The l'esearch in this Nport was supported by the Office of Naval
Reseal'ch under Contl'act No. N00014-67-A-0321-0002.
•
~ BAsIC ,RESl1.TS OF POINT PROCESS THEORY
by
M. R. LEADBETTER
Depctr'tment of Statistics
UnivePBity of North Carolina at Chapel BilZ
Institute of Statistics Mimeo Series No. 686
June$ 1970
On Basic Results ofPo:l.nt ProceSs Theory
M. R. Leadbetter
University of North Carolina at Chapel Hill
ERRATA
6
10
-7,
-6
8
ADD:
provided
lJ
is a-finite.
"a-finite" should be written before "measures" in each case.
Theorem 2.1 should include the condition that the principal measure
be a-finite.
9
-2
10
1
REPLACE:
''Pr{N (E) < Ql)}"
SHOULD READ:
M(E) .. Urn
by
-inft
n
"A (E) <
r
kfI
n
co".
x*
nk
4, "tim .inn" should be inserted before each right-hand expression.
n
= 0
7 ADD: Hence EM(T)... Um EM(E)
n
5
n+
:11
-4
"A(E) < ..."
should read
m
">.(E) >0".
•
ON BASIC RESULTS OF POINT PROCESS THEORY
by
M. R. LEADBETTER
Department of Statistics
UniveNity of North Carolina at Chapel HiZl
Sunmary.
Basic results of point process theory (such as Kbintcbine' s theorem,
Korolyuk's theorem, Dobrushin's lemma) are discussed in a general setting.
The treatment is based on an approach of Belayev, with emphasis on simpli-
e
•
city of proof.
Palm distributions for stationary point processes on the
real line are discussed (in essence along lines developed by Matthes) with
particular reference to their expression as limits of conditional probabilities.
A simple technique is used to prove known and obtain further results.
The research in this roeport 1J1aB supported by the Offioe of Naval
Research under Con~ract No. N00014-6?-A-0321-0002 •
.
ON BASIC RESULTS OF POINT PROCESS THEORY
M. R. LEADBETtER
DepaFtment of Statistics
Uni,verasi,ty of No1'th CaroUna
Chapel, Hi,l,l,~ Nopth CaroUna 27614
1.
INTRODUCTION. There are many existing approaches to the theory of point
processes.
Some of these -- following the original work of Khintchine
[9] are "analytical" and others (e.g. [15] [8]) quite abstract in nature.
Here we will take a position
so~at
in the middle, in describing the
development of some of the basic theory of point processes in a relatively
general setting, but by using largely the simple techniques of proof described for the real line in [11].
We shall survey a number of known re-
sults -- giving simple derivations of certain existing theorems (or their
adaptations in our setting) and obtain some results which we believe to be
new.
Our framework for describing a general point process will be essen-
tially that of Belayev [2], while that for the section concerning Palm distributions is developed from the approach of Matthes [14].
First we give the necessary background and notation.
There are vari-
ous essentially equivalent ways of defining the basic structure of a point
process.
For example t for point processes on the line t one may consider the
space of integer valuad functions
x{t)
with
x(O). 0,
and which increase
The 1'esearch in this reporrt b1aB suppopted by the Office of Naval,
Research under Contract No. N00014-87-A-0321-0002.
2
by integer jumps, of which there are a finite number in any finite interval.
•
The events of the process then correspond to jmnps of x(t).
One advantage
of such a specification is that multiple events fit naturally into the
framework.
To define point processes on an arbitrary space T,
propriate to consider the "sample points"
it is often ap-
to be subsets of
00
the point of view taken in [18], where each
00
T.
This is
is itself a countable sub-
set of the real line -- the set of points ''where events occur".
Sometimes,
however, a point process arises from some existing probabilistic situation
(such as the zeros of a continuous parameter stochastic process) and one
may wish to preserve the existing framework in the discussion.
ent structure for this is the following, used in [2].
a probability space and
(T, 1)
which the events will occurtl ) .
If for each
For each
oo€O,
let
(0, F, P)
Let
a measurable space.
(T
S
A convenibe
is the space "in
be a subset of T.
00
E€T
N(E)
•
N (E)
~(EnS)
W
•
OJ
is a (possibly infinite valued) random variable, then
:random set and the family {N(E): E€T}
S
a point process.
the process are, of course, the points of
is called a
00
The "events tl of
S.
00
The model may be generalized slightly to take account of muZtipZe events
that is the possible occurrence of more than one event at some
definition of N(E)
measure on T
S,
00
as
for each
and N ({t}) • 1
in T) •
00
~(EnS)
shows that
00
00.
As a measure,
for each
t€S
00
N (.)
00
N(E)
t€T.
The
is an integer-valued
has its mass confined to
(we assume, all one point sets are
To allow multiple events, we may simply redefine N (E)
integer-valued measure on T with all its mass confined to
00
S
00
to be an
and such
3
that
N ({tH ~ 1
w
occurs at
t.
for each
teS.
w
If
N ({t}) > 1 we say a mu'ttipZe event
w
If there is zero probability that any
tES
w
is multiple, we
say the process is ''without multiple events".
