Large deviations of Poisson cluster processes
Charles Bordenave
and Giovanni Luca Torrisi
Abstract
In this paper we prove scalar and sample path large deviation principles for a large class of
Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic
Hawkes point processes.
Keywords: Hawkes processes, Large deviations, Poisson cluster processes, Poisson processes.
1 Introduction
Poisson cluster processes are one of the most important classes of point process models (see Daley
and Vere-Jones (2003) and Møller and Waagepetersen (2004)). They are natural models for the
location of objects in the space, and are widely used in point process studies whether theoretical
or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or
Hawkes processes (Hawkes (1971a), (1971b); Hawkes and Oakes (1974)). From a theoretical point
of view Hawkes processes combine both a Poisson cluster process representation and a simple
stochastic intensity representation.
Poisson cluster processes found applications in cosmology, ecology and epidemiology; see, respec-
tively, Neyman and Scott (1958), Brix and Chadoeuf (2002) and Møller (2003). Hawkes processes
are particularly appealing for seismological applications. Indeed, they are widely used as statistical
models for the standard activity of earthquake series; see the papers by Ogata and Akaike (1982),
Vere-Jones and Ozaki (1982), Ogata (1988) and Ogata (1998). Hawkes processes have also aspects
appealing to neuroscience applications; see the paper by Johnson (1996). More recently, Hawkes
processes found applications to finance, see Chavez-Demoulin, Davison and Mc Neil (2005), and to
DNA modeling, see Gusto and Schbath (2005).
In this paper we derive scalar and sample path large deviation principles for Poisson cluster pro-
cesses. The paper is organized as follows. In Section 2 we give some preliminaries on Poisson cluster
processes, Hawkes processes and large deviations. In Section 3 we provide scalar large deviation
principles for Poisson cluster processes, under a light-tailed assumption on the number of points per
cluster. As consequence, we provide scalar large deviations for ergodic Hawkes processes. Section
4 is devoted to sample path large deviations of Poisson cluster processes. First, we prove a sample
path large deviation principle on D[0, 1] equipped with the topology of point-wise convergence,
under a light-tailed assumption on the number of points per cluster. Second, we give a sample
path large deviation principle on D[0, 1] equipped with the topology of uniform convergence, under
a super-exponential assumption on the number of points per cluster. In Section 5 we prove large
deviations for spatial Poisson cluster processes, and we provide the asymptotic behavior of the void
probability function and the empty space function. We conclude the paper with a short discussion.
∗INRIA/ENS, Départment d’Informatique, 45 rue d’Ulm, F-75230 Paris Cedex 05, France. e-mail:
charles.bordenave@ens.fr
†Istituto per le Applicazioni del Calcolo ”Mauro Picone” (IAC), Consiglio Nazionale delle Ricerche (CNR), Viale
del Policlinico 137, I-00161 Roma, Italia. e-mail: torrisi@iac.rm.cnr.it
http://arXiv.org/abs/0704.1463v1
2 Preliminaries
In this section we recall the definition of Poisson cluster process, Hawkes process, and the notion
of large deviation principle.
2.1 Poisson cluster processes
A Poisson cluster process X ⊂ R is a point process. The clusters centers of X are given by particular
points called immigrants; the other points of the process are called offspring. The formal definition
of the process is the following:
(a) The immigrants are distributed according to a homogeneous Poisson process I with points
Xi ∈ R and intensity ν > 0.
(b) Each immigrant Xi generates a cluster Ci = CXi , which is a finite point process containing
(c) Given the immigrants, the centered clusters
Ci − Xi = {Y − Xi : Y ∈ Ci}, Xi ∈ I
are independent, identically distributed (iid for short), and independent of I.
(d) X consists of the union of all clusters.
The number of points in a cluster is denoted by S. We will assume that E[S] < ∞. Let Y be a point
process on R and NY(0, t] the number of points of Y in the interval (0, t]. Y is said stationary if
its law is translations invariant, is said ergodic if it is stationary, with a finite intensity E[NY(0, 1]],
NY(0, t]
= E[NY(0, 1]], a.s..
By the above definition of Poisson cluster process it is clear that X is ergodic with finite intensity
νE[S]. In particular,
NX(0, t]
= νE[S], a.s.. (1)
2.2 Hawkes processes
We say that X ⊂ R is a Hawkes process if it is a Poisson cluster process with (b) in the definition
above replaced by:
(b)’ Each immigrant Xi generates a cluster Ci = CXi , which is the random set formed by the
points of generations n = 0, 1, . . . with the following branching structure: the immigrant
Xi is said to be of generation 0. Given generations 0, 1, . . . , n in Ci, each point Y ∈ Ci of
generation n generates a Poisson process on (Y,∞), say Φ, of offspring of generation n + 1
with intensity function h(· − Y ). Here h : (0,∞) → [0,∞) is a non-negative Borel function
called fertility rate.
We refer the reader to Section 2 in Møller and Rasmussen (2005) for more insight into the branch-
ing structure and self-similarity property of clusters. Consider the mean number of points in any
offspring process Φ:
h(t) dt.
