DISTANCES OF RANDOM VARIABLES AND POINT PROCESSES
by
D.J. Daley
Department of Statistics (I.A.S.)
Australian National University
Canberra 2600 Australia
ABSTRACT
The paper is basically a discussion and survey of some
literature concerning the concept of a distance function for random
variables and point processes.
v
Work of Kalzanov on the closeness of
renewal processes to Poisson processes is outlined, and there are
comments on various possible comparison methods for point processes
as given by Whitt.
Keywords:
Point processes, renewal processes, distance functions,
metrics, comparability of processes.
Research done whilst a visitor in the Department of Statistics,
University of North Carolina, Chapel Hill, and supported by the
Office of Naval Research under Contract NOOOI4-75-C-0809.
- 1 -
1.
INTRODUCTION
When are two point processes close to one another?
when are two stochastic processes close to one another?
Indeed,
These notes
are basically intended as a review of some of the literature concerning
the proximity of one point process to another, and necessarily therefore
also includes notions of one point process being in some sense larger
or thicker than another.
Since part of the difficulty in defining the distance
between point processes may be seen in defining the distance between
two random variables (as distinct from processes, which are indexed
families of random variables), we start with some ideas related to the
-e
v
distance of two random variables, then review some work of Ka1zanov
measuring the closeness of a stationary renewal process to a Poisson
process, proceed to ideas of comparison of point processes, and
finally come toa (brief) section covering some ideas on distance
functions for point processes.
2.
DISTANCE BETWEEN RANDOM VARIABLES
The distance between two points in d-dimensiona1 Euclidean
space
is a familiar enough idea, so that the distance between
d
realizations of two R -valued random variables (r.v.s)
defined on some common probability space
(n,F,p)
X and
Y ,
say, is sensibly
given by
!X(w) - Y(w)
I
(2.1)
- 2 -
for each
w in rl,
with
d
R
function in
1·1
here denoting the usual distance
As soon as we move away from just realizations
of two r.v.s and seek to measure the closeness or otherwise of the
r.v.s
~
mapping
se, by which we mean the measurable functions
rl
d
into
,
R
X
Y
we are forced either to perform some
operation on the random function
IX(w) - Y(w) I ,
or else to have
recourse to some other notion not necessarily related to the
distance of the realizations of the two functions involved.
Clearly the function
dE(X,Y) ~
J
(2.2)
IX(w) - Y(w) Ip(dw)
rl
is an obvious candidate for measuring the distance between the r.v.s
-e
X and
that
Y.
Provided the rovos
dE(X,Y) <
X and
Y have been given, and
the definition (2.2) is not easily disputedo
00,
However, it is not uncommon to be given not two rovos
but their distributions in
d
R
:
X and
for the present it will be enough
to consider the case of the distribution functions (dofos)
G
Y
of R1-valued rovos, so that for example
F
and
F(t)=P{X~t}o
One
metric on the space of dofos with finite first moment is the function
00
-J
IF(t) - G(t) Idt
(203)
0
_00
Manipulation shows that
ddf(F,G) = dE(X*,Y*)
y* , with dof.s
G respectively, defined by
F
and
-1
X*(w)
where
and
F
G
-1
and
and
for the rovos
Y*(w) = G (U(w»
-1
G
U(w)
X*
(204)
are the functional inverses of the dofos
is a rovo on
(rl,F,p)
with the uniform
F
and
- 3 -
distribution, so that
=t
p{U(w) $t}
for
0 $ t $ 1.
Thus, (2.3)
may be expressed as
1
ddf(F,G)
= dE(X*,Y*) = f IF-1 (u)
o
When the d.f.s
distributional sense,
then
X*
If
F and
of r.v.s
and
F(t)
~
G(t)
(all t in R),
and hence
=
lEX - EY\
= EY
- EX •
G cannot be ordered in this way, or if the pair
is not equivalent in distribution to
dE(X,Y)
(2.5)
G are partially ordered in the
F $d G , meaning that
$
Y*,
a.s.
ddf(F,G)
F
- G-1 (u) Idu •
(X*,Y*) ,
also includes some measure of the dispersion
For example, when
F
variances
0
1
and
extreme from
2
(X*,Y*)
and
< 1
f
1
o
then
of
X and
dE(X*,Y*)
= (1-0)(2/TI)~.