If we say a process may have multiple events, we shall be referring to
this framework.
In such a case, we shall srite N*(E)
and refer to N*(E)
multip.Ucities.
as the number of events in
(N(E),
E
for
CWLd{sfA) nE}
b1ithout N{/ard to their>
of course is the total number of events in
E.)
In the manner just described, a point process may be regarded as a
special type of !'f:lf'ldom measure.
This concept has been developed in consid-
erable generality for stationary cases (see e.g. [15», but this generality
will not be pursued here.
Another method of taking account of multiple events is to replace each
tES
e
w
by a pair
(t, k )
t
noting the multiplicity.
where
k
t
is a "mark" usociated with
t,
de-
This again is capable of considerable generalization
by considering rather arbitrary kinds of "marks" and the appropriate additional measure-theoretic structure.
These ideas have been developed by
Matthes (see e.g. [14]) for stationary point processes on the real line,
and provide an elegant framework for obtaining results, for example, in
relation to Palm distributions.
highly dependent on the set
In such cases, the marks are chosen to be
S
(e.g. translates of
w
5).
w
At the same
time, most results of interest can be obtained by using essentially these
techniques, but without explicit !'eference to marks.
Hence we here use the
framework previously explained.
For stationary point processes on the real line, there are several important basic theorems.
number of events in
(1)
Included among these are (writing N(s,t)
for the
(s,t]),
the theorem of Khintchine regarding the existence of the
parameter> A -timt +O Pr{N(O,t)
~
l}/t,
4
(ii)
Korolyuk's theorem which, in its sharpest form, says that
for a stationary point process without multiple events,
to the intensity
~
unit time;
I
(iii)
1.1. fN(O,l)
and
A is equal
(i.e., the mean number .of events per
may 15e infinite).
1.1
For the regular (orderly, ordinary) case (i.e., when
Pr{N(O,t) > l}
•
as
oCt)
t~O)
multiple events have probability
zero,
(iv)
if
A <
"Dobroshin's Lemma" -- a converse to (iii) -- stating that
and multiple events have probability zero, then the process
ell)
is regular.
Various analogues of these results have been studied for non-stationary
point processes on the real line in [20], [ 4], [S], largely by using properties of Burkhi1l integrals.
A clarifying and general viewpoint has been
more recently given by Belayev [2].
of the two constants
the space
T,
Specifically in [2], generalizations
are made in terms of measures
A, 1.1
instead of in terms of point functions.
sure 1.I(.} is defined (as customarily)
countable additivity of N
A(E)
•
T
4UP{ ~ Pr{N(Ei )
>
A(.)
simply by
a}: Eier,
1.1 (E)
Ei disjoint,
is easily shown to be a measure and it is clear that
for all
EeT.
1.I(E)· EN(E)
1.1(.).
On
is defined in [2] by
1
1..(.)
on
The principal mea-
guarantef7ing countable additivity of
the other hand, the pa:mmetric measure
(1.1)
on
1(·), 1.1(.)
UEi
• E}
1
A(E) S 1.I(E)
For a stationary point process on the real line
ME). Am(E) ,
= 1.Iul(E), where m denotes Lebesgue measure.
Using these definitions, it is possible to extend the basic results
e
quoted above to apply to point processes which may be non-s tationary, on
spaces
T more general than the line (including any Euclidean space).
These generalizations are systematically described in Section 2.
In Section
5
3 stationarity is discussed in general terms (with particular reference to
Khintchine's theorem) when
T is a topological group.
In both these sec-
tions the general lines of development are those of [2], with adaptation of
the results in presenting a somewhat different viewpoint, and with emphasis
on simplicity of proofs obtained by direct analogy with those of [11].
Finally, in Section 4, we discuss some basic results relative to Palm
distributions (and their expressions as limits of conditional probabilities),
for stationary point processes on the real line.
The approach is essentially
that of [14] (without explicit reference to marks), again with emphasis on
the simplicity of proofs obtained from the techniques of [11].
~.
•
THE BASIC GENERAL THEOREMS.
The notation already developed will be used throughout this section•
We shall systematically obtain the generalizations of the basic theorems
referred to in Section 1.
This development follows the same general lines
as [2] but with differences of detail and perspective.
All that is to be said in general, relative to Khintchine's theorem concerning the existence of the intensity, is contained in Belayev's definition
of the parametric measure (1.1) given in Section 1.
(For special cases
(when T has a group structure and the point process is stationary) it is
possible to say more that is directly analogous to the real line case -mention of this will be made later.)
It is shown in [2] that the truth of the generalized version of
Korolyuk's theorem, viz.
ACE)
= peE)
for all
EeT
without multiple events), depends on the structure of
any stationarity assumption.
(for a point process
T
rather than on
The proof given directly generalizes that of
6
[11] for stationary processes on the real line.
