ON POSITIVE AUTOREGRESSIVE SEQUENCES
by
Walter L. Smith
Department of Statistics
University of North Carolina
Chapel Hill, N~rth Carolina, 27514
.... _......--: ...
--
_.. -....
..-i.
Key Words:
_
--~--
-
__
~_
..
-_...- _
...
.....
. ~~_ .... ~.- . , '~-;.'" ...--.----
._~_
..
..-----,..... .. ....
,
-----.~
...
'.
Au toregressive ...p.tOcess8s,L;aplace~~nsTorms·of
_·w -..-..-'
__._
distributions.
-.-~ ••. _-----~.--~-_ ....,----
._,~
,
. . . .J,;--_.~--~",.-- :'-- ,,- ,.......~ ..._._.-,
This research is supported by Office .. of~aJlal Research-Grant No. NOOOI4-83K-0352.
SUMMARY
A stationary sequence of non-negative random variables {X n}, with
marginal df F, is to
Xn
=
2:~ al X n-J
+
En,
be generated by the familiar autorgressive model
where the
{En}
cases are excluded from consideration.
are iid with df G.
A class N
1
Certain periodic
of distributions which are
reasonably "pleasant" from a computational viewpoint is defined, and it is
shown that if both F and G belong to .N'l then k = 1; i.e. only the simple
Markov case of the autoregressive model permits both F and G to be
"pleasant".
Other questions related to the generation of non-negative {X n}
are also discussed.
page 2
11. INTRODUCTION
The main question considered in this paper is as follows.
Let a
stationary sequence {X n} be generated by the familiar autoregressive model,
of order k 2 1,
Xn = a1X n - 1
+ an-zX n- + ... +
Z
where the {€n} are assumed to be iid.
akXn-k
+ €n
(1.1)
Let F be the stationary marginal df of
the {X n} and let G be the df of the {€n}. It is required that the {X n} be
almost surely non-negative and that F and G be, in some reasonable sense,
"pleasant" distributions.
For example, it would be "pleasant" if both F and G
had rational Laplace-Stieltjes Transforms (L-S. T.) as defined in (2.1)
Later in this note we shall make explicit what, for present purposes,
we shall regard as a "pleasant" distribution; our definition will include much
more than merely those with rational transforms.
at
least
for
a
substantial
We shall then show that,
class of autoregressive
models,
the
desired
objective can, unfortunately, only be attained when (1.1) is a Markov model,
that is, when k
=
1.
This investigation was triggered by a colloquium given in Chapel Hill
by P. A. Lewis, in which he described extensive work which he and colleagues
have been doing for the computer generation of "interesting" stationary
sequences of dependent positive random variables.
Their object was to
obtain, with reasonably tractable algorithms, sequences {X n} which could be
used in simulation studies of queueing, inventory, and other related models
encountered in Operations Research.
For such applicaitons it is very
desirable that both F and G be tractable.
In particular, a df with a ratiuonal
L-S. T. is particularly attractive since it has a pdf expressible in terms of
page 3
exponential and polynomial funcStions; such a pdf is convenient for computer
use.
The feasibility of the autoregressive scheme (1.1) was considered by
Lewis, but only for the Markov case: k = 1.
Preliminary announcements of
this work was given in Lawrence & Lewis (1979).
However, the present author
conjectured at that colloquium that it is only possible to do what is wanted
if k
1 (and, of course, the process is Markov).
=
directed
at
the
substantiat~on
partial
of
that,
The present note is
perhaps
too
sweeping,
conjecture.
In «[2 we introduce the notion of a g-singularity and use this to define
the class
~'fl
of distributions which correspond to what we have loosely been
calling ·'pleasant".
F(x)
Very roughly, they are such that F*(sJ, the L-S. T. of
has at least one, but at most a finite number, of singularities on the
In 13 we examine general conditions which will need to be
negative real axis.
fulfilled if the {X'1.} are to be non-negative; not surprisingly we need the am
of
(3.~)
to be 2: 0 for all m.
We give a necessary condition on the
coefficients {ad, involving an infinite. sequence of determinants.
In
~3
we
also introduce Assumption A about the-. autoregressive scheme (1.1); by making
it we exclude from our rJ \...J._
discussion
certain
periodic models for which our
!
':) I;': -";
- -"
'~>'.;
e
~--
~:;"f.
present methods ~re it;!,a,4eqpat~. ,)n
•
...•.• i..
"
.-
' , . , , ' ..
f
~.
'A
~ .~~,
14 w~.-,eng~ge
•.
