..
Translation prepared under ONR Contract N00014-6 7-A-Q321-Q002.
ELEMENTS OF THE GENERAL THEORY OF RANDOM STREAMS OF EVENTS
by Yu. K. Bel,ayev
Translated by M. R. Leadbetter
Depcwtment of Statistics
University of North Carol,ina at Chapel, Bil,l,
Institute of Statistics Mimeo Series No. 703
Augus t, 1970
,1
Elements of the General Theory of Random Streams of Events
by
Yu. K. Belayev
Transla~ed
by M. R. Leadbetter
This report is a translation of the seeond of three appendiees by Yu. K.
Belayev to the Russian edition of the book Stationary & Related Stochastic
Process
by Harald Cramer & M. R. Leadbetter.
refer to this book.
begin with
e
1.
2,
References to tithe main text"
Since this is the second appendix, the equation numbers all
and these have been left unaltered in this translation.
Elements of the general theory of random streams,
by Yu. K. Belayev.
(Appendix No.2 to the Russian edition of Stationary & Related Stochastic ProC6SSelI,
by Harald Cramer & M. R. Leadbetter, published' by MIR, Moscow, 1969.)
From the contents of this book it is seen that the main ideas and results
for problems of "crossing" type are formulated in a natural way in terms of the
theory of random streams.
The theory of random streams, or, as they are also
called, point processes, developed principally in connection with various problems in the theory of mass service, of counters and so on.
One may see this in
more detail with the relevant references in the book of Cox and Smith [1].
In a
related direction, there is a series of works concerning generalizations of Palm
streams (see Slivnyak (1), Matthes [l], Kerstan and Matthes [1], Ambartsumyan
[1], Heeke [1]).
TrlG81at1ca ptepared under OUR Contract N00014-67-A-0321-0002.
2
work of McFadden [2], concerns various approaches to the definition of a stationary stream on the line.
A number of general results are also proved.
particular it is shown that if
~ (t)
denotes the random time from an instant
to the k-th following event of a stationary stream, then
t
n
I:
k=l
P{L.. (t) ;:; x}
it
In
is a convex non-decreasing function f.or
S (x)
III
n
x
E:
(O,ao).
We note that in the works of Zitek [1] and Fieger [1], integrals based on
functions of intervals on the line are used for generalizing Palm-Khintchine
f\D1ct:lons, the construction being a particular case of that used below in the
definition of a parametric measure ..
We restrict ourselves to these short bibliographic references.
e
We are now
interested in such generalizations of the main ideas of the theory of random
streams which are linked in a direct way with problems of "crossing" type ..
Further examples are given in the following paragraph.
Some of the results given
tn this paragraph appeared in a communication given by the author on the occasion
of a summer school on the theory of Probability in Palang in June 1967 and also
in seminars at Moscow University and the Mathematical Institute of the Academy
of Sciences (see Belayev [9]).
Let
(T,
M.r)
be a measurable space of values of the parameter
is a a-algebra of measurable sets,
elementary events
W E:
(n,~,
P)
tE:T.
M.r
an outcome probability space of
n. In questions of "crossing" type, the main objects of
study are subsets of points of the parameter space
T occurring in a random way.
We introduce the following definitions.
Definition 2.1.
By a random (point) set
is meant such a set for which, for any
S • {s}
W
a
A E:
MT ,
of points
the number
s
neAl
a
.. s (w)
a
E:
T
of points in
3
'e
Sw n a
is a random variable.
values
0, ·1, ••• ,
co.,
The random variables
n(L~).
A
£ ~
take integer
and possess the additivity property
co
In (at).
=
1
if
6 i n Aj
1Il~,
i rI- j.
The system of random variables
dom stream arising from the random sets
neAl
is called the ran-
S.
w
In this book r8lldom streams of arrivals, departures and crossings are
[T 'M.rJ
studied, arising from random sets on
where
T.. (_ClD,CO)
and
M.r
is the
class of Borel sets.
set
For example N (s, t) is the number of points of the random
u
~ (x) .. u} n (s, t) • Using the fact that the sum, difference and
S{J,) .. {x:
limit of (probability one convergent) random variables, are all random variables,
e
one may show that the number Nu{A)
of points of the set
a random variable for any Borel set
A.
If
A
€~,
neAl
[T,M.rl
is a random stream on
{s:
~(s)
• u} n 6
is
then corresponding to each set
one may look at the function
(2.1)
This function is a measure since
Ai
€
M.x.