At the opposite
is the pair
for which it may be shown that
IF-l(u) - G-l(l-u) Idu = 2Elz-~1
= Elx-~I+
~(F(t)
+ G(t»
EIY-~I
(2.6)
~
where the r.v.
Z has
median of
Intermediate between (2.6) and (2.5), when
Z.
as its d.f. and
Yare independently distributed r.v.s with d.f.s
E\x-YI
F
again vanishes (like (2.6) but unlike (2.5»
X = Y = constant a.s.: (2.5) vanishes if and only if
and
X and
G,
X
=d Y
dE
or
Rd -valued r.v.s, relates to the need of an analogue of the
construction of
(1981».
is a
if and only if
A major difficulty in extending the definition
to
(X*,Y*) •
Y •
G are normal d.f.s with zero mean and
= (F-l(U(W», G-l(U(w»)
(F-l(U(W», G-l(l-U(W»),
(X,Y)
Consider the following (cf. also Daley
- 4 -
Define
Problem 1.
distributed uniformly over a circle of unit area with centre at the
origin, and a square of unit area with centre at the origin,
respectively.
Amongst all r.v.s with these two specified distributions,
is there a construction of r.v.s
X*, y*
such that
A strategy that avoids the sample path problem, given two
distributions, is the use of metrics on the distributions on the
state space of the realizations induced by the r.v.s.
Thus we have
the variation metric
-
~
J
IF(dt) - G(dt)
I
for Rd-valued r.v.s with distributions
for integer-valued r.v.s
~
(2.7)
d
R
X and
L Ip{X=i}
and
F
G ,
and in particular,
Y ,
- P{Y=i} I
i
(2.8)
for appropriately defined r.v.s
in general different from
X and
(Xt,yt)
(X*,Y*».
{and this construction is
Also, for real-valued r.v.s
Y , we have the supremum metric
d
sup
I
(X, Y) - sup p{X ~
t
the Levy metric and others.
d - p{y ~ d
I,
(2.9)
- 5 -
3.
DISTANCE BETWEEN POINT PROCESSES:
INTRODUCTORY COMMENTS
Realizations of point processes on
R can be described
either as counting measures or by the distances of the nearest point
to the right of the origin and by the lengths of the intervals
between successive points enumerated respectively on each side of
that point.
For the present discussion we confine attention to
point processes on
R+
=
(O,~)
,
and assume without essential loss
of generality that every realization has
the counting measure.
point processes.
We use
N(R+)
N, N , N ,...
l
2
~
1
where
N denotes
as generic names of
The two descriptions referred to above are related
via
°,
and
xn = Sn
- S
(n = 1,2, ••. ):
n-l
time interval, and
= inf{t: N(O,t]
S
n
{x : n
n
between points so long as
n
~
.
'
n
(i=l)
and
{y}
n
(i=2) ,
the successive intervals
otherwise we may set
Given two point processes
{x }
(3.1)
,
is the forward recurrence
= 2,3, ••. }
S <
~n}
N
i
(i
= 1,2)
{x
n
- y }
n
=
~
, with intervals
the discussion of section 2 leads
to consideration as "natural" distance functions for
either functions of
x
INl(t) - N2 (t)
I
N
l
and
N
2
or else of the sequence
The simplest constructions of point processes usually
n
proceed via interval properties rather than counting properties
(though as a counter-example of this conventional construction,
recall Lewis and Shedler's (1979) discussion on realizations of
inhomogeneous Poisson processes).
N
l
and
elements
N
2
Accordingly, the distance of
may be studied via functionals of
( x* y*)
n' n
{x* - y*}
n
n
where the
are constructed successively from the sequence
- 6 -
{U*}
n
of independent identically distributed (i.i.d.) r.v.s uniformly
distributed on
N
i
(0,1).
Such a construction for renewal processes
is sensible, and also for point processes whose intervals are
stochastically monotone Markov chains or even certain semi-Markov
processes (see Sonderman (1980» •
However, when the successive
{X } are not monotonically dependent on
n
members of the sequence
their predecessors, there may not necessarily exist a mapping
linking
{x*}
n
and
4.
-e
Let
{y*}
n
that is unequivocally the optimum one.