As usual in the literature on Hawkes processes, throughout this paper we assume
0 < µ < 1. (2)
Condition µ > 0 excludes the trivial case in which there are almost surely no offspring. Recalling
that the total number of points in a cluster is equivalent to the total progeny of the Galton-Watson
process with one ancestor and number of offspring per individual following a Poisson distribution
with mean µ (see p. 496 of Hawkes and Oakes (1974)), the other condition µ < 1 is equivalent to
assuming that E[S] = 1/(1 − µ) < ∞. For our purposes it is important to recall that for Hawkes
processes the distribution of S is given by
P (S = k) =
e−kµ(kµ)k−1
, k = 1, 2, . . . (3)
This follows by Theorem 2.11.2 in the book by Jagers (1975). Finally, since X is ergodic with a
finite and positive intensity equal to ν/(1 − µ) it holds:
NX(0, t]
1 − µ, a.s.. (4)
2.3 Large deviation principles
We recall here some basic definitions in large deviations theory (see, for instance, the book by
Dembo and Zeitouni (1998)). A family of probability measures {µα}α∈(0,∞) on a topological space
(M,TM ) satisfies the large deviations principle (LDP for short) with rate function J(·) and speed
v(·) if J : M → [0,∞] is a lower semi-continuous function, v : [0,∞) → [0,∞) is a measurable
function which increases to infinity, and the following inequalities hold for every Borel set B:
− inf
J(x) ≤ lim inf
log µα(B) ≤ lim sup
log µα(B) ≤ − inf
J(x),
where B◦ denotes the interior of B and B denotes the closure of B. Similarly, we say that a family
of M -valued random variables {Vα}α∈(0,∞) satisfies the LDP if {µα}α∈(0,∞) satisfies the LDP and
µα(·) = P (Vα ∈ ·). We point out that the lower semi-continuity of J(·) means that its level sets:
{x ∈ M : J(x) ≤ a}, a ≥ 0,
are closed; when the level sets are compact the rate function J(·) is said to be good.
3 Scalar large deviations
3.1 Scalar large deviations of Poisson cluster processes
Consider the ergodic Poisson cluster process X described above. In this section we prove that the
process {NX(0, t]/t} satisfies a LDP on R. Define the set
DS = {θ ∈ R : E[eθS ] < ∞}.
With a little abuse of notation, denote by C0 the cluster generated by an immigrant at 0 and let
L = supY ∈C0 |Y | be the radius of C0. We shall consider the following conditions:
the function θ 7→ E[eθS ] is essentially smooth and 0 ∈ D◦S (5)
E[LeθS ] < ∞ for all θ ∈ D◦S . (6)
For the definition of essentially smooth function, we refer the reader to Definition 2.3.5. in Dembo
and Zeitouni (1998).
Remark 3.1 Since S ≥ 1 we have that the function ϕ(θ) = E[eθS ] is increasing. It follows that
S = (−∞, θ0) with θ0 ∈ [0,∞]. By the dominated convergence theorem we have that ϕ′(θ) =
E[SeθS ] and ϕ′′(θ) = E[S2eθS ], for all θ ∈ D◦S . Hence, if θ0 < ∞, to prove that ϕ is essentially
smooth it suffices to show that E[Seθ0S] = ∞. On the other hand, if θ0 = +∞, the function ϕ is
always essentially smooth.
It holds:
Theorem 3.2 Assume (5) and (6). Then {NX(0, t]/t} satisfies a LDP on R with speed t and good
rate function
Λ∗(x) = sup
(θx − Λ(θ)). (7)
where Λ(θ) = ν(E[eθS ] − 1).
It is easily verified that Λ∗(νE[S]) = 0. Moreover, this is the unique zero of Λ∗(·). Therefore the
probability law of NX(0, t]/t concentrates in arbitrarily small neighborhoods of νE[S] as t → ∞,
as stated by the law of large numbers (1). The LDP is a refinement of the law of large numbers in
that it gives us the probability of fluctuations away the most probable value.
Before proving Theorem 3.2 we show that the same LDP holds for the non-stationary Poisson
cluster process Xt,T with immigrant process empty on (−∞,−T ) ∪ (t + T,∞), where T > 0 is a
fixed constant. Furthermore, the LDP for Xt,T holds under a weaker condition.
Theorem 3.3 Assume (5). Then {NXt,T (0, t]/t} satisfies a LDP on R with speed t and good rate
function (7).