11{A) > 0,
co
co
11(U Ai)" tl}l(Ai),
I
Ai n A
j
III
~,
The reader will find the main ideas of measure theory used below in
Halmos [1].
If it is assumed that
tinuous with respect to the measure
11(A)
v (A) ,
is a a-finite measure, absolutely conthen by the Radon-Nikodym theorem
(2.2)
Definition 2.2. The measure 11{a) is called the pnncipat measure of the stream
neAl.
}let)
If
}leA)
is a-finite then the stream neAl
is called jtnite.
A function
satisfying the equality (2.2) is called the intensity funation of the
4
stFea11t reZative to the measuzaB v(ll).
'"
called the· intensity of the stream
The value of
101(t)
at the point
t
is
at t.
The direct calculation of the principal measure or intinsity function is
often accompanied by considerable difficulties.
In this connection, the intro-
duction of the idea of parametric measure of the stream, which generalizes
Khintc:.hine t s (1] idea of the parameter, turns out to be useful.
t:.
By a subdi1.JiBion of the set
a rI B.
Let
M.r
we will mean a collection of a not more
dell) COl {ll}
a
than countable number of sets
lla n lla COl.,
~
D(A)
such that
II
= Ua ll,
a
II a
~
AI ,
'-"T
be the class of all possible subdivisions of
the set
>.(ll)'"
(2.3)
4up
d(A)eD(A)
It may be shown that
101 (ll)
l
II fd(~)
a
A(ll)
P{n(t:. ) > O} •
a
is a measure on
is the principal measure defined by (2.1).
and if
101(A)
(2.4)
M.r
Definition 2.3.
The measure
>'(ll)
stream at
v(ll),
then
determined by equation (2.3) is called the
nell).
The function
called the fUnction of parametel'8 of the random stream
v(ll).
where
III >.(t)dv(t).
paramemc measure of the stream
measure
MA) S 101(ll)
Thus if the stream is finite
is absolutely continuous with respect to
A(ll)'"
and
The value
A(t)
at the point
t
A(t)
n(ll),
satisfying (2.4) is
relative to the
is called the paramete'P of the
t.
We note that in most problems a unique choice of
A(t)
or
101(t) -- deter-
mined exactly up to sets of zero v-measure, is proveded by continuity conditions.
The existence of the parameter
line T. (_,co)
A(t) _ A for stationary streams on the
was shown by Khintchine [1] in which he considered the function
5
e
of parameters relative to Lebesgue measure on
T,. (_,co).
Korolyuk showed (see
3.8 of the main text) that under very weak rest.rietions, for stationary proeess
on the line the intensity and parameter
generalizes this result.
coincide.
A theorem obtained below
However it is necessary to impose certain conditions
on the principal o-field M ,
T
for its formulation.
We shall say that theN is a fundamental. system
sets.in the space
I.
An, k
II.
n,
of disssating
[T,M..rlif
U A+
,
I kC: [0,1... ).
i€Iu,k n1i
,
n,
For any t rI t
there exists n" n (t l' t 2 )
l
2
A k-
n,
IV.
t2
V.
€
An,j'
sucll that
The minimal o-field containing
IA.r
tl
€
An,i'
i '" j.
C is
M•
T
In the case of m-dimensional Euclidean space, the space
eo-field
k}
MT, T· kU An, k.
6
.0 .~, k ri l
n,-\..
€
An, k n
III.
C· {A
of Borel sets, a fw.damental system
successive dissections of
rfl
m
T. R
and the
C may be formed by means of
into a countable number of non-intersecting m-
dimensional, right-angled parallelepipeds
A k
(with corresponding inclusion
n,
and exclusion of adjacent sides), the lengths of the edges tending to zero as
THEOREM 2.1.
a measure
[T ,t.<.r 1,
Let there exist a function of parameters
v(A),
for the random stream n(A)
IA.r
and let
}J(t)
..
except on a set
PRooF:
Let
6
arising from random sets
S
w
}J(t)
relative to
veAl
exists and
A(t)
A
O €
£
relative to
have a fundamental system C of dissecting sets.
Then the intensity
(2.5)
A(t),
C.
Mr'
V(A )"
O
o.
By properties II and III of the fundamental system there
on
6
for
exists a choice of indices
tit
any
k;;e O.
We define random variables on the sets
-
n n(An,k)
A
n,k
=
1
if
n(An,k) > 0
=
0
if
n(An,k)
= o.