THE MARGINAL COUNTING DISTRIBUTIONS
N
1
be a stationary renewal process for which
E N (O,l] = A and the lifetime d.f. is
l
are i.i.d. r.v.s with d.f. F and
=
AJ
F, so that
{Xn : n
=
2,3, ••• }
t
o
(4.1)
(l-F(u»du
N1 with the most
Kalzanov (1970, 1975a, 1975b) aimed to compare
"typical" (7) of point processes, namely, he took a Poisson process
N with rate parameter
2
A,
so that
(4.2)
His comparison was based on the probabilities
Pk(t)
k;?: 1
and
= P{Sk~t<Sk+1}
=
P{N1 (0,t]=k}
=
jt(F(k-1)*(t_u) - Fk*(t-u»dF (U)
1
o
=
Ajt(F(k-1)*(u) _ 2Fk*(u) + F(k+1)*(u»du ,
o
provided
{vk(e)}
for
k
=
0 ,
(4.3a)
- 7 -
(4.3b)
Kal~anov (1975a) claimed that
(k = 0) ,
(4.4)
(k
where
PI - supIF(t) - Fl(t)
= 1,2, ••• )
,
I
(4.5)
t
so that
Nl
P
1
=
i f and only i f
0
is a Poisson process.
and shows that
IPk(t) - vk(t)
At S R.n(l/ P )
At > R.n(l/P 1 )
l
Problem 2.
~
X + Y < b}
and hence
~
I
k
is bounded as asserted,
pda
~
X and
Y and
a < b ,
X < b}"
(*)
Investigate whether the bounds at (4.4) hold for
Can the coefficient of
P
1
= 0,1,2,3,
is erroneous, being based on the false
assertion that for non-negative r.v.s
"Pda
- e -At
= F1 (t) = 1
He does indeed establish (4.4) for
sup
but his proof for
-e
F(t)
k
~
4 •
be tightened?
...
It was also claimed by Ka1zanov (1975a) that
Problem 3.
(k
= 0)
,
(4.6)
(k = l, 2, ••• ) ,
00
AJ
where
AP
on
hold for
ok
2
=
o
IF(t) - F (t)ldt.
1
k
~
Investigate whether these bounds
3 •
Since these publications may not be so readily accessible,
an outline of the (correct) part of their derivation is given later
in this section (see around (4.15)).
- 8 -
Kalzanov's motivation is clear enough; he is comparing
with a Poisson process when the lifetime d.f.
forward recurrence time d.f.
F
l
F
N
l
and stationary
are close, and hence
F
is close
to the exponential d.f., because as shown in Kalzanov (1970),
F1 (t) - (1 - e
-At
) =
Jt (F1 (t-u)
- F(t-u»Ae
-AU
du,
(4.7)
0
so
supIFl(t) - (1 - e -At ) I
t
t
f IF I (t) - (1 - e -At Idt
0
and
sup\F(t)
(1 - e -At )
~
Pl(l - e
S
P (l - e
2
I ~Pl
-At
-At
)
~
PI
(4.8)
)
~
P2 '
(4.9)
+ sup IFl(t) - (1 - e -At ) I
t
t
=
2P l .
(4.10)
However, the bound (assumed true) is somewhat larger than is probably
likely, because it exceeds
'p
1
for
PI
~
.074778, as in the table below:
1
.01
.214207
.02
.372962
.03
.510787
.04
.635020
.05
.749146
.06
.855219
.07
.954593
.074778
Suppose the lifetime d.f.
1.0 7
6
F has a finite second moment.
It may then be of more interest to compare the marginal probabilities
or even the simpler probabilities
(4.11)
- 9 -
N~(.) with F as the d.f. of Xl as well as
of a renewal process
X2 ,
o
x3 ' ••• ,
Vk,y(t)
with the corresponding probabilities
their first two moments.
Y2' Y , •••
3
matching
to denote the standard normal d.f., and
P~(t) ~ V~,y(t).
X2 ' X , •••
3
in
Then for moderate (to large) values of
which is possible for all
t
~
o
2
=
var X
(n
n
k
~
2) ,
0 , and, of course, then
(P~(t) may have discrete jumps, but the approximation
will remain valid.).
X
n
or
of a stationary renewal or renewal process respectively with
gamma distributed intervals
using
Vk,y(t)
Indeed it is immediately obvious that, provided
has a finite third moment, the Berry - Esseen bound may be applied to
bounding
Ip~(t) - V~,y(t)1
uniformly in
t,
~ 0 as k ~
00
like
O(k-~) - though large k is needed before this is necessarily smaller
than one.