Proof . The proof is based on the Gärtner-Ellis theorem (see, for instance, Theorem 2.3.6 in Dembo
and Zeitouni (1998)). We start proving that
log E[e
θNXt,T (0,t]] =
ν(E[eθS ] − 1) if θ ∈ DS
+∞ if θ /∈ DS
For a Borel set A ⊂ R, let I|A = I∩A be the point process of immigrants in A. Clearly I|(0,t], I|[−T,0]
and I|(t,t+T ] are independent Poisson processes with intensity ν, respectively on (0, t], [−T, 0] and
(t, T + t]. Since I|(0,t], I|[−T,0] , and I|(t,t+T ] are independent, by the definition of Poisson cluster
process it follows that the random sets {Ci : Xi ∈ I|(0,t]}, {Ci : Xi ∈ I|[−T,0]} and {Ci : Xi ∈ I|(t,t+T ]}
are independent. Therefore, for all θ ∈ R,
θNXt,T (0,t]
Xi∈I|(0,t]
NCi (0,t]+
Xi∈I|[−T,0]
NCi (0,t]+
Xi∈I|(t,t+T ]
NCi(0,t]
Xi∈I|(0,t]
NCi (0,t]
Xi∈I|[−T,0]
NCi (0,t]
Xi∈I|(t,t+T ]
NCi(0,t]
We shall show
log E
Xi∈I|(0,t]
NCi(0,t]
ν(E[eθS ] − 1) if θ ∈ DS
+∞ if θ /∈ DS
log E
Xi∈I|[−T,0]
NCi(0,t]
= lim
log E
Xi∈I|(t,t+T ]
NCi (0,t]
= 0, for θ ∈ DS . (10)
Note that (8) is a consequence of (9) and (10). We first prove (9). With a little abuse of notation,
denote by C0 the cluster generated by an immigrant at 0. Since {(Xi, Ci) : Xi ∈ I|(0,t]} is an
independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have
Xi∈I|(0,t]
NCi(0,t]
Xi∈I|(0,t]
NCi−Xi (−Xi,t−Xi]
= exp
eθNC0 (−x,t−x] − 1
= exp
eθNC0 (−tz,(1−z)t] − 1
. (11)
Therefore if θ ∈ DS , the expectation in (11) goes to E[eθS − 1] as t → ∞ by the monotone
convergence theorem. Hence, for θ ∈ DS the limit (9) follows from the dominated convergence
theorem. For θ /∈ DS the expectation in (11) goes to +∞ as t → ∞ by the monotone convergence
theorem, and the limit (9) follows by Fatou’s lemma. We now show (10). Here again, since {(Xi, Ci) :
Xi ∈ I|[−T,0]} is an independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-
Jones (2003) we have
Xi∈I|[−T,0]
NCi (0,t]
Xi∈I|[−T,0]
NCi−Xi(−Xi,t−Xi]
= exp
eθNC0 (x,x+t] − 1
. (12)
Now note that, for θ ∈ DS ∩ [0,∞), we have
0 ≤ 1
log E
Xi∈I|[−T,0]
NCi(0,t]
eθS − 1
dx < ∞
and, for each θ ≤ 0,
eθS − 1
eθNC0 (x,x+t] − 1
dx ≤ 0.
By passing to the limit as t → ∞ we get that the first limit in (10) is equal to 0. The proof for
the second limit in (10) is rigorously the same. Hence we proved (8). Using assumption (5), the
conclusion is a consequence of the Gärtner-Ellis theorem.
Proof of Theorem 3.2. The proof is similar to that one of Theorem 3.3 and is again based on the
Gärtner-Ellis theorem. We start showing that
log E[eθNX(0,t]] =
ν(E[eθS ] − 1) if θ ∈ D◦S
+∞ if θ /∈ DS
By similar arguments as in the proof of Theorem 3.3, using the definition of X, we have
eθNX(0,t]
θNXt,T (0,t]
θ(NX(0,t]−NXt,T (0,t])
, for all θ ∈ R, t > 0.
By the computations in the proof of Theorem 3.3, in order to prove (13) we only need to check
log E
θ(NX(0,t]−NXt,T (0,t])
= 0, for all θ ∈ D◦S . (14)
It is easily verified for θ ≤ 0 (the argument of the expectation is bounded below by eθS and above
by 1). We only check (14) for θ ∈ D◦S ∩ (0,∞). Here again, for a Borel set A ⊂ R, let I|A = I ∩ A
denote the point process of immigrants in A. Note that
NX(0, t] − NXt,T (0, t] =
Xi∈I|(−∞,−T )
NCi(0, t] +
Xi∈I|(t+T,∞)
NCi(0, t], t > 0.
Clearly I|(−∞,−T ) and I|(t+T,∞) are independent Poisson processes with intensity ν, respectively on
(−∞,−T ) and (t + T,∞). Thus, by the definition of Poisson cluster process it follows that the
random sets {Ci : Xi ∈ I|(−∞,−T )} and {Ci : Xi ∈ I|(t+T,∞)} are independent. Therefore, for all
θ ∈ D◦S ∩ (0,∞),
θ(NX(0,t]−NXt,T (0,t])
Xi∈I|(−∞,−T )
NCi(0,t]
Xi∈I|(t+T,∞)
NCi(0,t]
Since {(Xi, Ci) : Xi ∈ I|(−∞,−T )} and {(Xi, Ci) : Xi ∈ I|(t+T,∞)} are independently marked Poisson
processes, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have
Xi∈I|(−∞,−T )
NCi(0,t]
Xi∈I|(−∞,−T )
NCi−Xi(−Xi,t−Xi]
= exp
eθNC0 (x,t+x] − 1
Xi∈I|(t+T,∞)
NCi (0,t]
= exp
eθNC0 (−x,t−x] − 1
= exp
eθNC0 (−t−z,−z] − 1
Now notice that since θ > 0 we have
eθNC0 (x,x+t] − 1 ≤ (eθNC0 (R) − 1)1{x ≤ L}, for all x ≥ T
eθNC0 (−t−z,−z] − 1 ≤ (eθNC0 (R) − 1)1{z ≤ L}, for all z ≥ T .
Relation (14) follows by assumption (6) noticing that the above relations yield
Xi∈I|(−∞,−T )
NCi(0,t]
≤ exp(νE[L(eθS − 1)]), for all θ ∈ D◦S ∩ (0,∞), t > 0.
Xi∈I|(t+T,∞)
NCi (0,t]
≤ exp(νE[L(eθS − 1)]), for all θ ∈ D◦S ∩ (0,∞), t > 0.