,..
We extend the definition of
nn if I
(2.6)
A
(U
For sets
n,
A
nn (A)
(2.7)
i)
to sets
UiE:I An,i
by putting
=
= UiE:In+k
=
nn
;n( U
iE:I
n
An+k,i
A
n,
we have
i)
From property (iv) it follows that with probability one
tim
(2.8)
n-+
=
nn(A)
n(A).
co
By monotone convergence, using (2.7) and (2.8) we find
tim Enn (A) = En (a)
(2.9)
n-+
=
~(A).
co
On the other hand, from the definition
E;n+k(A)
==
!
iE:In+k
Thus from (2.3) and (2.9) it follows that
(2.10)
A(A)
==
~(A)
nn (A)
and (2.6) we obtain
P{n(An+k,i) > o} .
7
e
for any
A
agree on
f:
C.
~.
Using property (v) we see that the measures
Hence
and 1J(t) == vet)
1J(A)
1J(A)
and
A(A)
is also absolutely continuous with respect to
veAl
a.e. (v).
From this result, we obtain a corollary:
.
In order that there exists an intensity function
relative to a measure
dom sets
S(j)'
on
v(A),
[T,M.:r] ,
function of parameters
for the
random stream
n(A)
1J(t)
arising from the ran-
it is necessary and sufficient that there exists the
A(t)
with respect to
holds everywhere except on a set
Ao'
v(A
o)
If this holds, then (2.5)
v(A).
== O.
A number of authors have studied the connection between the property of
being "ordinary" and the strict positivity of the interval between successive
events of stationary streams.
The lemma of Dobrushin cited in 3.8 (see also
Slivnyak (1] and Vasiliev [1)) will serve as an example of such a family of results.
In connection with the generality of the space
T,
on sets of which the
random stream is defined, we introduce a generalization of the ideas of an ordinary stream and of the positivity of the distance between successive events of
the stream.
Defi n1 ti on 2.4.
The random stream n (A) defined on
with respect to the fundamental system
(2.11)
P{n(A k) > l}
.tim .6up P{ (An, ) > O}
n t ~ k
n n,k
where if
P{n(A k) > O} == 0
n,
Definition 2.5.
==
we take
C == {A
n,
k}
[T, M.:r)
is called orodinary
if
0
P{n(A k) > l} I P{n{A k) > O} == O.
n,
n,
The random stream n(A)
defined on the space
regular with respect to the sequence of sets
{A },
k
A
k
C
[T,MTl
is called
Ak+l ••• Uk~ == T
if
8
P{n(~) < ~} ..
tit
~i,oo e
Mr'
1
~i,oo
for all
k
and with probability one there exist sets
~k such that 6i ,oo n 6j ,oo .. ~
c
(1 ~ j),
n(6 i ,oo)" 1,
n(~k) == Ln(~i,oo)·
The following two theorems show that the ideas of ordinary and regular
streams are closely related to each other.
THEOREM 2.2.
If the random stream n(6)
C" {~n,k}
fundamental system
Uk
'\= T,
ative to
PRooF:
is ordinary, relative to the
~ .. Uie~ ~nk,i'
and for sets
>..(~)
the parametric measure
is finite, then
n(~)
~ c ~+l'
is regular rel-
{~}.
The assertion of the theorem follows from
p{ U [n(~"k+t, i) > ll}
ie~
e
I
s
,l
p{n(~I\.+l,i) > l}
ielk,l
p{n(6~+l,i) > l}
s
4U.P
i
p{n(~Ilt+l,i) > oJ
P{n(~
s 4u.p P{
i
-+ 0
l
as
.
-+ co
lJ (~) == >.. (~) <
that
+l,i) > 1}
(6~
P{n(~~+l,i) > O}
ie~,.e
>"(~k)
) > o} •
n· ~+l,i
•
U I
A +D i in conformity with Property III
i e k,l n :(..,
k
Analogously to the conclusion of Theorem 2.1 we show
where
of a fundamental system.
I
•
A.
it
..
co, completing the proof of the theorem.
For the formulation of a second theorem, generalizing the lemma of Dobrushin
proved in 3.8, we introduce the following ideas.
C • {~n, k}
is called homogeneous if
n, k) > l}
P{n(A
of
k.