--
in some function theory and
_.
discover necessary conditions
on C (whether or not Assumption A holds true)
Mt:~
,)' <;.~~
., .... ~,;·.':tr;j3~'-'~1d~ ~ ;::.::"
if F E .N'I and the {X
nl. are onoT\:::negatiye.
W~ sh9W that C* can have no limit
L:," .. ", - ... ' : \. _...:..j
•
~,~
~
~--~.
·~t,
I
points of g-singulafiY:~~D~ t~~ n~~t~~~ ~real axis, we introduce a "powerfunction" which mea.~~;.e~ ~~, :;~m?iu~~ qf singularity" of C* in an interval,
and we obtain an
ord~r
, -: '"
of
'::-.,
ma~nitud~ .oLthis
~..,
1? .,:..'S:1
:',.
,'j:
"power-function".
1:-.., _
Finally, in
~S,
we prove the following:
.,
THEOREM
[J Assumption A
the {Xn} are a.s.
i~
valid, if both F and G belong to .N' h and if
non-negative.,t~n
we must have k
autoregressive scheme generates a Aitarkov process.
page 4
I, i.e. the
12. CLASSES OF "PLEASANT" DISTRIBUTIONS
For complex s with !Rs 2: 0 we denote the L-S. T. of a df F(x) thus:
I
00
F*(s) =
e-S'X
F(dx).
(2.1)
0-
It is well-known (cf Widder, 1946, p. 58) that F*(o5) will be analytic in some
half-plane lRs
> -c,
say, where c 2: 0 is real and where F*(o5) will necessarily
have a singularity at -c.
interval (-c+6',oo), for 6'
It
was
originally
For real s, F*(s) is bounded and decreasing in any
> o.
intended
to
take
as
our
class
distributions exactly those F for which F*(o5) is rational.
of
"pleasant"
However, such a
class excludes many distributions which would be found quite acceptable for
computer use. For example, the case
1
F*(o5) =
(2.2)
~1+o5
which corresponds to a pdf e-f I v'~2'7fx),x > 0, should by any reasonable
argument, be included in a class. branded' "plea'sant".
At this stage we mus('lnfrod'tice s6ri;;tti·nc6n~eritional terminology to
describe singularities of Ii ftinciioil~j(~)," ~ay,; "()i:{li~ c~·mplex variable o5.
Suppose !(s) to have an isolated singJlarity
.:.
to be a finite IJ.>O such that, ·'as
•
s-t,
-.
:
:br ?i~r6r'ar ~
;JJ,'-n ,~:,j;;
Is::....~1 1!(8) 1
't;:"l;'::~;~~
-.,:,:",
',~V':.•. ~'
"shall
r;. '.~
"'I
!'~
,~'
and suppose there
.~"
tends to a finite limit
?t::. ~ ... <'I t,., _ ~
say /(8) has a 'g'~singularity at ~,
.. n" ..,' . ,. ; '.1'
1.,:>.-: .',
and we shall call IJ. the power of 1(8) aC~.'''If the power'Is positive we may
.
~~:
~c'."" ':"l"'; ~rt·, r;~
,;.' .. ~_ "
say !(o5) has a g-pole at ~; if the power is ~riegafive we sa'y !(o5) has a g-zero.
different from zero.
In such a caS"e 'w'e
"..
·0 ' . '
In particular, for example, an ordinary zero of !(o5) at
-1; a pole of order 1<; (an
~
would yield a power
integ~'r ~1) at ( would yield a power
=
1<;.
However, as (2.2) above shows, non-inie'g'~~l ~c)wers can' arise: the example (2.2)
involves a g-pole at ~ = -1, of power ~.
page 5
We shall occasionally write
if
~
~[~,f(8)]
be a regular point of f(8), we have
for the power of f(8) at
~[~,J(8)] =
~;
thus.
O.
It is possible to generalize the preceding ideas considerably.
We could,
for example, seek a vector (fJ.l,iJ. 2,fJ.3j such that
(2.3)
8 -+~.
tends to a non-zero limit as
By introducing a dummy variable l\
could then define the power of f(8) as the polynomial fJ.l+fJ.2},.2+fJ.3},.3
say.
We could say
~
>
0 we
=
fJ.(},.),
is a g-pole if fJ.(},.) us strictly positive for all small },., a
g-zero if fJ.(},.) is strictly negative for all small}".
We believe the theory of
the present note could be developed to cover this more general concept of gsingularity and power, but, in the interest of simplicity, we shall proceed on
the earlier basis of a real-valued power. However, it might be mentioned that
other scales of growth could be used in (2.3) besides the logarithmic one
discussed.
For any small angle a.
>
0, defiM W(a.)
including Q. in this thin sector.
