Electron. Commun. Probab. 18 (2013), no. 96, 1–7.
DOI: 10.1214/ECP.v18-2851
ISSN: 1083-589X
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
Poisson point processes:
large deviation inequalities for the convex distance
Matthias Reitzner∗
Abstract
An analogue of Talagrand’s convex distance for binomial and Poisson point processes
is defined. A corresponding large deviation inequality is proved.
Keywords: Keywords: Poisson point process; convex distance; large deviation inequality..
AMS MSC 2010: 60F10; 60E15; 60G55..
Submitted to ECP on June 4, 2013, final version accepted on December 24, 2013.
Supersedes arXiv:1306.0693.
1
Introduction and statement of results
Concentration and large deviations have been active topics for many years. Apart
from theoretical interest much additional interest in these questions comes from applications in combinatorial optimization, stochastic geometry, and many others. For these
problems a deviation inequality due to Talagrand [10] turned out to be extremely useful. It combines the notion of convex distance with an elegant proof of a corresponding
dimension free deviation inequality.
Let (E, B(E), P) be a probability space. Choose n points x1 , . . . , xn ∈ E , x = (x1 , . . .
. . . , xn ) ∈ E n , and assume that A ⊂ E n is measurable. Talagrand defines his convex
distance by
X
dT (x, A) = sup inf
αi 1(xi 6= yi )
(1.1)
kαk2 =1 y∈A 1≤i≤n
where α = (α1 , . . . , αn ) is a vector in S n−1 . For A ⊂ E n denote the s-blowup of A with
respect to the convex distance by As := {x : dT (x, A) ≤ s}. Talagrand proves that for all
n∈N
P⊗n (X ∈ A)P⊗n (X ∈
/ As ) ≤ e −
s2
4
(1.2)
where X = (X1 , . . . , Xn ) is a random vector with iid random variables X1 , . . . , Xn .
To extend this to point processes denote by N̄ (E) the set of all finite counting meaPk
sures ξ =
1 δxi , xi ∈ E , k ∈ N0 , or equivalently finite point sets {x1 , x2 , . . . , xk }
eventually with multiplicity.
For a function α : E → R we denote by kαk2,ξ the 2-norm of α with respect to the
measure ξ . For two counting measures ξ and ν the (set-)difference ξ\ν is defined by
ξ\ν =
X
(ξ(x) − ν(x))+ δx .
x: ξ(x)>0
∗ Universität Osnabrück, Germany. E-mail: matthias.reitzner@uni-osnabrueck.de
(1.3)
LDI for Poisson point processes
As will be shown in Section 4, the natural extension of dT to counting measures η ∈
N̄ (E) with η(E) < ∞ acts on N̄ (E) and is defined by
dπT (η, A)
Z
=
sup
inf
kαk22,η ≤1 ν∈A
α d(η\ν)
(1.4)
for A ⊂ N̄ (E). Here the supremum is taken over all nonnegative functions α : E → R.
The main result of this paper is an extension of Talagrand’s isoperimetric inequality to Poisson point processes on lcscH (locally compact second countable Hausdorff)
spaces. If η is a Poisson point process then the random variable η(A) is Poisson distributed for each set A ⊂ E and the expectation Eη(A) is the intensity measure of the
π
point process. For A ⊂ N̄ (E) we denote by Aπ
s := {x : dT (x, A) ≤ s} the s-blowup of A
π
with respect to the convex distance dT .
Theorem 1.1. Let E be a lcscH space and let η be a Poisson point process on E with
finite intensity measure Eη(E) < ∞. Then for any measurable subset A ⊂ N̄ (E) we
have
2
P(η ∈ A)P (η ∈
/ Aπs ) ≤ e−
s
4
.
It is the aim of this paper to stimulate further investigations on this topic. Of high
interest would be an extension of this theorem to the case of point processes of possible
infinite intensity measure. On the way to such a result one has to extend the notion
of convex distance to locally finite counting measures with ξ(E) = ∞. It is unclear
whether (1.4) is the correct way to define convex distance in general, see the short
discussion in Section 4.