= Pn (1)
for any
n,
P{n(~
U,
The fmdamental system
k) > oJ .. p (0),
n
i.e. these probabilities are independent
9
THEOREM 2.3.
tit
Let
neal
be regular with respect to the sequence of sets
~ = Ui!I
6 ,i t ~ c 6k+l' ~ ~ .. T formed from the sets of a homogeneous
k nk
fundamental system C c {6 k} and lJ(A.) < CD. Then n(6) is ordinary with
n.
respect to
PRooF:
~
K
C.
It follows from Property III of a fundamental system that
= Ui!Ik,l
a~+l,i. Denote by Nk •l
the number of indices in the set
Ik,l.
The number is finite since otherwise it would follow from the fact that the system is homogeneous that
n (~) =
co.
Further,
(2.12)
~
If
l ..
co,
lJ(A).
then by regularity the left hand side tends to
A(~)
= lJ(6 )
k
which
is possib le only if
p (1)
(2.13)
0:_
n
oVU/l
()
n .. llD Pn 0
= 0•
Equality (2.13) is equivalent to (2.11) in the conditions for a fundamental system to be ordinary, and the proof is complete.
Thus the ideas of an ordinary and regular stream are close to each other.
However they do not coincide, as may easily be shown by considering a Poisson
stream with random parameter
A,
-a ,
peA S x) .. l-x
stream is regular, but not ordinary.
1 > a > 0,
We note that for it
lJ(A)"
x > 1.
co.
This
10
For studying possible applications to random streams arising from random
tit
fields and streams of crossings of hypersurfaces, we give the general ideas of
stationarity.
A stationary stream on the line may be considered as a stream for which the
joint distributions of the variables
cide for any displacement
t
n(A )
i
and
and any Borel sets
n(Ai+t),
Ai.
Here
i .. 1, ... ,m,
coin-
Ai+t .. {s+t: se:A i }.
We notice that the displacements form a group of transformations of the space
T .. (_co,co).
group.
In accordance with this we suppose that the initial space
T
is a
We do not restrict ourselves to the case of commutative groups, but de-
fine left and right translations, putting, respectively,
gA
..
{gt: te:A},
Ag
..
{tg: te:A}.
We shall suppose that the translations preserve measurability, i.e. if
e
then
gAe:M.r
and
Age:Ur.
Briefly we write
AEM.r
T<Ur> .. M to denote this property.
T
A typical example is the group of parallel translations of m-dimensional
Euclidean space
m
R.
Definition 2.6. The random stream neAl defined on [T,MT ) where T is a
group will be called left (right) homogeneous if
and any
i
Aie:Mx
= l, ••• ,m
(i
= 1,2,. .. ,m),
TUr"
M:r
the joint distribution of
coincide with those of
and for any
gET
n (8A i )[n (Aig)],
n(Ai).
In the theory of stationary streams, one of the central results is the
theorem of Khintchine [1] concerning the existence of the parameter (cf 3.8).
The parametric measure of a left (right) homogeneous stream is a left (right) invariant measure (Haar measure).
In this connection it seems natural to obtain a
general analogue of Khintchine' s Theorem from the lmiqueness (up to a multipl1cative constant) of the invariant measure.
This in its turn is connected with
properties of regular invariant measures (cf Ha1mos [1]).
11
We impose a number of restrictions on the group
It
locally compact Hansdorff topological group, and
the compact sets.
T.
We shall let
T be a
At:r the a-algebra generated by
The latter condition will be briefly written
M.r. MT(C) •
The principal assumption made is that for the (right) homogeneous stream,
its left (right)-invariant measure is regular.
a wide variety of spaces.
set is a
G6 •
For example, if
T
This assumption is satisfied by
is separable, then each compact
Consequently, the concepts of Borel and Baire measures coalesee.
As is known (cf. Halmos [I» Baire measures are regular.
spaces
T,
the parametric measure
A(A)
is regular.
measures differ only by a constant multiplier.
variant measure.
A(A) •
(2.14)
Let
Thus for separable
Two regular invariant
veAl
be the regular in-
Then
A'v(A).
By combining (2.14) and (2.4) we obtain the following result,
THEOREM
which
[T
2,4.
Let
T(M.r>;: M.r (e) ;:
,Mrl
measure.
and
A(A)
T be a locally compact Hausdorff topological group for
M.r.
Let
neAl be a left (right) homogeneous stream on
the corresponding regular left (right) invariant parametric
Then there exists the fmction of paTameters of the stream neAl
ative to the left (right) invariant measure
~
v(A),
rel-
exactly equal to the constant
satisfying (2.14).