=
{8!7I-a. :::;: Arg 8 :::;: 7I+a.},
We are now able to define the classes of
distributions which are'important for the; present work.
i.
Definition
all zeros and
co. -
1. A df'
• ','. ,
("'f
r?-t .
::;,,"'.
....
p" beio'~gs to the class No if, for some small a.
singulariir:~ oVIF*~{'~it~i~the sector
-j~:'£;';:<~
real axis.
Definition
~
i'·J
.':~<:''::::":,''
>
0,
W(a.) lie on the negative
~.
A df P, in the class No, belongs to the class N 1 if it has
at least one, but at most a finite number, of g-singularities in W(a.), for some
a.
>
0.
The class N 1 constitutes our class of "pleasant" distributions.
page 6
•
13 POSITIVE VARIABLES AND AUTOREGRESSIVE SEQUENCES
We shall be considering the autoregressive scheme (1.1) with real
coefficients all a2, ...
,
It will be assumed that (1.1) will generate a
stationary sequence {X n }. This, as is well-known, requires that the equation
Zk _
a l zk - I
a2 Z k-2
-
-
... -
ak = 0
(3.1)
have all its k; roots strictly within the unit circle.
It is then possible to represent the {Xn} in the following way
(3.2)
where the numbers {(J'n.} are calculable in terms of the roots of (3.1).
Indeed,
when the roots are all distinct, "11 "2' .... "k' say, one has the straightforward
formula:
(3.3)
Xn
<.
where the constants .4 11 .42,.,'. "A~. -~re . easy
tOj.o~t~jn..
compl~x"
D' ~~ ',)i '.:0""1,
(Box>,~a.nci
roots may not be distinct, some may be
j
~
':"
formula than (3.3) becomes necessary
,
.',
~':
' -i ... I
.)..:'
i~.~;
"\
general information on autoregressive models).
numbers "I
>
"2
> ... > "p,
j,
Unfortunately, these
and then a more elaborate
~'t.:
,
!.
Jenkins (1970) have much
;;.~ 7)
,:.1 .~ ,i L
' ;~, .
In general there will be real
say, and functions'+' I(m), '+'2(m), ... , wp(m) such
that
p
(J'm =
L}..7
"tj(m).
j=1
The functions "t j(m) have the form
page 7
(3.4)
t.j
'l'im) =
L
t=l
(3.5)
Ajt(m)e(Wjt m )
where: (i) each Ajt(m) is a polynomial in m, of degree d jt , say; (ii)
for either sin (.) or cos (.);
(iii)
of complex roots of (3.1).
However, let d 1
stands
the Wjt are periods, related to the arguments
~l
Because
evident from (3.4) that, as m
eo
exceeds the remaining
{~j},
it is
-+ 00,
max{d lh d: z, ... ,d lt }. Then eVidently, as m
-+
00,
'l'l(m) - mdlCl'l(m),
where Cl'l(m) has the form, for some constants {Clj},
(3.6)
This function Cl'lm) in important to the asymptotic behavior of crm; we shall
see below that, if the {Xn} are to"'be non-negative, we must have crT1/. ~ 0 for
all m. Thus we may never have Cl'l(m)
<
O.
The trigonometric terms in (3.6) only arise when there are roots of
::' ~)!:
(3.1) at points on the circle 1%1 =
When such roots exist
~he
'
~l (>0)
which are not at the real point
~1"
autoregressive process exhibits some curious
periodic (or almost periodic) aspects, and needs far more elaborate treatment
that we can manage in the.pre!i.;eJ)t.
note.
.~:
"
.,:' ' ..11
I..
Thus, so far as Theorem 1 is
'(.
concerned, we must introduce the following:
;
Assumptiol!
~
j;,
The roots.
oJ::."'!"
Q.l) of maximum modulus are all located
.' ; '-" .. 1'
at the positive real point
r
~l'
Consequently Cl'l(m)
page 8
=
Cll' a constant.
Subject to Assumption A we may then claim that, as m --
O"m -
m
m
~l
dl
00,
C ll .
(3.7)
Even if Assumption A does not hold, it is clear that we can always find a \,
o< \ <
1, such that O"m
Lemma
=
O(\m).
3.1 If g log +1 Eml <
00
then the series 0.2) is almost surely
absolutely convergent, and, under Assumption A, conversely.
Proof
For any h
implies that
>
0, if Slog +1 Eml <
z:; P{ I
En
I 2 e
almost surely, IEn I 2 e
Ot ~ m) for some 0
<
~
nh
<
nh
}
<
00,
then
z:; P{ log +1
Thus,
00.
En
1 2 nh} <
00;
this
by the Borel-Cantelli lemma,
for only finitely many values of n.