Recognizing from (4.3) that for
t
and that
P~(t)
~
1 ,
0
=
J Pk- l (t-u)dFl(u)
=
J
o
00
(Fk*(t) - Fk*(t-u»dF(u) ,
(4.13)
o
another possible approach to bounding
function of
k
F which behaves like
Pk(t)
is to use the concentration
k-~ for increasing k (see section
2.2 of Hengartner and Theodorescu (1973».
Alternatively, smoothing
inequalities on transforms may be useful (cf. Paulauskas (1971),
Petrov (1975».
For approximation purposes, observe that when
~(.)
denotes
the standard normal probability density function,
supl~(t) - O-l~(t/o)1 = (1-0')/0'
t
(4.14)
- 10 -
where
a' = min(a,a-1 ) •
..,
Kalzanov's (1975a) partial derivation of (4.4) comes from
(4.8) and (4.10) applied to
t
Pk(t) - vk(t) =
which holds for
k
J Pk- l (t-u)dF(u)
o
= 2,3, ••• ,
t
-
J vk _l (t-u)dG(u)
(4.15)
0
where we have written
G(u)
=
1 _ e- AU
and, using convolution notation,
2*
P1 (t) - v1 (t) • (F1
- F*F )(t)1
- (G - G )(t)
·
(4. 7) can be writ ten as
F1 - G· (F1- F) *G.
Then
Generally,
k
- vk+1 (t)1 ~ sup Ipk (u) - vk (u )1 + (G *+ G(k+l)*)*(GO+G)P l
u~t
so
.:0
~ Pl(F + 3G+4G2*+••• +4Gk *+ 3G(k+1)*+ G(k+2)*)(t) •
l
The coefficient of
k = 1,2,3.
PI
For all k
at (4.4)
~
and finite
12
t,
for
P1
~
.074778 so it holds for
the coefficient of
is at most
PI
PI (F1 (t) -G( t) + 4At)
in the
~
PI (PI+4At) •
.,
Kalzanov (1975b) endeavours to bound the difference
of the probabilities
renewal processes
Ni
on the lifetime d.f.s
(i=1,2)
F(i) (e) ,
of stationary
in terms of the metric
suP\F(l)(t) - F(2)(t)
t
assuming these to be absolutely continuous
d.f.s with bounded densities and finite second moments.
The argument
I
- 11 resembles that outlined above, and has the same appeal to the
erroneous statement (*) .
Az1arov and Ka1~anov (1976a) refers to Ka1~anov (1975a)
in deriving bounds on the difference of the stationary waiting time
d.f. for a GI/M/k from the same d.f. in an M/M/k queue with the same
arrival and service rates.
Bounds are given for each of the cases of
an IFR and DFR inter-arrival d.f. in GI/M/k.
Details of the proof
are omitted, so it is not clear whether (*) has been used. In
..,
Az1arov and Ka1zanov (1976b) the loss probabilities in pure loss
GI/M/k and M/M/k systems are compared, but only the abstract in
Mathematical Reviews is available.
While for example the sequence
-e
{Ok}
provides some
{P }
k
information on all the one-dimensional marginal distributions
as compared with those
could consider
{Ok}
{v }
k
of a Poisson process - or, indeed, we
for any two point processes via
{supIPr{N (0,t]=k} - pr{N 2 (0,t]=k\} ,
1
t
we may prefer, as an indicator of the "distance" between the processes
N ' some function that discounts behaviour for either large
i
large
°<
Z
k
or both, such as (for appropriate
£ >
° and
z
t
or
in
< 1)
00
L
zk
k=O
f
00
°
e-£tI Pr {N (0,t]=k} - pr{N (0,t]=kldt •
2
1
Alternatively, interest in large
k
may be measured by for example
sup k~lp~l)(t) _ p~2)(t)1 •
t
Trivially, since there exist point processes with distinct
probability distributions whose finite-dimensional distributions agree
up to some finite order
single member of
{Ok}
r
(cf. Szasz (1970) and Oakes (1974», no
,nor even the entire sequence, nor any function
- 12 -
based on finite dimensional distributions of bounded order, will be a
metric on the space of point processes.
Of course
supIF(l)(t) - F(2)(t)
I
t
is a metric on the space of renewal processes, as these are characterized
by their lifetime d.f.s.
An entry in Mathematical Reviews refers to Bo1'sakov and
Rakosic (1978) as having studied in detail Poisson and c1ose-to-Poisson
point processes.
The meaning of the phrase in the context can only be
guessed at.
5.