Therefore, (13) is proved. Now, if DS = D
S then the claim is a consequence of the Gärtner-Ellis
theorem and assumption (5). It remains to deal with the case DS 6= D◦S . We shall show the large
deviations upper and lower bounds proving that for any sequence {tn}n≥1 ⊂ (0,∞) diverging to
+∞, as n → ∞, there exists a subsequence {sn} ⊆ {tn} such that
lim sup
log P (NX(0, sn]/sn ∈ F ) ≤ − inf
Λ∗(x), for all closed sets F (15)
lim inf
log P (NX(0, sn]/sn ∈ G) ≥ − inf
Λ∗(x), for all open sets G, (16)
where Λ∗ is defined by (7) (then the large deviations upper and lower bounds hold for any sequence
{tn} and the claim follows). By assumption (5), there exists θ0 > 0 such that DS = (∞, θ0]. Let
{tn}n≥1 ⊂ (0,∞) be a sequence diverging to +∞, as n → ∞, and define the extended non-negative
real number l ∈ [0,∞] by
l ≡ lim sup
log E[eθ0NX(0,tn]].
Clearly, there exists a subsequence {sn} ⊆ {tn} which realizes this lim sup, i.e.
log E[eθ0NX(0,sn]] = l.
By (13) it follows that
log E[eθNX(0,sn]] = Λ̃(θ), θ ∈ R
where
Λ̃(θ) =
Λ(θ) if θ < θ0
l if θ = θ0
+∞ if θ > θ0.
Note that, irrespective to the value of l, Λ̃ is essentially smooth (however it may be not lower
semi-continuous). We now show that the Legendre transform of Λ and Λ̃ coincide, i.e.
Λ̃∗(x) = Λ∗(x), x ∈ R. (17)
A straightforward computation gives Λ̃∗(x) = Λ∗(x) = +∞, for x < 0, and Λ̃∗(0) = Λ∗(0) = ν. Now,
note that since θ0 < ∞, Λ̃∗(x) and Λ∗(x) are both finite, for x > 0. Moreover, since Λ and Λ̃ are
essentially smooth, if x > 0 we have that Λ∗(x) = θxx − Λ(θx) and Λ̃∗(x) = θ̃xx − Λ̃(θ̃x), where
θx (respectively θ̃x) is the unique solution of Λ
′(θ) = x (respectively Λ̃(θ) = x) on (−∞, θ0). The
claim (17) follows recalling that Λ̃(θ) = Λ(θ) = ν(E[eθS ] − 1) on D◦S . Now, applying part (a) of
Theorem 2.3.6 in Dembo and Zeitouni (1998) we have (15). Applying part (b) of Theorem 2.3.6 in
Dembo and Zeitouni (1998) we get
lim inf
log P (NX(0, sn]/sn ∈ G) ≥ − inf
x∈G∩F
Λ∗(x), for any open set G, (18)
where F is the set of exposed points of Λ∗ whose exposing hyperplane belongs to (−∞, θ0), i.e.
F = {y ∈ R : ∃ θ ∈ D◦S such that for all x 6= y, θy − Λ∗(y) > θx − Λ∗(x)}.
We now prove that F = (0,+∞). For y < 0, Λ∗(y) = ∞, therefore an exposing hyperplane satisfying
the corresponding inequality does not exist. For y > 0 consider the exposing hyperplane θ = θy,
where θy is the unique positive solution on (−∞, θ0) of E[SeθS ] = y/ν. Note that Λ′(θ) = E[SeθS ]
and Λ′′(θ) = E[S2eθS ] for all θ < θ0. In particular, since S ≥ 1, we have that Λ is strictly convex
on (−∞, θ0). Therefore, for all x 6= y, it follows
θyy − Λ∗(y) = Λ(θy) > Λ(θx) + Λ′(θx)(θy − θx)
= θyx − Λ∗(x).
It remains to check that 0 /∈ F. Notice that since E[SeθxS ] = x/ν, limx↓0 θx = −∞. Also, by the
implicit function theorem, x 7→ θx is a continuous mapping on (0,∞). Now assume that 0 ∈ F, then
there would exist θ < θ0, such that for all x > 0, −Λ∗(0) > θx−Λ∗(x). However, by the intermediate
values theorem, there exists y > 0 such that θ = θy, and we obtain a contradiction. This implies
F = (0,+∞) as claimed. Now recall that Λ∗(x) = +∞ for x < 0; moreover, limx↓0 Λ∗(x) = Λ∗(0) =
ν (indeed, limx↓0 θx = −∞). Therefore
x∈G∩F
Λ∗(x) ≤ inf
Λ∗(x), for any open set G.
Finally, by (18) and the above inequality we obtain (16).
3.2 Scalar large deviations of Hawkes processes
Consider the ergodic Hawkes process X described before. In this section we prove that the process
{NX(0, t]/t} satisfies a LDP, and we give the explicit expression of the rate function. Our result is
a refinement of the law of large numbers (4). The following theorem holds:
Theorem 3.4 Assume (2) and
th(t) dt < ∞. (19)
Then {NX(0, t]/t} satisfies a LDP on R with speed t and good rate function
Λ∗(x) =
xθx + ν − νxν+µx if x ∈ (0,∞)
ν if x = 0
+∞ if x ∈ (−∞, 0)
, (20)
where θ = θx is the unique solution in (−∞, µ − 1 − log µ) of
= x/ν, x > 0, (21)
or equivalently of
E[eθS ] =
ν + xµ
, x > 0.
Proof . The proof is a consequence of Theorem 3.2. We start noticing that by (3) we have
E[eθS ] =
(eθ−µ)k(kµ)k−1
and this sum is infinity for θ > µ − 1 − log µ and finite for θ < µ − 1 − log µ (apply, for instance,
the ratio criterion). If θ = µ − 1 − log µ the sum above is finite. Indeed, in this case
E[eθS ] = (1/µ)
e−kkk−1
= 1/µ.