Since O"m
=
1, it is clear that (3.2) will converge as claimed
because, with only a finite number of exceptions, iO"mEn-ml
s:
m
!O"m!e \
Conversely, if (3.2) is almost surely convergent, :O"m En-ml
a finite number of values of n.
>
1 for only
Hence, by the converse to the Borel-Cantelli
Lemma,
Z:~ P{ IEn-m I
> lO"ml-,l} <
J
.'
_:'
.;~.
Ol?
•
:':-,"-;}i~
If Assumption A holds, from (3.7) we can infer the existence of T7
that IO"m 1-1 < e nm for all large m. Hence':'-' ,,:,; :)[1,j '.:
~
.:
;;;~I'~r:.j
>
0 such
C}6
Because of this lemma we shall henceforth assume without comment
that Slog +1 Eml
<
00
and that (3.2) is
~rm~se s\i'ielY
page 9
absoluiely convergent.
Lemma 3.2
Let (1.1) produce a stationary sequence {X n
}
with df F(x);
let F*(s) have at least one singularity in the finite complex plane.
if the {X n
}
non-negative, it is necessary that
are almost surely
Then,
;Z
(jm
0
for all n ;Z O.
Proof
If the non-negative {X n } were a.s. bounded above then F*(s) would be
entire. But F*(s) is not entire, so the {X n } can be arbitrarily large.
Suppose that, with positive probability,
(jnE-'l
can be arbitrarily large
and negative. Evidently
co
L (jJLJ
J=O
(3.8)
J';'n
is a proper random variable, independent of E-n.
Thus there is a large
>
~
0
such that
P{Xo -
(jnE-n
<
~}
>
O.
But we may suppose
and hence, by independence, P,{Xo':::<'
(jnE-n
we use
must a.s. be
(jo =
-~}
>
O.
boun'de(b--be~".£ol" every ,in.
This contradiction shows
In particular, taking n
=
0
1 to infer that Eo (and hence every En) is a.s. bounded below.
If there were.. ;!,gX,n , fof".:w1}ich(jn
<
0 it would thus be necessary
that E-n be a.s. bounc;ied ;J~bo.Ye\,dsince·(TnE-n is bounded below).
This would
imply that every E-n is bounded above and thence that X o is a.s. bounded
above.
We have seen thisCannot:be.'so.
the lemma is proved. 0
The requirement
c;: ::.:
(jm
;Z
Thus
(jn
;Z
0 for every n.
Thus
S> T '
0, for all n, imposes restraints upon the {an}.
page 10
Indeed, it is possible to show (we omit the proof) that the {an} must be such
that every member of the following sequence of determinants must be nonnegative:
... , and so on
in this sequence one takes
Lemma 3.3
aJ =
0 for all j
>
k, of course.
Under the conditions of Lemma 3.2 the {En} must be a.s.
non-negative.
Proof
Let C(x) be the df of the iid {En}; let "'I be the esssential infemum of
the {En}.
We have seen that "'I
p{1 f
>
N(o)+l
-co.
Then choose 0
CTnE-nl
<
>
o} > O.
This must be possible because (3.2) is a.s. convergent.
< "'I+6'} >
hand, P{E-n
0 and N(o) so that
But. on the other
0 for all n. Thus
This implies. letting 0 t 0, that the essential infemumof.'Xo is S "'I L;;" CTn.
But the {X n
}
are assumed to be a.s.
non-nega'tiV~,~.Thus "t-'2. 0
and the lemma
is proved.D
Lemma 3.4
Let G( ~) be the df-Q,!;.the~'non;:.n~tiue.:{En }.
Let F*(s) be
analytic in Rs 2. -~ where ~ is a-:real i8ingJ.l~q;rftty>;()ljr'.F~(s). Then G*(s) is
analytic in, at least, Rs
Proof
>
-~.
,: k
':'~'"'
",
For any "'I<~ it must be true that 1 4"F(x) = O(e:-'J<:t'), as x ..... co.
(3.2) shows
E.o
s X 0,
from which it is immediate that 1 -
C(x) =
Thus C*(s) is finite for all real s > -~ and the lemma follows. 0
page 11
7
But
O(e- <:t') also.
In the sequel -~ will always refer to the real singularity of F*(s)
which is nearest to
o.
We shall always assume that such a finite
~
exists;
We shall assume ~ > 0,
for instance, it always does when F*(s) is rational.
and this implies an exponential rate of decay of 1 -
~
F(x) as x-
00.
SOME FUNCTION THEORY
Let {'Tn} be a rearrangement of the {ern} into no-increasing order.
Thus 'To:?: 1.