This is most clearly seen
for a non-stationary process on the real line (T
For then if
subintervals
Xni • 0
is an interval
E
n, i
(i • l •••n)
E
N •
n
we may divide
and write
Xui· 1
E
into
n
equal
N(Eni ) ~ 1,
if
Assuming there are no multiple events, it is easily
otherwise.
seen that
(a,b],
then being the Borel sets).
n
1:i=l
with probability one, as
)(i -+ N(E)
.'U
n -+
co, and hence
by Fatou 1 s lemma,
n
\.I (E)
s
Urn -i.n6 EN
n
= Um
s
A{E) s \.I(E)
But
Thus
and hence
A(E). \.I(E)
for all
•
T
E
of the form
(a,bl.
E.
for all Borel sets
For the above proof to be useful when
require
Pr{)(i-U
i=l·'U
ME).
= \.I(E)
A(E)
-i.n6 1:
T
is a more general space, we
to have sets playing the role of intervals.
A suitable definition
of such a class of sets is given by Belayev [2] and called a "fundamental
system of dissecting sets" for
Here we shall use a somewhat different
T.
definition to achieve the desired results.
a class
C = {Enk : n,k=l,2 ... } of sets
Specifically, we here say that
~k£T
is a dissecting system for
T if
(i)
C is a "determining class" (cf. [3D for measures on T.
That is two measures equal on
C are equal on . T
(e.g.
C may be a
semiring generating 1).
(ii)
For any given set
(a)
In
E£C
Corresponding to each integer
of integers such that
E-U
k£I
n
Fnk
n
~
1
are disjoint for
EnkCF£T
n
there is a set
k£In
and
7
where
F ...,
the empty set, as
n
(b)
Given any two points
ficiently large values of n
tl
E:
~kl' t 2
tIl t
E:
• 1,
of
l
E,
~
for all suf-
some
:I k 2 ,
n (t ,t
O 1 2
»,
such that
En~2.
with rational endpoints of length
quirement in (ii) (a) that
2
In' k
E:
For example, for the real line, we may take
(a,b]
1».
(i.e., all n
, k , k
E
, E
2
l
rik1 nk2
there are sets
n +
=.
lIn.
!uk to be any interval
We note also that the re-
tim Fn may
be replaced by
·
Pr{N(~F )=O}
n
but this, of course, depends on the process as well as the structure
of T.
The proof of Korolyuk' s theorem given for the real line now generalizes
as once to apply to a point process on a space
system.
•
This is easily seen from the following lemma•
LEMMA 2.1:
Consider a process without multiple events on a space T
sessing a dissecting system C
kE:l ,
n
T possessing a dissecting
write
Xnk· 1
if
1;1
{Euk}.
N(~k) >
0,
With the above notation for
Xnk" 0
otherwise.
Let
Then
N -+ N(E) S
(2.1)
n
Fur.ther, if
N(B) <
I»
I»
~k· 1
with probability one, as
when N(Euk) > 1, X~k • 0
n -+
1».
otherwise and if
with probability one, then
I
kE:l
n
X: + 0
k
with probability one, as
n +
01).
pos-
EE:C,
8
Proof.
It is clear that
there are points
't
l
N
n
••• t
m
~
N(E).
On the other hand, if
where events occur.
eventually contained in different sets
!uk
GO
For large
and hence N
n
~
~
N(E)
~
m,
n,
these are
m.
(2.1) fol-
lows by combining these results.
The second part follows by noting that since (with probability one)
only a finite number of events occur they are eventually contained in different
sets
Enk
when n
is large and thus for such
n,
N*
n • O.
By using the first part of this lemma, Korolyuk's theorem follows as
for the real line by a simple application of Fatou's lemma.
Stated specif-
ically we have (as contained in [2]),
THEOREM 2.1:
(Genera'tiaed Kol'oZyuk's theorem.)
For any point process, without multiple events, on a space T possessins
a dissecting system, the parametric and principal measures coincide.
If
or
A{.)
finite measure
is absolutely continuous with respect to some a-
lJ(·)
v on T,
then under the above conditions the densities
dA
dV'
coineide a.e.
.!!l!.
dv
This reduces again to the usual statement of KorolYuit's
theorem for stationary point processes on the line.
It was assumed in the above derivation that multiple events have zero
probability.
The statement is easily modified to apply to a point process
where multiple events may occur as follows:
COROLLARY to Theorem 2.1:
pIe events, on a space
Consider a point process which may have multi-
T possessing a dissecting system.
For
E€T,
let
9
N*(E)
denote the number of events in E coated without regard to multi-
plicity.
to
Let
ll*(E)
Proof:
Then the parametric:: measure
ll*(E)" fN*(E).
for all
A(E)
is equal
E£T.
A new point process
may be defined
in which one event occurs at
if there is at least one event of the original process at
tiple events are replaced by a single event).
(1.e., mul-
The parametric measure of
the new process is the same as that of the old;v1z.
measure of the new process is
t
t
:\(. h "The principal
Hence the result follows by applying
ll*{ • ).
Theorem 2.1 to the new process.
A
point process on Tis called regular (orderly, ordinary -- cf (20)
[5] and especially [2]) with respect to a dissecting system C· {Enk } if
•
P{N(E
) > U
) > O}
nk
Lim 4up P{N(E
n+ ClCI k '
uk
(For simplicity, we shall always assume
..
O•
P{N(Enk»O} tI 0
for any
n, k.)