This theorem generalizes the result of Khintchine [1] mentioned above Con-
ceming the existence of a parameter for a stream on the line
MT is the class of Borel sets.
T;: (-,01)
when
Another example of a homogeneous stream is that
considered by Belayev [8] and consisting of exits from the circle
222
T = {(xl,x ): Xl +x
;: r}
2
2
by a two-cimensional normal process.
homogeneous with respect to translations on the circle
measure
v (A)
is the length of the arc
A on
T,
T
This is
and the invariant
12
In §ll.l of the main text, special attention is paid to questions of def-
tit
inition of conditional probability relative to zero probability conditions such
as a "crossing of a level
u
at a time
tit,
and so on.
The following general
construction avoids the necessity of the passage to the limit.
random stream on
adjoin an event
events occur.
Then asA
s
is empty.
As
derived from the random sets
or its complement
As
A
SE:S,
~.
s
We suppose, of course, that the set
Definition 2.1.
Sw(A)
ure
",(A).
A(~),
SE:S
It gives rise to a random stream
exists a function of parameters
).(t)
for which
GIl
n(~·A)
n(~).
).,(~)
for the stream
moreover we may take
).,(t,~)
GIl
A
S
of the stream
n(~)
s
n(~),
S
W
dif-
takes place.
which may be
relative to a meas-
n(~,A)
is the parametric measure relative to
the function of parameters
n [S \s]
S
We suppose that there
Since the parametric measure of the stream
where
~
the set
W
tE:T.
is a random set in the sense of
regarded as a "thinned" substream of the stream
tit
of each point
We consider the random subset
consisting exactly of those points
GIl
SE:S w we
occurs if there are points of
s
s . in the neighbourhood
To each
~t
For example let there be a neighbourhood
The complementary event
S (A) c S
S.
w
be a
depending on which of these two
one may take the event that, for
ferent from
GIl
(T,AI]
.'1: ,
n(~)
Let
n(~,A)
",(~),
is
).(A,A) S
there exists
relative to
",(~)
and
).,(t,A) S A(t).
Definition 2.7. The conditional probability of the event A relative to the condition that an event of the stream
n(~)
occurs at
t,
is defined by the re-
1ation
(2.15)
....
P{A\t€S}
w
•
).,(t,A)/).,(t) •
As in Chapter 11, the conditional probabilities (2.15) have a natural statlstical interpretation for homogeneous ergodic streams.
I t is exactly such a
13
e
wide definition of conditional probability which made it possible to examine
asymptotic properties of envelopes, about which more detail will be given in §3. *
The calculation of the high moments
E{n(li l )n(li 2 )}
E{n (li)}k
and the covariance
etc. ate conveniently carried out by means of multi-dimensional
analogues of the principal and parametric measures.
An account is given below
of the general construction given by Belayev [7,9].
set
Let
n(li)
S.
We consider the direct product ofk
w
(t ,t , ••• ,t ),
k
l 2
of points
k
T =
TxXx... xX.
tiET,
In the space
k
T
Sw*
k
k
the set in
T
~
sai
~
Saj'
i
~
T,
formed by collections
We denote this product by
M.rk = tf.rxMrx ••• xM.r,
we introduce the a-algebra
li 1 xli 2 x ••• xli ,
k
S.
Ol
where
t.iEMr.
We denote
j,
sai E
SOl'
k
dif-
This means that all possible choices of
points are taken without repetition.
we have
arising from a random
of points obtained by all possible choices of
ferent points from the random set
k
spaces
i - 1,2, ••• ,k.
generated by rectangle sets of the form
by
[T ,Mr],
be a random stream on the space
Thus for any point
i,j
= l, ••• ,k.
shall call the k-th stream of the stream
The set
k
n*k(t. ),
duced is random, and gives rise to random stream
(sal' ••• 'Sak)
li
k
E
E SOl*k
sw*k
intro-
Mr k,
which we
n(A).
Using the idea of a k-thstream, it is easy to introduce the multi-dimenLet
sional analogue of the intensity parameter.
and
k
\lk(li )
the k-fold product measure of
\I (li)
be a measure on
on the space
\I
k
k
[T ,M ].
T
Definition 2.8. The k-th principaZ measure of the random stream n(t.)
from a random set
on the space
w
measure of its k-th stream n*k(L\)
(2.16)
S
arising
is defined to be the principal
given on
k
k
[T,Mr]
by the formula
k
llk(t. )
If the measure
to
[T ,M ]
T
[T,Mrl
\lk(t.k ),
is a-finite and absolutely continuous with respect
then by the k-th intensity function of the stream n(t.)