It is apparent, in view of the absolute convergence of (3.2), that
if we set
(4.1)
then Yo will be distributed like X o with df F(x).
From (4.1), for all s in the
half-plane lRs > -~, we obtain the crucial relation
(4.2)
F*(s)=TI G*('T JS).
J=O
....
Lemma 4.t If FENo, and
:
Proof
::-~
~'.~
L.1
.
~>O
then GEN o .
With no loss of generality we may suppose
~
change for the {X n } and the {lOn} will accomplish this).
(-00, -1] in the cdri1p'le;c"Jjfarte':
4
....
zero-free at every p6iht 'tif W(d:)
, '.-' 5 . . r: j
~: l~~
_....
:
Choose /3, 0 < /3 < a.
r~
=
1 (a suitable scale
Let 'J be the segment
L~t:ci>O be such that F*(s) is analytic and
!..!.i.'···
::,; ::: ...\ ~ ~
2 : ~; .
Then F*(s) is analytic in W(a) - W(/3).
Let us
also, for ease, write y(a,/3,r) for the set of all complex s in W(a) - W(/3) such
that Isl<r.
Lemma 3.4 shows that 6*'(s) will be analytic in lRs> -1.
C*(s)
cannot vanish in (-1,0). Thu~,'"it"d'''be;small enough, C*(s) must be analytic
and zero-free in y(a,/3,l).
page 12
It is possible that, for some integer P 2: 1, we have
T,)
= T 1 = ... = Tp _ 1 > Tp 2: Tp + l 2:
Because of this possibility we rewrite (4.2) as
{C*(TOS)}P
Ii C*(T jS)'
(4.3)
j=p
We then define
sup Tj
p
J ~p
To
<
(4.4)
1.
Evidently all C*(TjS) for j 2: P are analytic and zero-free in ~(a,I3,1IpTo)' It
is then routine to show that
is also analytic and zero-free in ~(a,I3,11pTo)'
free throughout W(a) - W(f3).
Thus we see from (4.3) that C*(ToS) must be
analytic and zero-free in ~(a,I3,I/pTo).
zero-free in
~(a,I3,11p)
But r*(s) is analytic and zero-
This implies C*(s) is analytic and
and hence, for all j :::: 'p,
t~~td*(TjS)
.. :~~:;~L\.;)
zero-free in ~(a,I3,1Ip2To).
is analytic and
1: ..
;:; :1
The argument can plainly recyc~~..: ~~; ~~\1all~ l~~~e, IIp(s) analytic and
zero-free in ~(a,I3,11 p2To ) and deduce that q.*(~) ),8 .'~,qaIY~ic and zero-free in
y( a,l3, 11 p2). It should be plain how this argument continues.
arbitrarily small, the lemma follows. 0
. ;,..
Since 13 may be
~.-
A comment or two is in order... lt must be understood that the
conclusion CENo is, of course, a consequence
of the over-riding assumption
,..
'"
; .2:J.;.j
that the autoregressive scheme is generating non-negative {Xn}.
It might
also be pointed out that the lemma could be proved in a very similar way if
page 13
~ = 00, which would mean tha t p*: 0'3) is entire.
However, we have no need for
such a result in the sequel.
Lemma 4.1
Let F & N
1
and
~
>0
and suppose that the singularities
which G*(s) may have on the negative real axis are g-singularities.
Then
these g-singularities of G*(s) must be, at most, countable; moreover they
can have no finite limit points.
The sum of the absolute powers of all the
g-singularities of G*(s) in (-x, 0), for x
>
>
0, is O( XC) for some constant c
O.
Proof
It follows from Lemma 4.1 that C & No, thus we are supposing that, for
small ex. > 0, whatever singularities C*(s) may have in W(ex.), they are on the
negative real axis; they are g-singularities.
number of g-poles and g-zeros in :/
Since F&N 1 , it has a finite
= (-00, -1]; we continue to let -1
location of the nearest singularity of F*(s) to O.
be the
Let the g-poles of F*(s) in
1 be at
and let the g-zeros be at
Let us also write
It is entirely possible that ~(s). have no g-zeros in 1; in that case that
: :.
.." ~ I '
;
_ .;:;.
argument which follows could be somewhat simplified.
page 14
Let :f(r)
= {sl lsi
<
r}
n W(ex).
Because
Tn
L 0, given any R
>
°
we
can find N( R) such that
Ii C*(TjS)
NCR)
is analytic and zero-free in :f(R).
This is because
and C*(s) is analytic and zero-free in (-1, 0];
T
j
Isl < 1 for all large j
the angle ex can always be
chosen small enough to ensure the validity of these claims.