This definition applies to a point process which may conceivably have multipIe events.
However, the next result shows that in fact regularity pre-
cludes the occurrence of multiple events UDder simple conditions on
THEOREM 2.2:
Consider a point process (allowing multiple events) on a
space T possessing a dissecting sys tem C.
Suppose the process 1s regular
A(E ) < ClCI.
n
bility one, the process has no multiple events.
and that there exist
Proof.
Let
T.
E!C
E £C, E tT
n
n
be such that
of mu Z.tip 7.e events in
E.
Then
such that
Pr{N(E)<ClCI}. 1.
Then, with proba-
Write M(E)
for the number
10
M(E)
I x:k
S
kc;I
~
(with the usual notation) where
or not.
n
is one or zero according as
N(E ) > 1
nk
Hence
fM(E)
s
I
k~I
Pr{N(E ) > II
nk
n
) > U
pr{N(E
s .6~p Pr{N(Enk)
nk
which tends to zero as
n -+
GO
>
O}
I
j~I
pr{N(E ) > o}
nj
n
since by regularity the first term tends to
zero, and the sum does not exceed
A(E) <
GO.
A converse result of Theorem 2.2 is "Dobrushin's lemma".
The general
form of this given in (2] assumes a ''homogeneous'' point process -- for which
T
•
possesses a fundamental system C of dissecting sets such that
Pnk(O) • Pr{N(E ) > Ol,
nk
but not on k
for
Pnk(l). Pr{N(E ) > l}
nk
Enk€C.
are each dependent on
n
This assumption does not imply stationarity of
the point process (indeed there may be no "translations" defined on
T) but
it may well be that the only interesting homogeneous processes are stationary
ones.
We give a less restricted result below.
It may still be of greatest
interest in the stationary case, but it does allow considerable variation
in the quantities
Pnk(O) J
Pnk(l)
for fixed
n.
Specifically to obtain Dobrushin' s lemma, we shall assume the existence
of a dissecting system
C. {E k}
for which there is a sequence {e} of
n
n
non-oegative real numbers, and a function 4J{ e) -+ 0 as e .. 0, such that
f.
for each. n
(2.2)
11
THEOREM 2.3:
(GeneNl:{,sed 'lJtJmon of Dobrushln'lJ lemma.)
Consider a point process without multiple events, on a space T
sessing a dissecting system
some
E£C.
Proof.
C satisfying (2.2).
A(E) < CllI
Using, the notation of Lemma 2.1, we have
~
1:
with probability one, as
n +
= A(E)
< CllI,
EN(E). IJ(E)
1:
(2.3)
k£In
+
0
n
Since both sums are dominated by
CII.
Pr{N(Enk ) > o}
1:
•
E{
1:
kEIn
Pr{N(E
) >
nk
ke:In
Xak}
+ IJ(E)
U
o.
•
A(E)
+
Hen)
+ 0
Hence by (2.2)
a
n
I
ke:I
and thus by (2.3)
Pr{N{E
nk
)
>
ol
~
n
an + 0
(since
A(E)<CllI).
Finally from (2.2) again
~up
k
as
n
+
00.
N(E)
it follows by dominated convergence that
and similarly that
•
for
Then the point process is regular.
kd
and
Suppose
pos-
[Pr{N(E ) > 1J/pr{N(E ) > Ol]
nk
nk
~
12
1.
STATIONARITY-GENERALITIES.
A very great deal of literature exists relative to stationa%y point
processes em the real 11De (see SeetiOll 4).
less about stationary poillt processes
OIl
ODe expects to be able to say
the plaDe, or in
an
(see e.g.
Howewr, there is qu.Ue a good deal that may be said even when T
[6]) •
just ass'&D8d to be a (locally compact) topological group.
is
In this section,
we COIIIIDe:Dt briefly on a few aspects of such results.
T is a locally cCllDpact (Hausdorff) group, the natural a-field 1
If
is the class of Borel sets -
generated by the
conWllient to assume (aud we here do) that
opeD
US»
assumed that
T
It is usually
is also a-compact, and then
T
T is also generated by the compact sets of T.
(e.g.
sets of T.
(It is in fact sometimes
is seCODd countable.
While this additional
assVlllption may be necessary for some purposes i t does, however, imply that
•
the group is also metrizable.)
For a point process
OIl
such a group T,
stationarity may be defined
in terms of the invariance of the joint distributions of
for
tE:T
where
n
SE:E},
ts
)
l1
I
is any fixed positive integer, and Hi
T (tE· {ts:
sets of
N(tE ) ...N(tl
are.y fixed
denoting the group operation).
If
T
is
not abelian this gives a COllcept of "left stationarity", "right stationarity"
being correspondingly defiDed.
Under (e.g. left) stationarity, the principal aDd parametric measures
11{·) ,;H
.)
are (left) invariant Borel measures which are [7,. Theorem ,.641 J
regular provided their values on compact sets are finite assume.
Thus
measure me·)
where
A( • )
on T,
and 11( • )
which we will
are just constant multiples of the Baar
A(E). Am(E),
pCE) .. ~(E), say,
for all
&1,
>. and p are exmstants, the paramete:re 8ftd the i.ntensity of the
stationary point process respectively.