Therefore DS = (−∞, µ − 1 − log µ]. The origin belongs to D◦S in that by (2) and the inequality
ex > x + 1, x 6= 0, we have eµ−1
> 1. The function θ 7→ E[eθS ] is essentially smooth. Indeed, it is
differentiable in the interior of DS and
E[Se(µ−1−log µ)S ] = ∞
because
E[Se(µ−1−log µ)S ] = (1/µ)
e−kkk
and this sum is infinity since by Stirling’s formula e
2πk. We now check assumption
(6). By the structure of the clusters, it follows that there exists a sequence of independent non-
negative random variables {Vn}n≥1, independent of S, such that V1 has probability density h(·)/µ
and the following stochastic domination holds:
Vn, a.s.
(see Reynaud-Bouret and Roy (2007)). Therefore, for all θ < µ − 1 − log µ, we have
E[LeθS ] ≤ E
= E[V1]E[Se
θS ].
Since θ < µ − 1 − log µ, we have E[SeθS ] < ∞; moreover, assumption (19) yields
E[V1] =
th(t) dt < ∞.
Hence, condition (6) holds, and by Theorem 3.2, {NX(0, t]/t} satisfies a LDP on R with speed t
and good rate function
Λ∗(x) = sup
(θx − Λ(θ)) = sup
θ≤µ−1−log µ
(θx − Λ(θ)).
Now Λ∗(x) = ∞ if x < 0, in that in such a case limθ→−∞(θx − Λ(θ)) = ∞. If x > 0, letting
θx ∈ (−∞, µ − 1 − log µ) denote the unique solution of the equation (21) easily follows that
Λ∗(x) = xθx − Λ(θx). (22)
It is well-known (see, for instance, p. 39 in Jagers (1975)) that, for all θ ∈ (−∞, µ − 1 − log µ),
E[eθS ] satisfies
E[eθS ] = eθ exp{µ(E[eθS ] − 1)},
therefore differentiating with respect to θ we get
E[SeθS ] =
eθ exp{µ(E[eθS ] − 1)}
1 − µeθ exp{µ(E[eθS ] − 1)} =
E[eθS ]
1 − µE[eθS ] . (23)
Setting θ = θx in the above equality and using (21) we have
E[eθxS ]
1 − µE[eθxS ] ,
which yields
E[eθxS ] =
ν + xµ
Thus, by (22) we have for x > 0
Λ∗(x) = xθx + ν −
ν + µx
The conclusion follows noticing that a direct computation gives Λ∗(0) = ν.
4 Sample path large deviations
Let X be the ergodic Poisson cluster process described at the beginning. The results proved in this
section are sample path LDP for X.
4.1 Sample path large deviations in the topology of point-wise convergence
Let D[0, 1] be the space of càdlàg functions on the interval [0, 1]. Here we prove that {NX(0,α·]
satisfies a LDP on D[0, 1] equipped with the topology of point-wise convergence on D[0, 1]. The
LDP we give is a refinement of the following functional law of large numbers:
NX(0, α·]
= χ(·) a.s., (24)
where χ(t) = νE[S]t. As this is a corollary of the LDP we establish, we do not include a separate
proof of this result. Letting Λ∗(·) denote the rate function of the scalar LDP, we have:
Theorem 4.1 Assume (5) and (6). If moreover DS is open, then {NX(0,α·]α } satisfies a LDP on
D[0, 1], equipped with the topology of point-wise convergence, with speed α and good rate function
J(f) =
{ ∫ 1
Λ∗(ḟ(t))dt iff ∈ AC0[0, 1]
∞ otherwise , (25)
where AC0[0, 1] is the family of absolutely continuous functions f(·) defined on [0, 1], with f(0) = 0.
While it is tempting to conjecture that the result above holds even if the effective domain of S is not
open, we do not have a proof of this claim. If we take χ(t) = νE[S]t, then J(χ) = 0. Moreover this is
the unique zero of J(·). Thus the law of NX(0, α·]/α concentrates in arbitrarily small neighborhoods
of χ(·) as α → ∞, as ensured by the functional law of large numbers (24). The sample path LDP is
a refinement of the functional law of large numbers in that it gives the probability of fluctuations
away the most likely path.
As in Section 3.1, denote by Xt,T the non-stationary Poisson cluster process with immigrant
process empty on (−∞,−T )∪ (t+T,∞), where T > 0 is a fixed constant. Before proving Theorem
4.1 we show that the same LDP holds for Xt,T . Furthermore, the LDP for Xt,T holds under a
weaker condition.
Theorem 4.2 Assume (5). Then {NXα·,T (0, α·]/α} satisfies a LDP on D[0, 1], equipped with the
topology of point-wise convergence, with speed α and good rate function (25).
To prove this theorem we need Lemma 4.3 below, whose proof can be found in Ganesh, Macci and
Torrisi (2005) (see Lemma 2.3 therein).
Lemma 4.3 Let (θ1, . . . , θn) ∈ Rn and let w1, . . . , wn ≥ 0 be such that w1 ≤ . . . ≤ wn. Then
i=k θiwi ≤ θ∗w∗ for all k ∈ {1, . . . , n}, for any θ∗ ≥ max{max{
i=k θi : k ∈ {1, . . . , n}}, 0} and
any w∗ ≥ wn.