Thus, if it is
known that IIp( s) has a g-singularity in :f(R) I then this g-singularity must
come from the product
n
NCR) - 1
*
(4.5)
C (TjS).
j=p
In particular, if IIp(s) has a g-singularity of power
fJ, (~o)
at some -E e :f(R)
then we must have, from (4.5), that as s-+-E
NCR) -
fJ,logls+EI +
1
L
10gIC*(TjS)1
j-p
tends to a finite limit.
If
- T jE
is not a g-singularity of C*(s) -then log IC*( T j s) I tends to a
finite limit as s -+ -E.
Let us therefore write , for the set of suffices j
such that p ~ j ~ N(R) -
1 and -TjE is a g-singularity of C*(s).
tends to a finite limit as S-+ -E.
L1 jJ.J
=
Let
<fh'-t,.
f.i.j,
C*CTjS)]
Then
say.
Then
< o.
Let us
fJ,.
Now suppose that C*(s) has a'g-pole at sobt'e real point
write CP[ -Xc, C*(s)] =
fJ,o.
Then C*(ToS) has a g-pole at
page 15
Xo
-Xo/To
with the
Let us us also write 'J> [ -xo /T O' F*(,s) ]
same power jJ.o'
that Vo
=
argument.
=
Vo.
It is possible
0, in which case minor changes are necessary in the following
It is also possible that
1/0 =
If we look at (4.3) we see that
pjJ.o.
this implies that IIp(s) has zero power at -xo /To:
the g-pole of F*(s) is
exactly "balanced" by that of {C*(ToS)}P at -Xo/To.
However it is possible that Vo
~
(4.3) to conclude that IIp(s) has power
In this case we are forced by
PjJ.o.
1/ 0 -
PjJ.o at -Xo/T,;;.
Thus, by the
argument we have already explained, there must be a set jo, say, of suffices
j such that
(4.6)
Moreover, if j
t
jo, then 'P[ -Xo/T o , C*(TjS)] = O.
then -TjXo/To is a g-singularity of C*(s).
On the other hand, if jEjo
We can say that a g-singularity
of C*(s) at -xo is "balanced" by one of F*(s) at -Xo/T o , together with the
finitely-many g-singularities of c*(s) at the locations
..
Since
T
j
:s;:
pT o , XOj
<
singularity of C*(S)
singularity
of
-TJXO/TO
,'.
p~ for all
ill
, -xoJ' say.
~
Jkfio.
Thus that part of the power of a g-
~x~ ;;'wfi'i~h is not immediately balanced by a g-
F*(8) at -~o/T~'
i's" bal~nced
by
a
singularities of C*(s) whichdie-> in! the interval [0, pXo]'
that C*(sl is analytic and zero-free in (-To ,0].
finite
number of
Recall that p
<
g-
1 and
Thus, if pxo :s;: To, the only
possibility is that the g-singularity ot C~"(s) at -xo is balanced exactly by
that of F*(s) at -Xo/T o '
Suppose, on the other hand. that pxo :2 To; then
there may well be numbers {-XOJ} in [-PXo, -To] at which C*(s) has gsingularities.
page 16
However, it should be clear that each g-singularity of C*(s) at the
loca tions {-xo j } is either wholly balanced by a g-singularity of F*(s) at
-Xo/T o or by further g-singularities of C*(s) at locations {-xu}, say, which
fall in [_p2>--O' -To], provided p 2t.o .2 To.
If i:'>--o
<
To then it must be that
every {-Xo/To} is a singularity of F*(s) and there can be no "second
generation" numbers {xu}.
Obviously, whatever the initial Xo
may be, this argument can be
repeated a finite number of times until a "generation" of singularities of
C*(s) is reached, each of which is balanced by a g-singularity of F*(s).
The first important consequence of this is that any such number Xo
must be of the form
(4.7)
where -EfTa is a g-singularity of F*(s), k is a finite integer, and all the
suffices {jd, amongst which there may be repetitions, satisfy j" .2 p. Thus
every factor (TofT j}
>
(1/ p).
Since we are supposing that r*(s) has finitely many g-singularities, it
follows that in any finite interval there can b.e.o)1ly a finite number of
numbers like (4.7).
This shows that t.h~,
g-singul!lrities of C*(s) on the
.'. . ;\.
negative real axis must be
enumeraN~:and
can h.av:e n9
fi~it~
;..Iext, for k = 1, 2, '" , let J" . be the, inter*d .[Jllp"-:-l
and
","'"
page 17
limit points.
:"'1/ pic).
let
The argument in which we have been engaged shows that for k= 1,
~,
...
,
(4.8)
From this one can infer
(4.9)
Since 'Pk.