Questions conceming the parameter
8l'lcl intensity :In such a setting have been disewaaed to some extent in [1].
The above gADeral liDe of argUlUftt: is that of [2].
13
If in addition T
possesses a dissecting system
C .. {E
nk
}
of, say
bounded sets (i.e., having compact closures) and if a stationary point
T is without multiple events, then Theorem 2.1 shows that
process on
A
III
This is Korolyuk's theorem in the stationary case.
lJ < ao.
in such a case it is not unreasonable to suppose that
m(E )
nk
are independent of
n
E
nk
the
k
P{N(E ) > O}
nk
are translates of each other).
r P{N(E 0) > o}
n
n
and
r
E
nO
n
Then using the notation of
for
E
nk
But since by definition of
•
A(E)
+
III
E!C
A m(E)
is the (necessarily finite) number of integers in the set
is any given
E-
and
(which will hold if, for example, for fixed
Theorem 2.3 we' have, from the proof of that theorem, for
where
Further,
In
k!I •
n
C,
U Enk
keI
C
F
n
n
~.
it follows that
r
n
m(E 0)
n
..
I
kEl
m(E_ f . ) + m(E)
n
~
and hence that
It is this latter property that the parameter satisfies in Khitchine's
enetence theorem.
We summarize this as a theorem.
statement, we will here call a dissecting system
distribution of
n.
N(E ),
nk
and
m(E )
nk
For convenience of
C homogeneous if the
do not depend on
k
for each fixed
14
THEOREM 3.1:
CoDsicier a stationary poirlt process without multiple events
on locally compact group T.
exist ccmstants
where
m(·)
Suppose
such that
A. P
T i8 also a-compact.
A(I). >.m(I),
is the Haar measure of T
Then there
p(E). pm(E)
E~T
for all
(0 S A $ p < CD).
Suppose, in addition, that T has a homogeneous dissecting system
e•
{E }
nk
of bounded sets
E
nk
Then the point process is regular,
•
A· P
and
lim PriNCE 0) > O}!m(E 0) .. A
n+ CD
n
n
COROLLARY: The stated results hold if the condition
~.
e
be a homogeneous
dissecting system is replaced by the requirement that for each fixed
•
the sets
E
nk
1'1,
are all translates of each other•
The above remarks have been concemed with a stationary process without
multiple events.
When multiple events are allowed, the appropriate gener-
alizations of the real line results occur.
T with
p(E) < CD
(e.g.,
E compact), and if N (E) denotes the number
s
which have "multiplicity" s . 1,2 ••• , then
of those events in I
Ps .. ENs(I)/l(E)
ips}
s".
For example, if E is a set of
is a probability distribution on the integers
1,2••••
may be interpreted as the "probability that an event has multiplicity
If in addition,
and we choose
X:k .. 0
T has a homogeneous dissecting system C· {E }
nk
IkC with I.1(E) <
Writing
OOt
X:k" 1
othezwise, then a1m:11arly to Lemma 2.1
I
~s.
bI""IlK
n
+
N (E)
s
as
n
+ 00,
1f N(Enk) · s,
15
with probability one.
The familiar argument of taking expectations and
using dominated convert;tence shows that
(Eno
is any
r Pr{N(E 0)
n
D
~
E
nk
and
l} +
~(E).
p
s
r
..
r Pr{N(E 0) .. s} + EN (E)
n
n
s
In) ~
the number of points in
u
Thus
lim Pr{N(E 0) .. siNCE O} ~ I}
n+
n
C11
n
giving intuitive justification to the description of
that an event has multiplicity
depend on
n
T .. R.
E).
Similarly
s
p
s
as the probability
(under these assumptions
ps
does Dot
Further questions of this type are considered in [16J whell
We note that the above calculation may also be considered as a
special case of that in the next section concerning Palm distributions.
•
!.
CONCERNING PALM DISTRIBUTIONS.
For a stationary point process the "Palm distribution"
Po
gives a
precise meaning to the intuitive notion of conditional probability "given
an event of the process occurred at some point (e.g.
the real line, we may write (for certain sets
(4.1)
When
T
is
EeF)
tim p{EINCC-~,O» ~ l}.
•
POCE}
t . 0)".
t5 ... 0
That is
POCE)
is then the limit of the conditional probability of
given an event occurred in an interval near
For example, if
terval
(Ott]
E
t .. 0
E
as that interval shrinks.
denotes the occurrence of at least one event in the in-
(i.e.,
{NCO,t)
~ U)
then
POCB). F (t),
1
the distribution
function for the time to the first event after time zero given an event occurred "at" time zero.
16
This kind of procedure for particular sets
E was used by Khintch1ne
[9] and is useful in providing an "analytical" approach to such conditional
probabilities (see e.g. [12]).
More sophisticated and general measure
theoretic treatments involving the definition and properties of
Po have
been given by a variety of authors (e.g. [18], [19], [14], [15], [17]).
In
this section, we shall use a ''middle of the road" approach to the definition
Po
(based essentially on [14]) which is capable of considerable gener-
ality.