Our method of proof consists of an extension of Talagrand’s large deviation inequality first to binomial processes and then to Poisson point processes. It would also be
of interest to give a proof of our theorem using only methods from the theory of point
processes. We have not been able to find such a direct proof for Theorem 1.1.
In general it seems that there are only few concentration inequalities for Poisson processes and Poisson measures. Apart from the pioneering work of Bobkov and Ledoux [2]
concerning concentration for Poisson measures, we mention the concentration inequalities by Ané and Ledoux [1], Wu [11], Breton, Houdre and Privault [4] and the recent
contribution by Eichelsbacher, Raic and Schreiber [5]. Concentration inequalities play
an important role in application, for an example considering intensity estimation see
Reynaud-Bouret [9]. As a general reference on recent developments we mention the
book by Boucheron, Lugosi, and Massart [3].
Our investigations are motivated by a problem in stochastic geometry. In [8] Theorem 1.1 is used to prove a large deviation inequality for the length of the Gilbert graph.
2
Binomial point processes
Assume that µ is a probability measure on (E, B(E)). The sets Â, B considered in
the following are measurable.
We call a point process ξn a binomial point process on E of intensity tµ with parameter n, 0 ≤ t ≤ 1, n ∈ N0 , if for any B ⊂ E we have
n
P(ξn (B) = k) =
(tµ(B))k (1 − tµ(B))n−k .
k
(2.1)
To link n iid points in E to a binomial point process ξn we consider the following
natural construction. We choose n independent points in E according to the underlying
probability measure µ and for each point we decide independently with probability t if
ECP 18 (2013), paper 96.
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LDI for Poisson point processes
it occurs in the process or not. To make this precise we add to E an artificial element
4 at infinity (containing of all points which have been deleted), define Ê = E ∪ {4} and
extend µ to Ê by
µ̂(B̂) = tµ(B̂\4) + (1 − t) δ4 (B̂) for B̂ ⊂ Ê.
Hence a random point Xi ∈ Ê chosen according to µ̂ is in E with probability t and equals
4 with probability 1 − t. Define the projection π of x ∈ Ê n unto N̄ (E) by ‘deleting’ all
points xi = 4, i.e.
π(x) = π((x1 , . . . , xn )) =
n
X
1(xi 6= 4)δxi ∈ N̄ (E).
i=1
and define the point process ξn ∈ N̄ (E) by
ξn = π(X) =
n
X
1(Xi 6= 4)δXi .
(2.2)
i=1
If B ⊂ E , any set of n iid random points X1 , . . . , Xn chosen according to µ̂ satisfies
n
P(ξn (B) = k) = P π(X1 , . . . , Xn )(B) = k =
(tµ(B))k (1 − tµ(B))n−k
k
and thus by (2.1) the process ξn is a binomial point process.
Assume that  ⊂ Ê n is a symmetric set, i.e. if y = (y1 , . . . yn ) ∈  then also
(yσ(1) , . . . , yσ(n) ) ∈ Â for all permutations σ ∈ Sn . Here SI is the group of permutations of
a set I ⊂ N, and we write Sn if I = {1, . . . , n}. It is immediate that a symmetric set is the
S
preimage of a set A ⊂ N̄ (E) under the projection π where π(Â) = y∈Â π(y) ⊂ N̄ (E).
As shown above for a random vector X = (X1 , . . . , Xn ) with iid coordinates we have
P(X ∈ Â) = P(π(X) ∈ π(Â)) = P(ξn ∈ A).
(2.3)
The essential observation is that the convex distance dT (x, Â) defined in (1.1) for
x ∈ Ê n is compatible with the projection π and yields the convex distance
hZ
dnT (ξn , A) =
sup
inf
αd(ξn \ν)
(2.4)
kαk22,ξn ≤1 ν∈A
+
(ν(E) − ξn (E))+
(n − ξn (E))
1
2
1
(1 − kαk22,ξn ) 2
i
on the space N̄ (E).
Lemma 2.1. Assume x ∈ Ê n and that  ⊂ Ê n is a symmetric set. Then for ξn = π(x)
and A = π(Â) we have
dT (x, Â) = dnT (ξn , A).