(The third appendix.)
relative to
14
the measure
k
stream n*
v (6) ,
(~)
11c (tl' ... , t k )
we mean the intensity function
relative to the measure
k
v
(~).
called the k-th intensity of the stream n(6)
corresponding to the points
t , ••• , ~ •
l
The value
of the k-th
11t(t , ••• , t )
k
l
relative to the measure
is
v(6),
The k-th intensity function is charac-
terised by the relation.
(2.17)
Definition 2.9. The k-th parametric measure of the random stream n(A)
from the random set
SfA) on the space
of the k~th stream n*k(6k )
[T,J\]
arising
is the parametric measure
k
).k (6 )
[Tk ,M k] • If \ (6k ) is a-finite and absoT
lutely continuous relative to the measure vk(Ak ), then by the function of k
on
pczttameters of the stream n(6)
e
tion of parameters
ure
k
v (6k) •
stream n(6)
relative to the measure
).k(tl, ... ,t )
k
The value of
of the streamn*k(Ak)
\ (t ' ••• , t )
k
1
relative to the measure
The function of
k
v(6)
veAl
we mean the func-
relative to the meas-
is called the k-parameter of the
corresponding to the points
t ... ~.
l
parameters is characterized by the relation
(2.l8)
We note that from the existence of a fundamental dissection of the space
[T ,Atr],
there follows the existence of a fundamental dissection of
Hence from Theorem 2.1, reformulated to the
CoROlJ.ARY 2.2.
k
stream n*k(6k ),
In order that the random stream
possess a k-th intensity function
n(A)
on
k
k
[T,Mr].
we obtain
[T ,~]
should
11t(t , ••• ,t ) relative to the measure veAl,
k
l
it is necessary and sufficient that there exist the function of k-parameters
).k(tl""'~)
relative to
v(6).
Then
15
(2.19)
up to a set
A
k
o
k
MT,
€
k
k
v (A o
) • o.
In a number of problems of interest in the supplement, we may calculate the
high moments of the number of events of the stream
A,
and also the mixed moment
a1
f{nCA 1)n(A 2)},
n
n(A)
a2
falling in an interval
..
41. The following
theorem connects the moments with the It-th principal measure of the stream.
THEOREM 2.5.
lJlt(Ak )
k· k
Let
l
= 1,2,...
+. u+kt where k i
be the k-th principal measure of the stream
are integers and let
n(A)
on
[T,M.r].
Then for
the rectangle sets
-It
A
•
Al x ••• x Al
x ••• x
~"-_.::.,...<'
It
where
Ai
(2.20)
PRooF:
l
€
MT ,
l
ki
(a l
x ... x
all
~
times
Itl
times
we have the identity
}
E{ n n (n(A t ) - j + 1] •
i=l j=l
•
We denote the coordinaties of the random points of the set
Si,l' ••• , si,ni'
ai
i · l, ••• ,l,
€
MT ,
ni
= n(Ai)'
Sji ~ srp
S
w
when
by
j ~ r
k
occurring in A is equal to
w
the number of all possible ordered choices without l'epetition, k
from each set
i
or
Sw
i:l p.
n Ai
The number of points of the set
=
{Si,l' ••• 'Si,ni}'
from the set
Sw n Ai
= l, ••• ,l.
i
of size
k
k
i
S "It
The number of different choices
without repetition is
i
n
(nCal) - j + 1].
j-l
Since the sets
Ai
are mutually disjoint, the points of the set
obtained from any combination of choices
(si i , ••• ,si i
, 1
' kl
)
S "k n
w
Ait
i = 1,2, ••• ,l
are
16
,I"
... ,
Ak
The total number of points of the set
l
ki
rr
rr
n S *k
w
is
(n(A ) - j + 1).
i
i-I j-1
The identity (2.20) follows from (2.16), i.e. the fact that the k-th principal
measure
in
~
llk (A)
k
S!AI*
is the expectation of the number of points of
occurring
Ak •
We formulate two important cases of Theorem 2.5 as a corollary.
CoROUARY 2.3.
i r/: 1,
(2.21)
k" k l ,
If in the conditions of the previous theorem,
A,. AI'
llk (AxAx ••• xA)
'"--,,,,.... ~
ki
III
0,
then
..