Our main pUrPOse will be to give simple proofs for formulae such as
of
(4.1), and its generalizations to include "conditional expectations" of
functions.
Such results have application, for example, to the evaluation
of the distributions of the times between events in terms of conditional
moments ([13]).
We give the construction of the Palm distribution
Po for stationary
point process on the real line in the manner of [14], though from a some•
what different viewpoint.
The construction generalizes to apply to point
processes on groups (cf. also [15]) but we consider just the real line case
since the later results of this section apply to the real line.
Consider, then, a stationary point process (without multiple events,
for simplicity) with finite parameter
simplicity we take the sample points
T ... Rl ,
of
l.e.
BeT.
w to
. be themselves the subsets
For any real
(~{wn(s,t]})
t,
and
wd),
w translated to the left by
t;
= Nw(B)
n making N(B)
Finally, we shall again wrl te
number of events
Again for
T will denote the Borel sets of T. RI
A < 00).
will be the smallest a-field on
each
on the real line.
Sw
w 1s a countable set of real numbers (without finite
limit points, since
f
A,
N(s, t)
for
measurable for
N{ (s , t ]} ,
in the semiclosed interval
let
w
t
i.e. if
(en)
and
the
(s,t].
denote the set of points of
w - {til,
w • {t -t}.
1
t
If
F
17
F and lild'l,
is any set of
precisely those points
"thinIt
lil
ti£w
l1li
{til,
say, we define
for which
translated to
t
as origin) is in
i
That is, to form
Wti£F.•
w by retaining only the points
t
F.
such that
i
The
to consist of
w*dl
w*
lil*
(i.e.
lilti
w
define a stationary point
process formed from some of the events in the original point process.
example, if
events
t
i
F
l1li
{w:N(O, t)
~
U, the new process contains precisely those
t
(i.e. no later than
ti+t).
Write
of events of the thinned process in the interval
cMcl{w* n (O,l)}.
•
i.e.
Then the thinned process has intensity
F.
is countably additive as a function on
f
is a measure on
for the number
N'...l;'
(0,1)
Now this procedure may be carried out for any
F
For
of the old process which are followed by a further event within
a further time
N
we
. EF
A " ENp •
F
and, for fixed
til,
It follows at once that
and hence that
(4.2)
is a probability measure on
F (A" An
l1li
EN(O,l».
Po
is the desired
Palm dlstnbutian.
To give
Po
such as (4.l).
intuitive interpretation one wishes to prove relations
Equation (4.l) is not universally true, however, as can be
seen by considering a "periodic" stationary point process in which the
events occur at a regular spacing
t
l1li
0
h,
the distance to the first one after
being a uniform random variable on
(O,h) •
For this process take
to be the occurrence of at least one event in the open interval
Clearly
Pr{FjN{-o,O) ~ U .. 1 when
IS < TI.
But
NF(O,l)" 0
(h-l1, TI).
and hence
PO(l!') .. O.
We give now a class of sets for which (4.1) does hold.
we
shall call
11
set
Specifically,
F€f righ-t continu.ous if its characteristic function
P
18
X (l1l)
F
is such that
x,(l1l ) ... x,(l1l )
s
t
W
t
e:F
then
W
s
as
X, (w )
t
is continuous to the right in
s i- t.
Equivalently, this means that for any
EF when
is sufficiently close to
8
t
t;
that is
C,
if
on the right, and
conversely.
THEOREM 4.1:
Proof.
Let
as
CD.
m'"
interval
rEF is a right-continuous set. Then
Suppose
6m be any sequence of non-negative numbers converging to zero
Write
r
for the integer part
m
into
(0,1)
r
intervals of length
m
terval of length less than
N«i-l)o , i6 )
m
m
•
Then
~
1,
"mi
1
[15- ]
m
c5
CI
0
m
left over).
otherwise
N denotes the number of intervals
m
event and such that the translate
c5
m
Write
of
Xm·
( i · 0,1••• r ).
m
«i-l)o. io]
m
m
w
is in
iom
F.
wio €F
containing
fidently large.
«i-l)o. io]
m
m
1
if
Let
containing an
But by the right con-
to
for that interval
Divide the
(with perhaps an in-
tinuity assumption, if an event occurs at
m
1
15- •
m
then
Further, with probability one, when
l1ltOEF
to'
if and only if
when
m is suf-
m is sufficiently
large the events all lie in different intervals and there is no event in the
last short interval.
Since
that
N
m
~
N(O,l),
ENm '" AF
as
Hence, with probability one,
and
m'"
00.
EN(O,l) <
That is
00,
N '" N as m'" 00.
F
m
it follows by dominated convergence
19
or, by stationarity,
But
Pr{Xmo .. 1,
Hence, since
r
m
"" 0
W E
F}
-1
Pr{FIN(-o ,0) ~ I} Pr{N(-o ,0) ~ I}.
..
m
Pr{N(-o ,0)
m,
m
~
m
l} - Ao ,
m
as required.
As an example, consider the set F .. {w: N(O,t)
~
r}
(r == 1,2, ... ).'
This is easily seen to be right continuous and hence the theorem applies.