0 for all large k, it follows that 0. 1
=
>
Hence if, for all x
+
c"z
+ ... +
c"k
0, we now write
~
c"(x)
1'P[y,C*(s)]1
y<.( -;x.O]
it is possible to claim C,,(1/pk) = O(i").
that Q(x)
=
O(X
C
)
From this it is an easy deduction
where
c
10g2
=
;-lo-g-:":(l'-:/~p)
This proves the lemma.O
~
PROOF OF THEOREM
Let F&.N'l' and l~t F~('~J n~V& g~poles at _e<l, _e<z, ... , -e<P on the
negative real axis.
W~. insist i '.that: F:IIJt.(s) have at least one such g-pole.
Moreover, as we have already explained, we can assume with no loss of
generality that ~
e"
=
To.
=
e<l have any convenient value, so we shall arrange that
nl
Let F*(s) also have 'g~zeros lit _e
(
c:.<....
,
72
_e
, ... ,
_e
nq
;
there may be
no such g-zeros, but if there is one at all, it must be true that e 1 > eEl.
T)
page 18
Write
L
GJ (Xj
'P[y, F*,s;]
y"[-X,O]
and
y i:Xj
L
=
'P[y, C*, sJ] •
,,"[-X,O]
The discussion of
~4
then shows that
00
GJ(xJ
=
L
(5.1)
y(TjX),
.J-V
for all x.
that
y (x)
But GJ(x) = 0 for all x
=
0 for all x
Let us now set
Then <I>(t) = \(fit) =
°
<
T) =
<
T:"
from which we may deduce from (5.1)
1.
e -Kj, so that K j Too, and also write
for all t
<
0, and (5.1) gives
<I>(t) =
i: 'i'(t -K
(5.2)
j ).
j-O
From Lemma 4.2, since F<:.N'l, we can: infer that-' 'II(tJ
,'-Co;
O(ect ).
Thus \(fit)
admits a Laplace-Stieltjes Transform' for' all,s,'sach',thaFlRs :> c, and (5.2)
yields that very useful result
. i'
"',
'.- , . : ' - ,
;: .
This result (5.3) has depended only on the hypothesis that F<:.N' l '
page 19
However,
let us at this point invoke the additional hypothesis of the theorem that
Cs oJ'\[ j also.
Suppose that C*(sl has g-poles at _eJ:'j
g-zeros at -e Y 1
>
-e Y2 :> .. ,
>
>
> ... >
_e.rz
_e:rm, and
_e ym •
Let us also write
-b J
=
GJ>[-eyj , C*(s)],
Then (5.3) can be rewritten as:
(5.4)
which implies
(5.5)
Our argument will actually make no use of the powers
depends only on the "locatio~s", e<j ,en:, etc.
pole of F*(s) the term
(Xle-S<l
(X.j'
(3j' aj, b j , etc., but
Since there is at least one g-
on the left of (5.5) must be balanced by a sum
of terms (or by a single termlon'.the-t:ight of (5.5).
Since Ej
~
'r/j for every
j, we deduce there must be. 3.t'.. 1east.,QJ1e term in the sum L~aje-s.rJ, i.e. C*(s)
must have at least one g-pole.
Let
ale-
sorl
page 20
be such a term.
Since
Kj
.....
00,
a
term like
ule-S(XI
+ ICjJ, which arises on
the right of (5.5), can only be
balanced, for K j large, by a term of the form
bu
eS(J/u
+ xVI.
Thus there must be at least one g-zero of C*(s).
It follows that neither of
the sets (Yl' Yz , ... , Yn) nor (Xl' Xz, ... , Xm) is empty.
as if Yn
if Yn >
<
Xm
We shall now proceed
Xm; it should be clear how the ensuing argument could be changed
(it is impossible to have Yn
=
Xm).
Then, for each value of j there must correspond integers u and 1c such that
Xm
+
(5.6)
Kj
Thus (5.6) provides a correspondence between the suffix j and the suffixes u
and 1c.
s:
Let' be ~ infinite set of positive integers. Let u· be fixed, 1
n. Let Ku·) be the set of all j
Xm
E' for which a 1c can be found so that
s:
u·
+ KJ
At least one of the sets '(1), '(2), ... ,
,en)
•. )
!
must be infinite.
Let '(u) now be
such an infinite set.
Under Assumption A we have for 1
>
~l
>
~2
>
etc.,
,\
where Al(j) is a polynomial of degree "di
degree d z with, however, some
in·"j~,' A:!(j)
.
is a "polynomial" of
cOerriCeTi'~::possibly involving
terms, and similarly for A3 (j), A4 (j),' and' sO"on.
page 21
cosine and sine
Let us set "(
10g(1It- t ).
