lResearch sponsored in part by the Office of Naval Research under
Contract N00014-67A-032l-0002, Task NR042214 with the University
of North Carolina.
A LIMIT THEOREM FOR THINNING OF POINT PROCESSES
.
Olav Kallenberg
1
University of Goteborg, Sweden
University of North Carolina, Chapel Hill
Institute of Statistios Mimeo Series No. 908
February, 1974
A LIMIT THEOREM FOR THINNING OF POINT PROCESSES
OLAV KALLENBERG
I
University of Goteborg
University of North Carolina
ABSTRACT
Let
;
;
and
n be point processes and let p€[O,I].
is a p-thinning of n,
if it is obtained from
independently with probability l-p.
with Pn +0
~n
;
n
n by deleting the atoms
It is shown that, if PI,P2, ... €(O,I]
is a pn -thinning of some
for each n€N, then
converges in distribution in the vague topology if and only if Pnnn
Furthermore, denoting the limits by
does.
~
that
by
and if
We say that
n.
~
and
n respectively, we have
is a subordinated (or doubly stochastic) Poisson process directed
These results include some limit theorems by
R~nyi
and others on
thinnings of a fixed process as well as a characterization by Mecke of the
class of subordinated Poisson processes.
KEY WORDS AND PHRASES:
Thinning of Point Processes; Subordinated Poisson
Processes; Convergence in Distribution; Vague Topology; Random
Measures.
IResearch sponsored in part by the Office of Naval Research under Contract
NOOOI4-67A-0321-0002, Task NR042214 with the University of North Carolina.
1
Let
B be the a-ring of bounded Borel sets in
let
~
S be a locally compact second countable Hausdorff space and
S.
By a random measure
on S we shall mean a mapping of some probability space (Q,A,p) into
M of Radon measures on (s,B) such that
the space
B€B.
variable for each
lfuen
~
~B
is a random
is a.s. integer valued, it will also be
called a point proaess.
If
~
and
n are point processes while p
we shall say that
~
is a real number in [0,1],
is a p-thinning of n, provided it is obtained from
by deleting the unit atoms independently with probability l-p.
of size
k
~
2 is considered as the superposition of k
subordinated Poisson process direated by
n,
~
n
(abbreviated
(Here an atom
unit atoms.)
more, if n is an arbitrary random measure, we shall say that
n
~
Further-
is a
SP(n)) if, given
is conditionally distributed as a Poisson process with intensity measure
4
n.
The existence of such thinnings and SP-processes is easily verified, using
.Harris' existence theorem in the formulation of Jagers (1972).
The limiting behavior as
p+O
of p-thinnings of a fixed point process
has been considered by Renyi (1956), Nawrotzki (1962), Belyaev (1963) and
Goldman (1967) (see also [5], [8] and [11]).
In the present note, we shall
prove a general limit theorem containing the results of these authors as well
as the following interesting characterization of the class of SP-processes,
due to Mecke (1968) (see also [7], page 34):
A point process is SP
if and
only if it can be obtained as a p-thinning for each pE(O,l].
Before proceeding, we introduce some further terminology and notation.
Let us write
F for the class of bounded measurable functions on
compact support, and let
S with
c be its sub-class of continuous functions.
F
Vague
2
~ ~~
of measures in M means that ~ f~~f for any f€F.
.
n
c
~f = Jf(s)~(ds). Convergence in distribution of random elements
convergence
Here
n
d
~.
(cf. [2]) is written
For random measures, the underlying topology in
M is taken to be the vague one (see [4] and [7]).
We are now able to state
our main result.
Theorem.
for each n€N
Let
Pl,P2' •.. E(O,l]
with
Pn~O,
and suppose that
is a p -thinning of some point process
n
n.
n
Then
~n
~n ~ some
~ if and only if Pnnn~ some n, and in this case, ~ is SP(n).
Proof.
We shall need the facts that, if
Ee -~f
(1)
while if
~
~
is SP(n), then
. { -n(l-e -f) }
= E exp
is a p-thinning of
n,
Ee -~f
. = E exp { nlog(l - p(l-e -f)) }
(2)
These formulae were established by Mecke (1968) for S
for general
S are similar.
us first suppose that
= R,
and their proofs
To prove the assertions of the theorem, let
pn nn ~ n.
We have to show that
~n~ ~ where ~ is
SP(n) , or equivalently, by (1), (2) and [4], that for any
f€F,
c
E exp{n log(l - p (l_e- f ))} ~ E exp{-n(l-e- f )} •
n
n
(3)
But this will follow by Theorem 5.1 in [2], if we can show that
(4)
where
random
g
= l-e -f .
nn
and
By Theorem 5.5 in [2], it suffices to prove (4) for nonn satisfying pn nn ¥ n, and since
we may even assume the
(J
Pnnn
are random elements in
and
g has bounded support,
n to be probability measures.
S with distributions
pn nand
n
If
(J
n
and
n respectively,
3
we may then write (4) in the form
and since
g is bounded and continuous, this will follow by Theorem 5.1 in
[2], provided we can show that
_p-1 10g(1 -p g(o )) ~ g(o)
(5)
n
n
n
By another application of Theorem 5.5 in [2], we may assume that the
o
are non-random elements in
satisfying on+
S
well-known result from calculus.
0,
on
and
in which case (5) is a
This completes the proof of the sufficiency
and the last assertion.
Conversely, suppose that
To prove that
it suffices to show that the sequence
tightness, any sequence
P n ~ some nil as n
n n
already proved, it follows that
nil
is unique.
is tight.
In fact, assuming
N'c N must contain some further subsequence
such that
of
{Pnnn}
pn nn~ some n ,
+
00
~
through
is
Nil.
Nil
By the part of the theorem
SP(n"), and so, by (1), the distribution
The convergence of Pn nn
now follows by Theorem 2.3
in [2].
To prove tightness of
show that
a
{PnnnB}
B we have
{Pnnn}' it suffices by Lemma 1.2 in [6] to
is tight for any
B€B
with
~dB
=0
a.s.
But for such
~n B ~ ~B by Theorem 1.1 in [6], or equivalently, in terms of
Laplace transforms
Here the right-hand side is continuous in
some
u
= l-e-t€(O,l)
and nO€N
such that
t, so there exist for any
E >
0
4
E exp{nn B log(l - Pnu)} > 1 - 8/2,
1/
n > nO .
,
Writing n l = max{n: Pn > 1/2} , it follows by Cebysev's inequality and an
elementary estimate that, for n > nO v n l '
p{p n B > u-1log 2} ~ P{-n B log(l-p u) > log 2}
n n
n
n
This shows that
..
{pn nn B}
is tight, and hence completes the proof.
5
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[1]
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Probab. Appl. 8.1 165-173.
Theor.
[2]
BILLINGSLEY, P. (1968)
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Wiley,
[3]
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[4]
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[5]
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[6]
KALLENBERG, O. (1973) Characterization and convergence of random measures and point processes. Z. WahrsaheinZiahkeitstheorie verw. Gebiete
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[7]
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[8]
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[9]
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•
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[11]
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