In this case,
•
PO(F)
..
lim Pr{N(O,t)
o ..
~ rIN(-6,0) ~ I}
0
is interpreted as the distribution function for the time to the r-th event
after time zero, given an event occurred "at" time zero.
least
r
events occur in
(0, t)
after time zero does not exceed
F
(Note that at
if and only if the time to the r-th event
t) •
is right continuous, leading to what could naturally be termed the joint
distribution function for the time to the rl-st, r -nd, ... , rk-th events
2
after the origin, given an event occurred at the origin.
The convergence in Theorem 4.1 does not occur for all
However, we may regard the probability space
n
Fe:F
in general.
as consisting of real in;;'
teger valued functions increasing by unit jumps where events occur, and
20
consider it £os a subspace of
D,
the space of functions with discontinuities
of the first kind (see [3]) where
D
has the "Skorohod topology".
Theorem 4.1 may be shown to imply weak convergence of
to
Then
Pl'l. P(·IN(-o.O) > 0)
PO.
A slightly different definition given by Matthes ([14]) does give con-
vergence similar to Theorem 4.1 for all
FeF.
Specifically let
denote the time of the first event prior to the origin.
Po(F) • P{w: w~FIN(-l'l,O) > O}
we may consider
That is the "origin is moved" slightly to the point
an QVent
!'.~r.crs.
Then instead of
= P{w:
P*(F)
s
s · sew)
of
ws€FIN(-l'l,O) >
(-0,0)
where
Then the following theorem (which is virtually identical
to that of [10, Sec. l(f)] holds.
THEOREM 4.2:
•
For each
Fe F
(4.2)
(hence the total variation of
Proof:
P~-P0
tends to zero as
This can be proved as in [10, Sec. 1].
o.
However, a very easy proof
follows by a simplification of the method of Theorem 4.1.
In fact -- using
the notation of that proof -- we consider N* (instead of
m
8
mi
0).
N) ,
m
denoting the position of the last event prior to (or at)
contribution to the
S\DD
only occurs if this event is in
Then, with probability one,
for which
Wti€F.
tionarity that
~
m
(A
«i-l)l'l, 10 ].)
m
m
converges to the number of events
rmPr{Xmo = 1, Ws€F} .. A ,
F
l} -
i6 •
tie (0,1)
It follows as before by dominated convergence and sta-
follotN'S (for any sequence
Pr{N(-o ,0)
m
N:
where
AIrm•
0 . 0)
m
from which the desired result
on using the fact that
ol.
21
The fact that the limit in (4.2) holds for all
FeF
is, of course,
more satisfying than that in (4.1) which requires "continuity sets".
P~
ever, the definition of
is more complicated than that of
limit in (4.1) may be more useful in practice.
and Pg
PIS
How-
and the
The difference beDieen
P <5
is, of course, slight (but we feel it worthy of exploration).
Theorems 4.1 and 4.2 concerned conditional expectations of the function
X"~
given an event near the origin.
hold for other functions.
One may ask whether similar results
To answer this in relation to Theorem
call a measurable function
~(w)
continuous to the right in
t.
continuous to the Pi,ght if
4.1~
~(Wt)
we will
is
Before stating the generalization of Theorem 4.1, we give a lemma
(the result of which is contained in [14 J) and which is useful in a number
•
of contexts •
LEMMA:
If
cf>
is a measurable function on
or integrable with respect to
PO'
PO(F).
cf>
is either non-negative,
then
The statement of this lemma, when
of
nand
~
0=
X
F
(FE F)
is just the definition
Its truth for non-negative measurable or PO-integrable
~
fol-
lows at once by the standard approximation technique.
The following result generalizes Theorem 4.2.
THEOREM 4.3:
such that
Le~
\fl
be
measurable (on
1~(Wt)1 < ~(w)
E{ct>IN(-<5,O)
for all
~
I}
~
m,
t£(O,l)
f ct>dPO
continuous to the right and
f{~N(O,l)} <~.
where
as
<5
~
0 •
Then
22
Proof:
for
The pattern of the proof of Theorem
X •
F
Specifically
rm
8
m
J
4.1 applies, with 41 written
= L
X
4I(wi6 )
mi
L{.(wt
+
m
i-l
with probability one.
But
):
t
j
E W
j
IS I S 1/J(w) • N{O,l)
n
n (O,l)}
which has finite expec- .
tation and thus, be dominated convergence, and the lemma,
A- 1 tim rmE{xmO.(w)}
•
m+
=
since
Pr(Xmo = 1) - A15
m
tim E{~lxmO
m+ CID
- Ar;l.
N(-6 ,O) ~ 1
m
for
XmO· 1).
COROLlARIES:
(i)
If
sult. holds.
For if
(by stationarity)
oo
= l}
This is the desired result (writing
41 is a bOlmded, right continuous ftmction, the reI~I ~ K
we may take
1/J(w)
= K and E{1/JN(O,l)}::I
This corollary is similar to a theorem of Ryll-Nardzewski [18] (their
two-sided continuity of
t
1).
k
Suppose EN +1 (O,T) <
replaced by
(li)
4I(w )
N(-6,c5)
is required and the condition
N(-c5,O) ~ 1
~
00
for some positive integer
k,
T > O.