Then, for jc:1( u) we have from (5.6) the
relation
:em. - Yu
=
-j"(
+
Xm
O(;t-:/t-1J
)}
+ O([t-Z/t-lt)}
log{ A j ( k)
+k"( -
It is evident, since
+
10g{Aj(j)
...
J
and Yu are fixed, that k
-+
if j
00
(5.7)
-+
.:xl
through 1(u).
Furthermore,
Thus we deduce from (5.7) that, as j
tha t, also as j
-+
j/k
00,
-+
1.
Hence we may infer
-+ 00,
A1(j)
At(k)
+ O([t-z/t-i)
+ O([t-z/t-d k )
1.
Hence (5.7) can be rewritten:
. rXm as j
-+ 00
Yu = (k - j ) "(
+ 0(1),
through the set,(u).
From (5.8) we are forced to make conclusions:
be some fixed
(ii)
(5.8)
illte.g~a~,multi:pJ~1;of:."(, say Xm -
Yu =
(i)
that
Tu"(
that for large;j in :'(u): it:,>JDust be true that k - j
Xm
-
Yu must
(for integer
=
Tu
Tu);
exactly.
Therefore, for all large j in '(u) we may deduce from (5.7) the equation
(5.9)
Since
Xm
>
Yu
it is necessary that
Tu
>
page 22
O.
But (5.9) shows that Aj(j) -
.
At(j + Tu) is a polynomial in j which tends to zero as j ....
in
~(ul.
This can only be the case if At(x) -
polynomial, which requires that At(x)
must have C t
>
0 if
(7m
::0
through values
At(X+Tu) is the zero
C t , say, a constant.
Evidently we
:2 0 for all m.
j£~(u),
Thus we can rewrite (5.7) in more detail and deduce that for
,J'I '")
1'.2.'12\J
+.
,fA'"'
1'.3 (JJ
+
\ -ru{ , J+ru
1'.2
I'.t
This implies that, for all j
£
Az ( J+Tu
.
)
\ J+r'.!. .'13\J+Tu)
A
+
I
1'.3
•
,
+ ...
J(u),
(5.10)
By the nature of Az(j), there will be an integer d z :2 0 such that
Az(j)
J
---:d
z
where
~2 (j)
_
~z (j), say,
-'
is a bounded periodic or, possibly, almost-periodic, function.
From (5.10) we can therfore claim that, as j ....
I ~z(j) + + I
~z(j+Tu)
0(1)
0(1)
-+
through values in J(u),
00
(Xl}..,)ZClf,
2.
•
(5.11)
We introduced an infinite set f earliet,oFwh'ieh'J(u) is an infinite
subset, but left
~
undefined.
Let us therefore no\¥· define' as consisting of
~,'
any subsequence {jll} such that
l~z(jll)1
-+ lim sup
1~2(j)1
-
1;
.u,say,
J .... oo
where the supremum refers to j increasing through integer values.
page 23
Suppose 'fJ.
> o.
Then, as j
-+
oc through the set j(u)
lim inf i <11 2(j) I
..
fJ.
=
J--+oo
and
~
lim sup i<ll2(j +Tu)!
fJ..
J--+oo
Therefore
'f
, In
I 1m
;_00
However, }..2
<
and Tu
}..1
>
I
+
then <11 2 (j)
= O.
=
jd 2
<11 2 (j)
+
=
(To
=
<
o.
I 'L
1•
(5.12)
Thus (5.12) contradicts (5.10
1.
In other words, if
terms involving lower powers of j,
This forces the conclusion that A 2 (j)
conclusions for A 3 (j), A.. (j), and so on.
we require
+
0, so (}..2/}..1{
and we are forced to conclude that fJ.
A 2 (j)
+
<11
0(1)
. .2 (j)
<11 2U TtL)
0(1)
Thus
I, it follows that C 1 =
(TJ =
= 0, and hence similar
C 1 }..{ for all j; and since
1 and that the {X n } are generated
by the first-order Markov equation
Xn -
}..lX n -
1
=
€n.
Subject to Assumption A this is the only autoregressive model which allows
both F and G to be in N I •
page 24
REFERENCES
1.
Box, G. E. P. and Jenkins, G. M. Time Series Analysis, Holden-Day,
Inc., San Francisco, California (1970).
....,
Lawrence, A. J. and Lewis, P. A. W. "Simulation of some autoregressive
markovian sequences of positive random variables." Report NPS55-79024, Naval Postgraduate School, Monterey, California (1979).
3.
Widder, D. V. The Laplace Transform, Princeton Unversity Press,
Princeton, New Jersey (1946).
page 25