Proof of Theorem 4.2. With a little abuse of notation denote by C0 the cluster generated by an
immigrant at 0. We first show the theorem under the additional condition
NC0((−∞, 0)) = 0, a.s.. (26)
The idea in proving Theorem 4.2 is to apply the Dawson-Gärtner theorem to ”lift” a LDP for the
finite-dimensional distributions of {NXαt,T (0, αt]/α} to a LDP for the process. Therefore, we first
show the following claim:
(C) For all n ≥ 1 and 0 ≤ t1 < . . . < tn ≤ 1,
NXαt1,T (0, αt1]/α, . . . ,NXαtn,T (0, αtn]/α
satisfies
the LDP in Rn with speed α and good rate function
Jt1,...,tn(x1, . . . , xn) =
(tj − tj−1)Λ∗
xj − xj−1
tj − tj−1
, (27)
where x0 = 0 and t0 = 0.
Claim (C) is a consequence of the Gärtner-Ellis theorem in Rn, and will be shown in three steps:
(a) For each (θ1, . . . , θn) ∈ Rn, we prove that
Λt1,...,tn(θ1, . . . , θn) ≡ lim
log E
θiNXαti,T
(0, αti]
(tj−tj−1)Λ
where the existence of the limit (as an extended real number) is part of the claim, and Λ(·)
is defined in the statement of Theorem 3.2.
(b) The function Λt1,...,tn(·) satisfies the hypotheses of the Gärtner-Ellis theorem.
(c) The rate function
Jt1,...,tn(x1, . . . , xn) ≡ sup
(θ1,...,θn)∈Rn
θixi − Λt1,...,tn(θ1, . . . , θn)
coincides with the rate function defined in (27).
Proof of (a). For a Borel set A ⊂ R, denote by I|A = I ∩ A the Poisson process of immigrants in
A. Since, for each t, I|(0,t] and I|[−T,0] are independent, it follows from the definition of Poisson
cluster process that, for each i, the random sets {Ck : Xk ∈ I|(0,αti]} and {Ck : Xk ∈ I|[−T,0]} are
independent. Therefore,
θiNXαti,T
(0, αti]
Xk∈I|(0,αti]
NCk(0, αti]
Xk∈I|[−T,0]
NCk(0, αti]
where we used the independence and the assumption that NC0(−∞, 0) = 0 a.s.. In order to prove
(28), we treat successively the two terms in (29). Viewing I|(0,αti] as the superposition of the i
independent Poisson processes: I|(αtj−1,αtj ] on (αtj−1, αtj ] (j = 1, . . . , i) with intensity ν we get
Xk∈I|(0,αti]
NCk(0, αti]
 = E
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
, (30)
where in the latter equality we used the independence of {Ck : Xk ∈ I|(αtj−1,αtj ]} (j = 1, . . . , n). Since,
for each j, {(Xk, Ck) : Xk ∈ I|(αtj−1,αtj ]} is an independently marked Poisson process, by Lemma
6.4.VI in Daley and Vere-Jones (2003) we have
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
∫ α(tj−tj−1)
θiNC0(−αtj−1 − s, α(ti − tj−1) − s]
 . (31)
We now show
log E
Xk∈I|(0,αti]
NCk(0, αti]
(tj − tj−1)Λ
 (32)
for each (θ1, . . . , θn) ∈ Rn. We first notice that by (30) and (31) we have
log E
Xk∈I|(0,αti]
NCk(0, αti]
(tj − tj−1)Jj(α),
where
Jj(α) =
α(tj − tj−1)
∫ α(tj−tj−1)
θiNC0(−αtj−1 − s, α(ti − tj−1) − s]
 ds.
Now suppose that (θ1, . . . , θn) ∈ Rn is such that
i=j θi ∈ DS for each j ∈ {1, . . . , n}. Then by
Lemma 4.3 it follows that there exists θ∗ ∈ DS such that θ∗ ≥ 0,
i=j θi ≤ θ∗ for all j ∈ {1, . . . , n},
θiNC0(−αtj−1 − s, α(ti − tj−1) − s] ≤ θ∗NC0(R), a.s..
By (33) and the dominated convergence theorem, we have
Jj(α) = ν
i=j θiS
Hence we proved (32) whenever (θ1, . . . , θn) ∈ Rn satisfies
i=j θi ∈ DS for every j ∈ {1, . . . , n}. Now
suppose that (θ1, . . . , θn) ∈ Rn is such that
i=j θi /∈ DS for some j ∈ {1, . . . , n}. We have that
Jj(α) is bigger than or equal to
α(tj − tj−1)
∫ α(tj−tj−1)
1{θi < 0}θiS +
1{θi > 0}θiNC0 [0, α(tj − tj−1) − s]
1{θi < 0}θiS +
1{θi > 0}θiNC0 [0, α(tj − tj−1)(1 − z)]
 − 1
 dz.
The expectation in the latter formula goes to E[exp(
i=j θiS) − 1] as α → ∞ by the monotone
convergence theorem. Therefore, by Fatou’s lemma we have
Jj(α) ≥ νE
 = ∞.