COMPARABILITY OF POINT PROCESSES
The idea of studying the distance of one object from another,
is an endeavour by implication to compare the two objects.
Comparisons
in stochastic analysis usually refers to the partial ordering of the
objects, and consideration of the variety of definitions proposed for
the comparability of point processes may be helpful.
a partial ordering on the space
Let
~
denote
X of (real-valued) r.v.s on (Q,F,p):
Stoyan (1977) surveys some of the ideas and properties of some partial
orderings on
X, mostly expressed in terms of partial orderings on
functions of the probability measures
P
x
induced on
(Q,F, P)
via
XEX.
For point processes, work of Schmidt (1976) that Stoyan reviews
is covered in Whitt (1981) whose survey is the source of the definitions
below; Whitt's numerical numbering of definitions is replaced by a more
suggestive notation.
write
N(O,t]
= N(t)
terminology:
if
say that
is
N
1
The definitions are interspersed with some comments;
~ -thicker than
N
1
for brevity.
~
N
2
for some partial ordering
~-thinner
N •
1
Stoyan (1977) has an appealing
than
~
on point processes,
N2 ' or, equivalently, that
N2
is
- 13 -
Define
(5.1)
the forward recurrence time d.f. at
Ht
history
of
up to time
N(o)
t
conditional on the entire
t.
Then say
(5.2)
if
x > 0
holds for all
H
and all histories
t
In other words, the
.
N
conditional forward recurrence time is stochastically larger in
than
N
for all realizations on
2
(O,t)
and for all t .
In the particular case that (say)
with lifetime d.f.
process with
F, and
EN (O,t]
2
sup
N
2
= A(t),
l
N
l
is a renewal process
is a possibly inhomogeneous Poisson
(5.2) implies that
(F(u+x)-F(u»/(l-F(u»
:5: 1 - e-(A(t+x)-A(t»
• (5.3)
O:5:u:5:t
This condition is more easily written via the representation, valid
f or a 11
u
F(u) -- 1 - e-~(u)
F()
u < 1 , t ha t
f or whi c h
non-decreasing function
~(o)
,
~(u)
-+
00
(u-+ oo )
for some
Then (5.3) is
equivalent to
~(u+x)
-
and it follows that
set is non-empty.
such
~(u)
(all 0 :5: u :5: t) ,
:5: A(t+x) - A(t)
A(t+O)
=
00
for
inf{t: ~(t+O) > ~(t-O)}
if such
This would lead to a Poisson process exploding at
t , as would be the case also if
~(x)
-+
00
for
x -+ x
o<
00
•
Eliminating these cases, we are left with the (interesting) case that
A(t) :5: Ct
(all t) for some finite constant
provided the limit exists.
h(u)
(all u)
met)
sUPO<u:5:t h(u) , and then we may take
C
~ limv~O(~(u+v)-~(u»/v
-
Indeed, it suffices to write
- 14 -
=f
A(t)
t
o
m(u)du •
This is the smallest function
A(e)
satisfying (5.3); Miller (1979)
also showed that such a nonhomogeneous Poisson process can be constructed
so that
N will bound the renewal process
2
=1
F(x)
- exp(-
in the sense that
Poisson process
N
l
NO
f
with
with lifetime d.f.
x
o
h(u)du)
N2 •
~
N
l
He also shows that an inhomogeneous
t
ENO(O,t] • ~ (infOsusxh(u»dx satisfies
Returning to general point processes
N
l
and
N2 ' suppose
that their realizations satisfy
•
for all 0 < x <
00
,
is also a jump in
as in
N •
l
i.e., every jump in the counting function
N (x,w) ,
2
and its size in
N2
Nl(x,w)
is at least as large
When (5.4) is satisfied, write
(5.5)
{SCi)}
n
since the definition is precisely that the sets
(3.1) satisfy such an inclusion relation.
by any thinning operation, then clearly
on which
If
N
l
N
l
and
~
If
N
l
as defined by
is obtained from
N
2
~~ N2 •
N ' then (there exists a common probability space
2
N may be defined such that) * Nl
2
converse need not hold.
