if we are to determine the {B }.
n
Let us set
log {-K (8)}
= C1 (i8)
+
.C"2
.
iT (18)
2
+ •••
40
Then, from (4.13), it follows that
K(e)
But
(-1) r+ 1
J a:: ~ G(a, e)1
[aa
00
1:
=
ja=o
(ie)j
• I
J.
j=O
llr+1,j
so that
00
-K(e) =
L
[-z (e)]
1
r!
r=O
00
=
r
00
{
1:
• 0
J=
[-z (e)]r-1
L
r=l
1
(r - 1)!
(i8)j
. I
J.
}
llr+1,j
00
{
(i8)j
• I
J.
j=O
L
II .}
rJ
On comparing this expression with (4.14) and (4.15) we see that formulae
for
el , C2,
... , can be obtained
2, ... , by replacing
from those for Ci, C
product moment llaa at every appearance in (4.16) byall .
aa
we can infer
From (4.11) and (4.12) we see that
.
Br
= C'r
C"r
r
= 1;2,
..•
every
Thus, for example,
41
Thus we ca~ deduce some easy rules for obtaining the' {Bn } from the formulae
(4.16) for the' {C 1}.
n
Eliminate terms in
(H)
C~ involv~ng
only product moments of the type
A term'in C' involving only one product moment
n'
(possibly mUltiplied by one or more
(iii)
A term in
and y
~
C~
~la);
~.
Q
a~
with a >.2
-
gets multiplied by
involving two product moments
~la'
~aa' ~yo'
~(a ~
1).
say, with a > 2
0, gets multiplied by - (ay -.1).
It should be obvious how these rules arise and how they may, when necessary,
be extended.
They lead to the following results: -
- l5~11~22 - 10A3~11~2l -
5
2
A4~11~20 + lO~11~30
- l5~12~21 - 5A3~12~20 - 5~20~13 +
45
:2
~20~2l +
15
2
:2 A3~20
...
42
We close this section with the comment, promised earlier, of the effect of
the assumption vI·
they stand when VI
= O. The formulae given for {An } and {Bn } are correct as
= O.
If VI
;t
0 then the correct values for
{An}n~2
can be
obtained if every product moment ~rs used in the formulae
for the {An } is
. .
. r
s
.
replaced by ~rs = EXl(Yl-vlX l ) ; it should be borne in mind that the time and
2
."y" scales have been adjusted to make ~l = EX I = 1 and E(Yl-VlX l ) = 1,
something which can be achieved with no loss of generality.
~
0 we have provided no results about the {Bn }; it is not an
impossible task to obtain them by the methods we have used earlier and
If vI
B ,B ,B ,B should not be too complicated.
l 2 3 4
shunned their calculation.
For the present, however, we have
43
§5.
Checks on the calculation.
It is desirable to devise checks on the formulae obtained in the previous
section since there is plainly ample opportunity for slips in the calculation
of the {A} and {B } for higher values of n.
n
n
If the' {X } are tid, with pdf e- x and the {y } are independent of the {X }
n
n
n
then it is easy to see that Wet) is a compound Poisson process and, indeed, that
(5.1)
In this case it is clear that
(5.2)
A
n
= vn
n = 1,2, .•..
Furthermore, if we write {K } for the successive cumulants of H, then (5.1)
n
makes it plain that
(5.3)
B = K
n
n
Thus, if we set
V
1
= 0,
n
= 1,2, . . . .
and
(5.4)
for all relevant values of ~ and 6, then the formulae of §4 for {A.l
and {Bn }
n
should agree with (5.2) and (5.3). We call this the Compound Poisson Check.
All our formulae pass this check satisfactorily.
every product-moment like
pendent and v
1
= 0),
~~1
Unfortunately, because
must be set to zero (because Xj and Yj are inde-
this check give no reassurance at all that terms involving
such product-moments have correct coefficients.
44
A more elaborate test, which we call the DoubZePoisson Cheak is as
Let the
Y'
n
:=
fOllOWS.~
n be distributed as for the Compound Poisson Check, but let
1- X.
For. this model one has
n
X
Wet)
= N(t)
+ 1 - SN(t)+l
= N(t)
+ 1 - t - Set),
(5.5)
where N(t) is a random variable with a Poisson distribution, E N(t)
= t,
and
x
~(t) is a non-negative random variable, independent of N(t), with pdf e- .
Thus (5.5) gives
(5.6)
E eieW(t)
=
ie
1 + i8
e
exp t(e
i8
- 1 - ie)
From (5.6) we see that
(5.7)
An
=1
n
= 2,3, ...
and
(5.S)
(_l)nB
n
= (n
- 1)1
n
= 2,3, ...
For this check it is necessary to calculate a fair number of the moments
(5.9)
A table of the necessary values is given below, since their computation :.is
tedious once the values of a and
a get
moderately large.
e
45
Valuesofproduct~moments·~ae
for Double Poisson.check
When the values of the product-moments
~ae
given in this table are used in
the formulae for {A } and {B } of §4, the values obtained are in full agreement
n
n
with those predicted by (5.7) and (5.8).
This Double Poisson Check involves
all coefficients involved in our formulae and the fact that they pass this
check strongly suggests that none of these formulae contain an error.
46
e
§6.
Brief·caJillnertts art Class B Cumulative Processes.
Suppose Wet) to be a Class B cumulative process instead of Class A as we
have been supposing throughout this paper.