Proof. Since  is a symmetric set for any function f
inf f (y1 , . . . , yn ) =
y∈Â
inf
f (yσ(1) , . . . , yσ(n) ),
y∈Â,σ∈Sn
and we can rewrite the convex distance on Ê n given by (1.1) as
dT (x, Â)
=
sup
inf inf
X
kαk2 ≤1 y∈Â σ∈Sn 1≤i≤n
ECP 18 (2013), paper 96.
αi 1(xi 6= yσ(i) ).
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LDI for Poisson point processes
We write ξn = π(x), ν = π(y). It is immediate by the symmetry of  that dT (x, Â) is
invariant under any permutation of x1 , . . . xn . Hence we assume w.l.o.g. that xi are
sorted in such a way that xi 6= 4 for i = 1, . . . , ξn (E) and xi = 4 for i ≥ ξn (E) + 1.
dT (x, Â) = sup
inf inf
n (E)
h ξX
kαk2 =1 y∈Â σ∈Sn
n
X
αi 1(xi 6= yσ(i) ) +
i=1
i
αi 1(yσ(i) 6= 4)
i=ξn (E)+1
Here the second summand is zero if ν(E) ≤ ξ(E). For fixed x and y we decrease
the summands if we assume that the permutation acts in such a way that the maximal
number of 4’s in x and y coincide. If ν(E) ≤ ξn (E) this means that the minimum over Sn
is attained if yσ(i) = 4 for all σ(i) ≥ ξn (E) which coincides with the fact that the second
summand in this case vanishes. If ν(E) > ξn (E) then yσ(i) = 4 implies σ(i) ≥ ξn (E). To
make things more visible we take in this case the infimum over additional permutations
τ ∈ S[ξn (E)+1,n] of the second summand.
dT (x, Â)
=
sup
n (E)
h ξX
inf inf
kαk2 =1 y∈Â σ∈Sn
+
αi 1(xi 6= yσ(i) )
(2.5)
i=1
n
X
inf
τ ∈S[ξn (E)+1,n]
i
αi 1(yτ (σ(i)) 6= 4)
i=ξn (E)+1
The second summand equals the sum of the (ν(E) − ξn (E))+ smallest αi ’s in {αξn (E)+1 ,
. . . , αn }. We set αξn (E)+i = βi for i = 1, . . . , n − ξn (E) and denote by β(1) ≤ · · · ≤
β(n−ξn (E)) the order statistic of the βi . We obtain
dT (ξ, A)
=
sup
n (E)
h ξX
inf inf
kαk22 +kβk22 =1 y∈Â σ∈Sn
Pξ
(ν(E)−ξn (E))+
X
αi 1(xi 6= yσ(i) ) +
i=1
β(j)
i
j=1
(E)
where from now on kαk22 = 1n
αi2 . The βj2 sum up to 1 − kαk22 so that the sum of the
2
k -th smallest is at most (1 − kαk2 )k/(n − ξn (E)). Hölder’s inequality yields
dT (x, Â) ≤
sup
n (E)
h ξX
inf inf
kαk22 +kβk22 =1 y∈Â σ∈Sn
αi 1(xi 6= yσ(i) )
i=1
(ν(E)−ξn (E))+
21 i
X
2
+ (ν(E) − ξn (E))+
β(j)
j=1
≤
sup
inf inf
kαk22 ≤1 y∈Â σ∈Sn
n (E)
h ξX
αi 1(xi 6= yσ(i) )
i=1
+
(ν(E) − ξn (E))+
(n − ξn (E))
1
2
i
1
(1 − kαk22 ) 2 .
On the other hand if we take βj2 = (1 − kαk22 )/(n − ξn (E)) we have kβk22 = 1 − kαk22 and
the supremum is bounded from below by setting βj equal to these values.
dT (x, Â) ≥
sup
inf inf
kαk22 ≤1 y∈Â σ∈Sn
n (E)
h ξX
αi 1(xi 6= yσ(i) )
i=1
+
(ν(E) − ξn (E))+
(n − ξn (E))
ECP 18 (2013), paper 96.