However, if
Nl
~
~
N2 and both
N2 ' but the
Ni
are Poisson
processes, then the converse will hold, because the intensity measures
Ai
will satisfy
*This
~d
Al(A) S A2 (A)
for all bounded Borel sets
parenthesized remark is to be understood in defining
~
s~2 •
A, and for
Nl ~ N2 ' Nl Sint N2
- 15 -
inhomogeneous Poisson processes,
Ht •
independent of
If both
=1
F(xIH )
t
N
i
- exp(-A(t,t+x])
are doubly stochastic Poisson
processes for which the sample realizations of the intensity functions
satisfy
Al(A,w) s A2 (A,w) a.s.
we have
N
for all Borel sets
l C N2 ' we need not have
Ht ::: {N i (s):
0 < sst}
N
l
C
N
h 2
A, then while
because the "history"
Ai (. ,w).
is not equivalent to
This statement
could be rephrased in measure-theoretic language, but an illustration
is probably worth more in word-value:
Ai(·,w)
have the densities
Ai(·,w)
given by
0
{
for t s 1
2
for 1 < t
2
-{
for all t
2
for
Al(t)
co
A (t)
2
Then
let
= 0IN
pdN (2,x]
l
pdN (2,x]
2
= 0IN
l
2
2 for t > 1
S
with probability .5,
2 , 1 otherwise with probability .5,
with probability .5,
t s 2 , 1 for t > 2
with probability .5.
(0,2] -O} .. (e- 2 (x-2)+ e- l e-(x-2»/(l+e- l )
(0,2] -O} _ (e- 2 (x-2)+ e-(x-2»/2
so that
F(1)(xIN (0,2] -0) > F(2)(xIN (0,2]-0), contra
l
2
to have
Nl Sh
(5.2) if we were
2 •
N
Particularly as inputs to queueing systems, there is much
interest in the intervals
{Xn } following (3.1).
For
N
i
defined on a
common probability space, say that
(all n .. 1,2, ••• ) a.s.
In other words, the points of
If
which
N
i
F(x) S G(x)
larger than
Nl
are more spaced out than those of
are renewal processes with lifetime d.f.s
(all x)
(5.6)
F, G for
(in the language of Stoyan, the d.f.
G) , then there are realizations
{X(i)}
n
N2 •
F is
for which (5.6) is
- 16 necessarily satisfied.
then
It need not be the case that if
N1 Sint N2 ' nor conversely, although if
processes satisfying
N
1
N'
N
i
i
distributed like
~
Ni
N
1
~
N '
2
are renewal
N2 ' then there will be renewal processes
In terms of models
for point processes, the ordering
S
int
arises naturally only when
the process is conveniently specified by its intervals (cf. also the
comments in section 3).
It will be evident that all three orderings presented so
far have the property of implying that
N
1
(all t)
~
N yields
2
a.s.,
(5.7)
and when (5.7) holds, we shall say that
(5.8)
To justify the notation as suggesting that some underlying probabilistic
inequality may suffice, recall the equivalence of events
{Ni(O,X ] s n j (j = 1, ••• ,k)}
j
= {S~~) ~
x j (j
and that the multivariate analogue of the inequality
r.v.s
=
1, ••• ,k)} ,
X sd Y between
X and Y is that
(5.9)
for all functions
expectations exist.
f
increasing in each argument such that the
If for all such
•
f
we have
(5.10)
then by an extension of a result of Strassen (see Chapter 17 of Marshall
and 01kin (1979), or, for more detail, Kamae, Krengel and O'Brien (1977»
there is a probability space on which
- 17 -
(all n)
a. s. ,
(5.11)
and hence (5.7) holds also.
Since (5.11) for
=1
n
implies that
xiI) ~ x?)
i t follows immediately that for renewal processes,
Nl $int NZ '
Nl
alternating renewal process with lifetime d.f.s
Gl(x)
~
Jx Gl(x-u)d
o
(take
F(x)
S P N implies
Z
but this result need not be true in general.
may be a renewal process with lifetime d.f.
for all x,
N
l
= Gz(x+a)
~
F(x)
GZ(x)
Z*
GZ(u) ~ F
,
Gl(x)
F,
G
l
a.s.,
and
and
For example,
N an
Z
GZ for which
and
(x) ;
this is an easy construction
= GZ(x+a+b)
for
0 < b < a) •
The last inequality Whitt lists is just the marginal
distribution property that
(all t) •
(5.lZ)
The quasi-Poisson processes of Szasz (1970) or Oakes (1974) afford
simple examples of processes satisfying
other partial orderings.