Thus, since the quantities J1(α), . . . , Jn(α) are bounded below by −ν, we get (32) also in this
case. We now show
log E
Xk∈I|[−T,0]
NCk(0, αti]
 = 0 (34)
for all (θ1, . . . , θn) ∈ Rn such that
i=j θi ∈ DS for each j ∈ {1, . . . , n}. By Lemma 4.3 we have
that there exists θ∗ ∈ DS such that θ∗ ≥ 0,
i=j θi ≤ θ∗ for all j ∈ {1, . . . , n} and
Xk∈I|[−T,0]
NCk(R) ≤
Xk∈I|[−T,0]
NCk(0, αti] ≤ θ∗
Xk∈I|[−T,0]
NCk(R), a.s.,
where θ− ≡
i:θi<0
θi and θ− ≡ 0 if {i : θi < 0} = ∅. Therefore, using again Lemma 6.4 VI in
Daley and Vere-Jones (2003), we have
νT (E[eθ−S ] − 1)
Xk∈I|[−T,0]
NCk(0, αti]
 ≤ exp
νT (E[eθ
∗S ] − 1)
Equation (34) follows taking the logarithms in the above inequalities and passing to the limit. The
conclusion follows putting together (29), (32) and (34).
Proof of (b) and Proof of (c). Part (b) can be shown using assumption (5) and following the lines
of the proof of part (b) of Proposition 2.2 in Ganesh, Macci and Torrisi (2005). The proof of part
(c) is identical to the proof of part (c) of Proposition 2.2 in Ganesh, Macci and Torrisi (2005).
End of the proof under condition (26). By claim (C) and the Dawson-Gärtner theorem, {NXα·,T (0, α·]/α}
satisfies the LDP on D[0, 1], equipped with the topology of point-wise convergence, with speed α
and good rate function
J̃(f) = sup
(tk − tk−1)Λ∗
(f(tk) − f(tk−1)
tk − tk−1
: n ≥ 1, 0 ≤ t1 < . . . < tn ≤ 1
The conclusion follows noticing that J̃(·) coincides with J(·) in (25), as can be checked following
the same lines as in the proof of Lemma 5.1.6 in Dembo and Zeitouni (1998).
Removing the additional condition (26). The general case is solved as follows. Since Ck is almost
surely finite, there exists a left-most extremal point Yk ∈ Ck such that NCk(−∞, Yk) = 0 a.s.. Note
that, given the immigrants, Yk − Xk is an iid sequence. Therefore, by a classical result on Poisson
processes we have that {Yk} is a Poisson process with intensity ν. Viewing Xt,T as a Poisson cluster
process with cluster centers Yk and clusters Ck, the conclusion follows by the first part of the proof.
Proof of Theorem 4.1. The proof uses similar steps as in the proof of Theorem 4.2. Here we sketch
the main difference. Assume the additional condition NC0((−∞, 0)) = 0 a.s. (the general case can
be treated as in the proof of Theorem 4.2). Define the following subsets of Rn:
(θ1, . . . , θn) ∈ Rn :
θi ∈ DS for all j ∈ {1, . . . , n}
(θ1, . . . , θn) ∈ Rn :
θi /∈ DS for some j ∈ {1, . . . , n}
We start showing that for all n ≥ 1 and 0 ≤ t1 < . . . < tn ≤ 1
Λt1,...,tn(θ1, . . . , θn) =
j=1(tj − tj−1)Λ
i=j θi
for (θ1, . . . , θn) ∈ A1
+∞ for (θ1, . . . , θn) ∈ A2,
where
Λt1,...,tn(θ1, . . . , θn) ≡ lim
log E
θiNX(0, αti]
and Λ(·) is defined in the statement of Theorem 3.2. Using the definition of X and the assumption
NC0((−∞, 0)) = 0 a.s., we have
θiNX(0, αti]
Xk∈I|(0,αti]
NCk(0, αti]
Xk∈I|[−T,0]
NCk(0, αti]
Xk∈I|(−∞,−T )
NCk(0, αti]
As noticed in the proof of Theorem 4.2 we have
θiNXαti,T
(0, αti]
Xk∈I|(0,αti]
NCk(0, αti]
Xk∈I|[−T,0]
NCk(0, αti]
Therefore, by the computations in the proof of Theorem 4.2, to prove (35) we only need to check
log E
Xk∈I|(−∞,−T )
NCk(0, αti]
 = 0, for all (θ1, . . . , θn) ∈ A1. (36)
Since {(Xi, Ci) : Xi ∈ I|(−∞,−T )} is an independently marked Poisson process, by Lemma 6.4.VI in
Daley and Vere-Jones (2003) we have
Xk∈I|(−∞,−T )
NCk(0, αti]
 = E
Xk∈I|(−∞,−T )
θiNCk−Xk(−Xk, αti − Xk]
= exp
i=1 θiNC0 (x,αti+x] − 1
Take (θ1, . . . , θn) ∈ A1. By Lemma 4.3 we have that there exists θ∗ ∈ DS such that θ∗ ≥ 0,
i=j θi ≤ θ∗ for all j ∈ {1, . . . , n} and
θ−NC0(R) ≤
θiNC0(x, αti + x] ≤ θ∗NC0(R), a.s.
where θ− ≡
i:θi<0
θi and θ− ≡ 0 if {i : θi < 0} = ∅. Thus,
i=1 θiNC0 (x,αti+x] − 1 ≤ (eθ∗NC0 (R) − 1)1{x ≤ L}, for all x ≥ T
i=1 θiNC0 (x,αti+x] − 1 ≥ (eθ−NC0(R) − 1)1{x ≤ L}, for all x ≥ T
The limit (36) follows by assumption (6) noticing that the above relations yield, for all (θ1, . . . , θn) ∈
Xk∈I|(−∞,−T )
NCk(0, αti]
 ≤ exp(νE[L(eθ∗S − 1)])
Xk∈I|(−∞,−T )
NCk(0, αti]
 ≥ exp(νE[(eθ−S − 1)(L − T )1{L ≥ T}]).