1
2
i
1
(1 − kαk22 ) 2 .
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LDI for Poisson point processes
Both bounds coincide so that dT equals the right hand side. Define the function α : E →
R by
αi
if x = xi
α(x) =
0
otherwise
so that kαk22,ξ =
R
α2 dξ =
Pξn (E)
i=1
αi2 = kαk22 and by the definition (1.3) of ξ\ν
ξn (E)
X
inf
σ∈Sξn (E)
Z
αi 1(xi 6= yσ(i) ) =
αd(ξ\ν).
i=1
This proves
dT (x, Â) =
sup
inf
hZ
kαk22,ξn ≤1 ν∈A
αd(ξ\ν) +
(ν(E) − ξn (E))+
(n − ξn (E))
1
2
i
1
(1 − kαk22,ξn ) 2 .
By (2.3) we have P(X ∈ Â) = P(ξn ∈ A) for any measurable symmetric subset  of
E n . Recall that ξn = π(X) and A = π(Â). Lemma (2.1) shows that
dT (X, Â) ≥ s iff dnT (ξn , A) ≥ s,
so that X ∈
/ Âs iff ξn ∈
/ Ans . Here we denote by Ans the blowup with respect to the
n
/ Âs ) = P(ξn ∈
/ Ans ). Combining this with
distance dT . Again by (2.3) this yields P(X ∈
Talagrand’s large deviation inequality (1.2),
P(X ∈ Â)P(X ∈
/ Âs ) ≤ e−
s2
4
we obtain a large deviation inequality for the binomial process.
Theorem 2.2. Assume ξn is a binomial point process with parameter n on E . Then we
have
2
P(ξn ∈ A)P (ξn ∈
/ Ans ) ≤ e−
s
4
for any A ⊂ N̄ (E).
3
Poisson point processes
We extend Theorem 2.2 to Poisson point processes using the usual approximation
of a Poisson point process by Binomial point processes. Assume that the state space
E is a lcscH space (locally compact second countable Hausdorff space) and that µ is a
probability measure on (E, B(E)). As usual, the space N̄ (E) is endowed with the σ -field
N (E) generated by the evaluation mappings η 7→ η(B) with η ∈ N̄ (E) and B ∈ B(E),
see [7, Chapter 12]).
Fix some t > 0 and recall that µ is a probability measure on E . Set tn = t/n for
n ∈ N, t ≥ 0 and assume that n is sufficiently large such that t/n ≤ 1. The following
Lemma is known in great generality (see Jagers [6], or Theorem 16.18 in Kallenberg
[7]), we include its proof for the sake of completeness.
Lemma 3.1. The sequence of binomial point processes ξn defined in (2.2) with intensity
measure tn µ and parameter n converges in distribution to a Poisson point process η with
intensity measure tµ as n → ∞. I.e.,
P(ξn ∈ A) → P(η ∈ A).
(3.1)
for any A ∈ N (E).
ECP 18 (2013), paper 96.
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LDI for Poisson point processes
Pn
Proof of Lemma 3.1. Put ξni = 1(Xi ∈ E)δXi such that ξn =
i=1 ξni . Here Xi is in E
with probability tn = t/n. Since ξni , i ∈ {1, . . . n} are independent for given n and
supj E(min{ξnj (B), 1}) ≤ tn → 0 for all measurable B ⊂ E , the random measures
ξni form a null array. By Theorem 16.18 in Kallenberg [7] on an lcscH space E the
P
point process
ξni converges to a Poisson point process η with intensity measure tµ
P
P
if
P(ξ
(B)
>
0) → tµ(B), and i P(ξni (B) > 1) → 0. Both is immediate for all
ni
i
B ∈ B(E) from the definition of ξni .
The distance dn
T depends on n and has to be extended from binomial to Poisson point
processes as n → ∞. As stated in the introduction we use as a suitable definition
dπT (ξ, A)
Z
=
sup
inf
kαk22,ξn ≤1 ν∈A
αd(ξ\ν)
This is motivated by the more detailed investigations in Section 4. For A ⊂ N̄ (E) let Aπ
s
be the blowup of A with respect to the distance dπ
T . It is immediate that
dπT (ξ, A) ≤ dnT (ξ, A).