N $d N but none of the
l
Z
(As Whitt notes, (5.lZ) is not in fact a
partial ordering on the space of point processes, because, as for
example with the quasi-Poisson processes, we can have
N $d N
l
Z
Nl Sd NZ and
N having a common (joint) distribution.) Note
i
that if (5.lZ) holds, then, by the equivalence as below (5.8),
•
without
S S (Z) (all n) •
d n
Schmidt (1976) formulated the definition (5.lZ), unnecessarily
restricted to stationary point processes, and also formulated a
distributional version of
Sint for stationary point processes via the
inequality
(5.13)
- 18 with
f
as at (5.13)
and
EO
denoting expectation with respect to
the Palm probabilities, for which
are then stationary sequences.
By the extension of the Strassen result, there then exist point processes
NO
i
,
for which
these processes having the Palm distribution
of the statiQnary processes
Note that if
N
i
N •
i
are renewal processes for which
N
1
~d
N '
2
then since
we must also have
N1
processes, neither
<
~int
-int
N .
2
nor
But if the
N
i
are stationary renewal
need imply the other, as Schmidt
~d
showed.
In subsequent work, Whitt (1981b) has taken a more pragmatic
'e
view and compared some point processes numerically by studying how
different processes may have different effects on a queueing system
when fed as inputs into the system.
More work of this nature has been
done by Whitt and his colleagues.
6.
DISTANCES BETWEEN POINT PROCESSES
For the time being we pass over the comparisons
for suggesting distances between point processes.
~h
and
C
Coming to
and recalling (2.3), we could use
00
•
e
d.
1(N ,N ) ::sup n- 1 J \Pr{s(l) >t}_pr{S(2) >dldt
l 2
1nt,
~l
0
n
n
where the factor
n
-1
(6.1)
has been introduced to average out the effect of
S(l)_ S(2) being asymptotically like n(EX(l) _ EX(2)) i f the sequences
1
1
n
n
{X(i) } are stationary. However, i f N ~d N and N ~d N ' as at
1
2
2
1
n
(5.11) and below, we should have
dint,1(N1,N2) = 0 , for (6.1) depends
- 19 -
only on the marginal distributions of
distribution over a set of values of
= SUP!EN1(t)
dE (N ,N 2)
l
t>O
has a similar drawback.
N(i)(t) , not the joint
t.
- EN (t) I/(t+l)
2
(6.2)
However
(6.3)
does not have this drawback:
problem of requiring
N
l
instead we are confronted with the
and
N
2
to be defined on the same
probability space, and, as noted in section 2, the appropriate
construction is far from evident as soon as we dispense with renewal
processes.
As an alternative to (6.3), and one that is better suited
to processes constructed by the interval properties, we have
(6.4)
For renewal processes, defined via sequences
as in section 3,
dint = dint,l'
{(X(l)*
n
'
although the proof is not immediately
obvious.
Since renewal processes
renewal functions
U (·) ,
i
N
i
are characterized by their
we could use, as an alternative to (6.2)
(but only for renewal processes) the function
supIUl(t) - U2 (t) I/(t+l) .
(6.5)
t~O
Matthes, Kerstan and Mecke (1978) (e.g. in their Sections
1.9 and 1.12) use the variation distance for point processes, and
T.C. Brown (private correspondence) has written of using compensators
to obtain total variation results for the distance of point processes
- 20 from Poisson processes.
I have been told of an unpublished review by
R.L. Farrell that may have relevance to this section.
Mark Brown (1981
and correspondence) has compared renewal processes with decreasing
failure rate (DFR):
in the notation of section 2, he has shown that
for distributions with the weaker property of increasing mean residual
life,
(6.6)
for DFR d.f.s it is easily checked from a sketch of the tails of the
d.f.s involved and the results of section 2, that the maximum on the
left-hand side of (6.6) is in fact
Pl(F,F ) •
l
He has also studied
metrics in terms of the hazard functions being close.
We have not attempted to assess the effect of common
operations on point processes (thinning, superposition, translation,
cluster-formation, etc.) on the distance functions given above.
ACKNOWLEDGEMENT
I am greatly indebted to Ross Leadbetter for his interest.
REFERENCES
v
AZLAROV, T.A. & KALZANOV, K. (1976a)
certain system with waiting time.
(In Russian).
(1976b)
II
II 4354.
An estimate of the stability of a queueing
v
system.
Izv. Akad.
Fiz .-Mat. Nauk ~, no. 5, 68-69. MR
Nauk. UzSSR Ser.