The arguments showing that Wet)
has cumulants of the asymptotically linear form exemplified in Theorem 3.1
will go through much as for the Class A case; the remainder terms will have
the same properties.
Furthermore, the calculations of §3 leading
to (3.3) can
be imitated, and will lead to the result
(6.1)
o
Ma (s)
1 - F(s)
= -s......[..;:;.l-G.....(~s"""',e=)~r
It is clear from this, as it should be intuitively in any case, that the coefficients {An} will be exactly the same as for the Class A process, obtained by
expanding the root zl (a) of the equation G(zl (a) ,a)
..
ference, however, in the coefficients {B}.
n
00
ep(a)
r
=
n=l
= 1.
There will be a dif-
If we once again set
B
-2!. (ia)n
n!
.
then it can be shown from (6.1) that
(6.2)
e
ep (a)
where K(a) is as given in (4.13).
=
1 -
F(Zl (e))
,.
zl (e)K(6)
This result (6.2) should be compared with
(4.12) obtained for the case of the Class A process.
•
Nothing but the need for careful and tedious
computat~onis
needed to
extract from (6.2) the desired coefficients· {Bn }, but we shall not proceed
.}
further along such lines in the present report.
47
References
Goursat) E. (1904)
A course in mathematicaZ anaZysis" translated by E. R.
Hendrick) Ginn) Boston.
•
Kendall) M. G. and Stuart) A. (1958)
The Advanced TheoPyofStatistics" 1,
London: Charles Griffin.
Sheps, M. C. and Menken, J. A. (1973)
Mathema:ticaZ ModeZs of Conception and
Birth" Chicago) Univ. Press.
Smith,
w.
L. (1955), Regenerative Stochastic Processes, Proc. Roy. Soc. A, 232"
6-31.
(1958), Renewal Theory and its Ramifications, J. Roy. Statist. Soc.
B, 20" 243-302.
".
e
(1959), On the cumulants of renewal processes, Biometrika" 46"
1-29.
(1966),
•
A theorem on functions of characteristic functions and its
application to some renewal theoretic random walk problems, Proc. Fifth
Be~keZey
Symposium on MathematicaZ Statistics and ProbabiZity, Univ. of
California Press, Vol, 2, 265-309 •
•
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4. TITLE (. .d SUb"tI.)
S. TV"E 01' RI"ORT II .. 1:111100 COYIEIIlED
rrECHNICAL
On the Cumulants of Cumulative Processes
e.
PIRI'OIllMINO ORO. 1Il1.. 0RTNUM.ER
Mimeo Series No. 1257
7. AUTHOIll(a)
•
I. CONT"ACT OR GRANT NUMBlft(a)
I
Walter L. Smith
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to. PROORAM ELEMENT. "ROJECT, TASK
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ARIA II WORK UNIT NUMBERS
Department of Statistics
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t t. CONTROLLING OI'I'ICI: N'AIltI AND 1.00'-.'
Statistics and Probability Program
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t2.
'
RIE~01t.i-,;!ATE __ ,_., -------
October 1979
t ••
NUM.I!R.J;ll"~~GES
48
t•• I.CUlltITV CLASS. (01 "',. ,."ort)
'4. MOIIITORtNG AGI!NCV NAMI! II ADOR.II(II dille..., 1_ CCIfltrOlUn, 01110.)
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Approved for Public Release: Distribution Unlimited
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""ac' en'.red'n Bloole 20, "dllt....' ' - Report)
17. DISTIIlIBUTION STATEMENT (01 "'• •
18. SUPPLEMENTARY NOTES
t9. KEY WORDS (Continua on
,.V.,•••Id. II n.o....", ..d Idantl"
by bloole nUlllb.,)
Renewal Theory, Cumulants, Cumulative Processes, Limit Theorems
20. A.STIIlACT (Contlnu.
Oft
,.v.,•• ald. II n.o•••.", and 'dantll)' by "'oolc nUIII".')
Let {(Xn ,Yn )}, n=1,2, ••. , be a sequence of iid 2-vectors such that pix > O} = 1
n
and write S = o and S = Xl + X +
+
>
n
1.
Let
N(t)
be
the
maximum
k
X
'
0
n
2
n
such that Sk ~ t. A C1ass A cumulative process is
:: N(t!+l Y.
Wet)
1
J
...
.
DD
1413
EDITIO" 0 ' I MOV e~ .. O.SO.LlTE
UNCLASSIFIED
se:CUIIlITY CLAlII1'ICATION 01' THIS PAGE (IPIIan
.
D.'. En'ered)
UNCLASSIFIED
SECURITY CLASSIFICATION
20.
oar THISPACJ.~M
Del. JrnteNdJ
(continued)
I
l
Let K (t) be the mth cumulant of Wet). Under general conditions involving the
m
existence of various product moments and the hypothesis that the {Xn } are
non-lattice, it is shown that there exist constants Am and Bm such that, as
t +
00,
= Amt
B + R (t) .
m
m
m
Here R (t) + 0 as t +00, and various things can be said about'Rm(t), depending
m
on hypothesis on the joint df of (Xn ,Y).
Expressions are obtained for Am,
n
B , m = 1,2, ... ,6 in terms of the product moments
m = 1,2, ... ,8 and ·m
11
= E lY. y S , and methods of checking these complicated formulae are
K (t)
to
•
+
as
n n
discussed.
•
~
..
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