En(A)[n(A)-l) ••• [neAl - k + 1].
k-times
If in the conditions of the previous theorem k
i
III
i " 1, ••• ,l.,
1,
k· l
then
(2.22)
As on the one hand functions of
k
parameters are generally easier to de-
termine than k-th intensity functions, and on the other hand these functions coincide, we give the following result from (2.19), (2.21), (2.22).
CoROUARY 2.4.
to the measure
v(A),
Let the function of
k
parameters
exist for the random stream neAl
k-th order factorial moment of
neAl
Ak(tl"u,~) relative
on
and the mixed moment for
[T,~].
Then the
neAl) ... n(At)
17
are given in terms of the function of k-parameters by the relations
(2.23)
J (6)
•
k
En(6)[n(6) - 1] ••• [n(6) - k
·
f
Ak(tl,···,~)dv(tl)
ti~6,ti"tj
+ 1]
••• dv(t k )
and
Ct (6 1 ,···,6t )
( 2.24)
•
En(6 1) ••• n(bt )
. I
At(t!' ... ,tt) dv (t l ) ... dv(tt)
t e6 , l1 06j"1I<jl
i i
i
1'ij, i-l, ... ,t
We 1llustrate the above results by examples.
For the first example, consider a stream of renewals (cf. Cox and Smith (1»,
e
n(6)
T
1
=
def1ned on
t
1
iables,
-t _ ,
i l
T
= (O,~)
i · 1,2, ... ,
P{T S x)
i
= F(x),
by random sets
where
F(o+)
T
i
= O.
Sw· {til,
o· to < t 1 < t 1+ ,
l
are mutually independent random varLet
B(t)
= En(O,t)
when the k-th prin-
cipal measure for an element of volume in the neighbourhood of the po1nt
k
(tl' ••• '~) e R,
where
t l < t 2 < •••
<~,
is
(2.25)
If there exists the probability density
h(t) • H'(t).
f(t)
= F' (t),
then there exists
In this case the k-th intensity function relative to Lebesgue
measure has the form
Corresponding to (2.23) and (2.25) we obtain the k-th factorial moment of the
number of renewals on
(0, t)
as
18
En(O,t)[n(O,t) - 1] ••• [n(O,t) - k
+ 1] • k!
~*(t)
where
The second example concems the intersection of two levels by the sample
~ (t)
functions of a process
is given a random set
•
We suppose that on T = R\ uR ' 2 '
SI.ll = {tl,i} u {t2 ,i}
where
ti,j
E:
R'i'
R' i • (-eo , co)
i · 1,2.
R'i
may be interpreted as two parallel horizontal lines passing through the points
(O,U ).
i
Denote the l-intensity functions by
2-intensity functions by
ij
AZ
(ti,t ),
j
t
i
E:
Ai(t )
i
,
R i' t
t
j
E:
i
E:
R'i
R'j.
and the
We note that these
intensities coincide with the functions of parameters.
In calculating factorial
moments of the second order, we take into account that
Txt
planes
R' i xR ' j •
points
0 < ti,j < t,
is formed from four
Hence from (2.3) t the second factorial moment of the number of
i
= 1,2
is
i,j=l,2
It is useful to keep in mind that the factorial moments may be used for an
approximation to the probability of occurrence of at least one event of the
stream n
= n(~).
From calculation of the first three factorial moments we have
m«x[O, En -
~
~ En(n-l)] ~ Pin> o}
min[l, En, En -
i En(n-l) + ~ En(n-l)(n-2)].
Analogous inequalities may be written down by calculating factorial moments of
higher order.
We consider a particular question about the calculation of the k-th intensity for the "summedu stream
19
m
nI (6)
= l
ni (6)
i=l
when the single streams
sities
llktl(tlt •••• ~)
two-intensity of
ni (6)
are mutually independent and have k-th inten-
relative to the measure
n (A)
v(6).
The one-intensity and
exist and are given by
I
m
.
m
=
l
ll2.i(t l .t2 ) +
i=l
l
lllt i (t l )lll,j(t2 ) •
i#j
Finally in the case of the k-intensity we have
r
k l +· ..+km=k
k ~ 0
~l,l(tl,l,···,tl,kl)
••• ~
. K ,m (tm, l,···,tm, k)
m
m
i
Where the summations are over all possible partitions of the set
disjoint subsets
(tl, ••• t k )
into
(t l l, ••• ,t l k) ••• (t l, ••• ,t k). If the subset corre,
, I
m.
m, m
sponding to the i-th stream turns out to be empty for such a partition, (ki-O),
then
llO,i
= 1.