--- & ---
Estimation of the stability of a
(In Russian).
v
Taskent. Gos. Univ. Naucn. Trudy Vyp. 490
Voprosy Matematiki (1976), 21-26. MR 58 # 3092.
===
v
v
BOL'SAKOV, I.A. & RAKOSIC, V.S. (1978)
(In Russian).
Applied Theory of Random Flows.
Izdat. "Sovetskoe Radio", Moscow.
MR 80g: 60055
- 21 -
BROWN, M. (1981)
Approximating DFR distributions by exponential
distributions, with applications to first passage times.
Ann. Probab. 9
DALEY, D.J.
(1981)
The closeness of uniform random points in a
square and a circle.
Tech. Report, ••••
Mimeo Series,
Statistics Dept., University of North Carolina, Chapel Hill.
HENGARTNER, W., & THEODORESCU, R. (1973)
Concentration Functions.
Academic Press, New York.
KALZANOV, K. (1970)
(In Russian).
The closeness of the Palm and Poisson flows.
Izv. Akad. Nauk UzSSR Sere Fiz-Mat. Nauk 14
===
(1970), no. 3, 69-71.
(1975a)
MR 44 II 1073.
=-=
The stability of a Poisson flow.
(In Russian).
=
Izv. Akad. Nauk UzSSR Sere Fiz.-Mat. Nauk _
1975,
no. 2, 75-77.
_ _ _ _ _ _ _ _ _ _ _---c.-'--_---c._-'--
MR 55 II 11431.
==
(1975b)
The nearness of two Palm flows.
(In Russian).
Izv. Akad. Nauk UzSSR Sere Fiz.-Mat. Nauk 1975, no. 4, 25-30.
--------------------~
MR 55 II 11432.
-=
KAMAE, T.,KRENGEL, U. & O'BRIEN, G.L. (1977)
Stochastic inequalities
Ann. Probab. 5, 899-912.
on partially ordered spaces.
=
LEWIS, P.A.W. & SHEDLER, G.S. (1979)
Poisson processes by thinning.
Simulation of nonhomogeneous
Naval Res. Logistics Quart.
26, 403-413.
===
MARSHALL, A.W. & OLKIN, I. (1979)
...
Inequalities:
Majorization and Its Applications.
Academic Press, New York •
MATTHES, K., KERSTAN, J. & MECKE, J. (1978)
Point Processes.
OAKES, D. (1974)
Theory of
Infinitely Divisible
John Wiley & Sons, Chichester.
A generalization of Moran's quasi-Poisson process.
Studia Sci. Math. Hungar.
2,
433-437.
- 22 PAULAUSKAS, V. (1971)
On a smoothing inequality (in Russian).
Litovsk. Mat. Sb. 11, 861-866 (= Selected Transl. Math. Statist.
=-=-
Probab. 14 (1977), 7-12).
=-==
PETROV, V.V. (1975)
.\
Sums of independent Random Variables
'
Springer-Verlag, New York.
(=
Translation from original
Russian edition, 1972).
SCHMIDT, V. (1976)
On the non-equivalence of two criteria of
comparability of stationary point processes.
Zast. Mat.
15, 33-37.
=-==
SONDERMAN, D. (1980)
Operate Res.
STOYAN, D. (1977)
~,
Comparing semi-Markov processes.
110-119.
Qualitative Eigenschaften und Abschatzungen
stochastischer Modelle.
SZASZ, D. (1970)
Math. Hungar.
WHITT, W. (198la)
I:
Akademie-Verlag, Berlin.
Once more on the Poisson process.
~,
Studia Sci.
441-444.
Comparing counting processes and queues.
Adv. Appl. Probab.
(198lb)
Math.
~
(to appear).
Approximating a point process by a renewal process,
A general framework.
Operate Res.
~
(to appear).
,
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TECHNICAL
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19. KEY WORDS (Contlnu .. on ,ev"rse .• i.1e il "" ...•.•.• 11')' ,,,,d Idp",Hy by blo,h ""mbp,)
point processes, renewal processes, distance functions, metrics, comparability
of processes
ABSTRACT (Conllnu..
20.
•
0" ,.. ve,s"
sid.. It "e' ..... a'y a"d
I"..
"tlty hy blo('h numl,,,,)
The paper is basically a discussion and survey of some literature
concerning the concept of a distance function for random variables and point
processes. Work of Kalzanov on the closeness of renewal processes to Poisson
processes is outlined, and there are comments on various possible comparison
methods for point processes as given by Whitt.
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