(3.2)
n
This implies Aπ
s ⊃ As and thus for a binomial point process ξn
P(ξn ∈
/ Aπs ) ≤ P(ξn ∈
/ Ans ).
(3.3)
The following theorem is an immediate consequence of Theorem 2.2 and formulae (3.3)
and (3.1).
Theorem 3.2. Assume η is a Poisson point process on some lcscH space E with
Eη(E) < ∞. Then we have
P(η ∈ A)P (η ∈
/ Aπs ) ≤ e−
s2
4
for any A ⊂ N̄ (E).
4
The convex distance
π
In this section we collect some facts about the convex distances dn
T and dT on N̄ (E).
n
To start with we show that dT not only gives the lower bound (3.2) for the distance
dπT defined in (1.4). We also prove that dnT → dπT as n → ∞ which shows that there is
essentially no other natural choice for dπ
T.
We start with the representation (2.4) of the convex distance for binomial point
processes. Assume ξ ∈ N̄ (E) satisfies ξ(E) < ∞. Set
Dα = inf
hZ
ν∈A
αd(ξ\ν) +
(ν(E) − ξ(E))+
(n − ξ(E))
1
2
1
(1 − kαk22,ξ ) 2
i
2
so that dn
T (ξ, A) = sup{Dα : kαk2,ξ ≤ 1}.
Since ξ is finite, the map ν → ξ\ν can take only finitely many ‘values’ µ1 , . . . , µm ∈
N̄ (E). Write A1 , . . . , Am ⊂ A for the preimage of the measures µi under this map.
Denote for i = 1, . . . , m by νi one of the counting measures in Ai with minimal ν(E).
Note that these minimizers are independent of α. Assume that νi (E) ≤ · · · ≤ νm (E).
We compute the infimum over ν ∈ A by taking the infimum over ν ∈ Ai and then the
minimum over i = 1, . . . , m.
Z
Dα
=
min
αdµi +
i=1,...,m
Z
=
min
i=1,...,m
inf ν∈Ai (ν(E) − ξ(E))+
1
1
(1 − kαk22,ξ ) 2
(n − ξ(E)) 2
1
(νi (E) − ξ(E))+
2
2
αdµi +
(1 − kαk2,ξ ) .
1
(n − ξ(E)) 2
ECP 18 (2013), paper 96.
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LDI for Poisson point processes
Since νi (E) is bounded by νm (E) we have
Z
min
i=1,...,m
Z
αdµi ≤ Dα ≤
min
i=1,...,m
αdµi +
(νm (E) − ξ(E))+
1
(n − ξ(E)) 2
which shows that for n → ∞ the distance dn
T converges to
dπT (ξ, A)
Z
=
sup
inf
αd(ξ\ν).
kαk22,ξ ≤1 ν∈A
Note that this is only a pseudo-distance since dπ
T (ξ, A) = 0 does not imply that ξ ∈ A.
For dπ
(ξ,
A)
=
0
it
suffices
that
A
containes
some
counting measure of the form ξ + ν
T
with ν ∈ N̄ (E) because then ξ\ν = 0.
It would be nice to have a definition of dπ
T which is a distance for counting measures
and which indicates extensions to point processes with possibly unbounded Eξ(E). One
could also use the distance dπ
T given in (1.4) as a definition but we could not relate it to
the distance dT for binomial processes. In applications it would be of high importance
to have such a representation and a large deviation inequality at least for Poisson point
processes on Rd . To the best of our knowledge even recent results like the one by Wu
[11] or Eichelsbacher, Raic and Schreiber [5] cannot be easily extended to our setting.
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Acknowledgments. This work was partially supported by ERC Advanced Research
Grant no 267165 (DISCONV). The author thanks Imre Barany and the Renyi Institute
of Mathematics, Hungarian Academy of Sciences, for the kind invitation and a fruitful
meeting in Budapest, where this work got started.
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