20
References
The following references which are quoted in this appendix are extracted
with the same numbers from the complete reference list for the (translated)
book.
AMBARTSUMYAN, R.V. [1]
DAN
Al'm.
"On a relation for stationary point processes",
SSR 42 1966, 141-147.
BELAYEV, Yu. K. £7]
"On bursts and shines of random fields",
DAN SSSR 176 3(1967), 495-497.
---- [9] "On properties of random streams"
Tear. Vert i Prim. 13 3(1968), 578-580.
BEUTLER, F.J. and LENEMAN, O.A.Z. [1] "The theory of stationary point processes"
Acta Math.~ 116 3-4(1966), 159-197.
COX, D.R. and SMITH, W.L. [I]
FIEGER,
RenetJal.
Theo1!Y~
Sov. Radio, M., 1967.
w. [1] "A generalization of Palm's formulae for a general process of
calls" Math. Scand. 16 2(1965), 121-147.
HALMOS, P.R. [1]
Measure
Theorty~
Van Nostrand, 1950.
KERSTAN, J. and MATTHES, K. [1] "Stationary random point series II",
Jahr. D.M.V. 66 (1964), 106-118.
KHINTCHlNE, A. Ya. [1]
Mathematical. Methods in the Theory of
Queueing~
Griffin, London, 1960.
LEADBETTER, M.R. [4] "On three basic results in the theory of stationary point
processes", Proc. Amer. Math. Soc. ~ 19 1(1968), 115-117.
McFADDEN, J.A. [2] "On the lengths of intervals in a stationary point process",
J. Roy. Statist. Soc (B)~ 24, 1962, 364-382.
MATTHES, K. [1]
"Stationary random point series I",
Jahr. D.M.V. 66 (1963) 66-79.
MECKE, J. [1] "Stationary random measures on locally compact Abelian groups",
Z. Wahr. 9, 1(1967) 36-58.
SLIVNYAK, I.M.
"Some properties of ordinary stationary streams of events",
Prim.~ 7 (1962) 347-352.
Tear. Vert i
21
VASILIEV, P.I. [1] "On ordinary stationary streams",
Uchen. Zapic. Kishinev. Gos. (Jni,v •., 82 (1965) 44-48.
ZITEK, F. "On the theory of ordinary streams",
Czech. Mat. r1ouzrn. 8, (1958) 448-459.
TINer
ASSIEliP
Department of Statistics
University of North Carolina
f4CPo,,'f1w,~ ailtl, }le.lft Qal'eli:fta
'"
I~b.
GltOUP
I
Elements of the general theory of random stream of events
•• O••C'''''T.va HOTal
(ty"..,,.,,..
_II liteM'" . . . .)
,
Technical Report
i
Malcolm R. Leadbetter
1
:•• "C"O"T OAT.
I
August, 1970
: ... CON TItACT 0 . . . .AHT HO.
.
j
NOOl4-67-A-032l-0002
i,
.. -"cuac,. NO.
NR042-2l4/l-6-69 (436)
Institute of Statistics Mimeo Series
No. 703
c.
10• ...-T"'."'T'OH 'TAT"'."'T
. i .
Unlimi ted
I ••U ............HTA• ., NOTU
U. '-NIO""'G M..... TAft., ACT.V.T.,
Logistics and Mathematical Statistics
Branch Office of Naval Research
n
-,-
n r.
~
')n~lln
This report is a translation of the second of three appendices by Yu. K.
Belayev to the Russian edition of the book Stationary and X'eZated stoahastic
ppocesses by Harald
Cram~r and M.R. Leadbetter.
theory of point processes with special
~mphasis on methods useful in dealing with
"crossing problems" for stochastic processes.
e
The contents concern the general'
•
I
~------OD
t'~~M •• 1473
UNCLASSIFIED
se~urit)' Cl..."UicallOn
•
•••
KCY WORO.
•
&' ... 1< A
"OL&
&oINIC
'" 'HI< •
"OLe
W'f
IIIOLl[
"'T
c:
WT
Stochastic processes
Point processes
.
Random streams
Zero crossings
....
,
•
.
.
•
,.. .
,
UNCLASSIFIED
Securit, CI_utlie_Uon
- -_.
__._--