Then
tim
6 +0
where
~
E{Nk (O,T)IN{-6,O)
~ 1}
=
Ep
Nk(O,T)
0
EpO denotes expectation with respect to the Palm distribution.
is, the k-th moment of
N(O,T)
That
with respect to the Palm distribution, is
23
simply the k-th conditional moment (defined as a limit) given an event flat"
the origin).
The proof is immediate on noting that
,(w). Nk(O,T)
w
is
continuous to the right and, for all . t£ [0,1] ,
where
E{1/J(w)N(O,l)}' s fNk+lCO,l+T).
This latter quantity is finite since
it is easily seen by MInkiwski's inequality and stationarity that
ENk+lCO,s) <
=
for all
s > O.
Finally, we note the corresponding generalization of Theorem 4.2.
this, the condition required above that
,
be continuous to the right can
be omitted, but the origin must "be moved" to measure from the time
the first event prior to zero.
e
THEOREM 4.4:
tf:(O,l)
where
Let
where
S·
sew)
,
co.
s
of
We state this formally.
be measurable and such that
E{1/JNCO,l)} <
For
I.(w ) I < 1/J(w)
t
for all
Then
denotes the position of the first event prior to
t · O.
24
REFERENCES
[1]
.'
R. A. AGNEW,
''Transformations of uniform and stationary point
processes,"
[2]
YU. K. BELAYEV,
Ph.D. thesis, Northwestern University, 1968.
"Elements of the general theory of random streams,"
Appendix to Russian edition of Statianarry Be Zated Stochastia
'Pztocesses by H. Cramer and M. R. Leadbetter, MIR, Moscow, 1969.
(English translation under preparation for University of North
Carolina Statistics Mimeograph Series.)
[ 3]
PATRICK BILLINGSLEY,
Convergence of Pl'Obabi Zity MeansUNB,
John Wiley & Sons, New York, 1968.
[4]
W. FIEGER,
"Eine fur belliebige
Call-Prozesse geltende
Verallgemeinerung der Palmschen FormaIn," Math. Scand.
16,
(1965), pp. 121-147.
[5]
W. FIEGER,
"Zivei Verallgemeinerungen der Palmschen Formaln,"
Pl'ans. 3rd Prague Conf. on Inforrmation
Theory~
Czech. Acad.
Sci. Publishing House, Prague, 1964, pp. 107-122.
[6]
J. R. GOLDMAN,
"Infinitely divisible point processes in Rn , If
J. Math. Anal. AppZ.
17
(1967), pp. 133-146.
[7]
P. R. BALMOS,
MBQBuztB 11&eory~
[8]
T. E. HARRIS,
"Random measures and motions of point processes,"
Van Nostrand, Princeton, 1950.
University of Southern California, Los Angeles, California.
25
[9]
Y. A. KHINTafINE,
Mathematical. Methods in the The01'Y of (iueueing#
Griffin, London, 1960.
[10]
1).
ICONIG & K. MATTHES,
I,"
Fomeln,
[11]
M. R. LEADBETTER,
"Veral1gemeinerungen der Erlangschen
Math. Naahtt.
26
(1963), pp. 45-56.
"On three basic results in the theory of stationary
point processes, n
'Pztoo. Arneza. Math. Soo.
19
(1968), pp. 115-
117.
[12]
M. R. LEADBETTER,
"On streams of events and mixtures of streams,"
J. Roy. Statist. Soc. Series B,
[13]
M. R. LEADBETTER,
[14]
BuZZ. Ins1i. 'Intemat. Stcrt.
In8~
(19~9),pp. 309-319.
IC. MATTHES,
"Stationilre zufiillige Punktfo1gen
Jb~rt Deutsch. Math. V62'ein.
[15]
(1966), pp. 218-227.
"On certain results for stationary point processes
and their application,"
43
28
.J. MECKE,
66
I,"
(1963), pp. 66-79.
"Station5re zufii11ige Masse auf 1okalkompakten Abelsehen
Gruppen,"
Z. WahzasoheinUohkeitstheoztie und YeN. Gebiete.
9
(1967), pp. 36-58.
[16]
R. K. MILNE,
"Simple proofs of some theorems on point processes,"
Mimeograph report, Australian National University.
[17]
J. BEVEU,
"Sur 1a strueture des processus ponctue1s stationnaires,"
C. R. Aoad. Sci. Pattie A
e
[18]
C. RYLL NARDZEWSKI,
267.
(1968) pp. 561-564.
"Remarks on processes of calls," Prao. Fouzrth
Bezak. Symp. on Math. Stat. & ProbabiUty
2 (1961),
pp. 455-466.
26
(19)
I.H. SLIVNYAK.
"Stationary streams of homogeneous random events,"
Vestnik Baztkov. Gos. Univ.
[201
F. ZITEK,
32
(1966)· pp. '73-116 (Russian) •.
"On the theory of ordinary streams."
CaeohosZorJak Math. tl.
8
(1958), pp. 448-458 (Russian).
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