Now since DS is open, the claim follows by applying first the Gärtner-Ellis theorem in R
n to get
the LDP for the finite-dimensional distributions, and then the Dawson-Gärtner theorem to have
the LDP for the process (argue as in the proof of Theorem 4.2 for the remaining steps).
4.2 Sample path large deviations in the topology of uniform convergence
In the applications, one usually derives LDPs for continuous functions of sample paths of stochastic
processes by using the contraction principle. Since the topology of uniform convergence is finer than
the topology of point-wise convergence, it has a larger class of continuous functions. Thus, it is of
interest to understand if {NX(0, α·]/α} satisfies a LDP on D[0, 1] equipped with the topology of
uniform convergence. In this section we give an answer to this question assuming that the tails of
S decay super-exponentially.
Theorem 4.4 Assume
E[eθS ] < ∞ for each θ ∈ R (37)
E[LeθS ] < ∞ for each θ ∈ R. (38)
Then {NX(0,α·]
} satisfies a LDP on D[0, 1], equipped with the topology of uniform convergence, with
speed α and good rate function (25).
In this section, without loss of generality we assume that the points of I are {Xi}i∈Z∗ , where
∗ = Z\{0}, Xi < Xi+1, and we set X0 = 0. As usual, we denote by Xt,T the non-stationary
Poisson cluster process with immigrant process empty on (−∞,−T ) ∪ (t + T,∞), where T > 0 is
a fixed constant, and by C0 the cluster generated by an immigrant at 0.
Before proving Theorem 4.4 we show that the same LDP holds for Xt,T , under a weaker con-
dition.
Theorem 4.5 Assume (37). Then {NXα·,T (0, α·]/α} satisfies a LDP on D[0, 1], equipped with the
topology of uniform convergence, with speed α and good rate function (25).
To prove Theorem 4.5 above we use the following Lemma 4.6, whose proof is omitted since it is
similar to the proof of Lemma 3.3 in Ganesh, Macci and Torrisi (2005). Let {Sk}k∈Z be the iid
sequence of random variables (distributed as S) defined by Sk = NCk(R).
Lemma 4.6 Assume (37), NC0(−∞, 0) = 0 a.s., and define
(Sk − NCk−Xk(0,Xk]), n ≥ 1.
It holds
log P (An ≥ nδ) = −∞ for each δ > 0.
Proof of Theorem 4.5. We prove the theorem assuming that NC0(−∞, 0) = 0 a.s.. The general
case is solved as in the proof of Theorem 4.2. As usual denote by I|A the restriction of I on the
Borel set A ⊂ R. Define
C(t) =
Xk∈I|(0,t]
Sk, t > 0.
We prove that {NXα·,T (0, α·]/α} and {C(α·)/α} are exponentially equivalent (see, for instance,
Definition 4.2.10 in the book of Dembo and Zeitouni, (1998)) with respect to the topology of
uniform convergence. Therefore the conclusion follows by a well-known result on sample path large
deviations, with respect to the uniform topology, of compound Poisson processes (see, for instance,
Borovkov (1967); see also de Acosta (1994) and the references cited therein) and Theorem 4.2.13
in Dembo and Zeitouni (1998). Define
CT (t) =
Xk∈I|[−T,t]
Sk, t > 0.
Using Chernoff bound and condition (37) can be easily realized that the processes {C(α·)/α}
and {CT (α·)/α} are exponentially equivalent with respect to the topology of uniform conver-
gence. Therefore, it suffices to show that {CT (α·)/α} and {NXα·,T (0, α·]/α} are exponentially equiv-
alent with respect to the topology of uniform convergence. Note that the assumption NC0(−∞, 0) =
0 a.s. gives
NXt,T (0, t] =
Xk∈I|(0,t]
NCk(0, t] +
Xk∈I|[−T,0]
NCk(0, t] t > 0, a.s..
Therefore, we need to show that
log P (Mα > δ) = −∞, for any δ > 0, (39)
where
t∈[0,1]
CT (αt) −
Xk∈I|(0,αt]
NCk(0, αt] −
Xk∈I|[−T,0]
NCk(0, αt]
Since
Mα ≤ M (1)α + M (2)α , a.s.,
where
M (1)α =
Xk∈I|[−T,0]
Sk and M
t∈[0,1]
Xk∈I|(0,αt]
(Sk − NCk(0, αt]),
the limit (39) follows if we prove
log P (M (1)α > δ/2) = −∞, for any δ > 0 (40)
log P (M (2)α > δ/2) = −∞, for any δ > 0. (41)
The limit (40) easily follows by the Chernoff bound and condition (37). It remains to show
(41). Since the random function t 7→ NCk(0, αt] is non-decreasing, it is clear that the supremum
over t is attained at one of the points Xn, n ≥ 1. Thus
M (2)α =
n≥1:Xn≤α
(Sk − NCk(0,Xn]).
Note that
M (2)α ≤ M̃α where M̃α =
n≥1:Xn≤α
(Sk − NCk−Xk(0,Xn − Xk]) a.s..
Therefore (41) follows if we show
log P (M̃α > δ/2) = −∞, for any δ > 0. (42)
Since Xn, n ≥ 1, is the sum of n exponential random variables with mean 1/ν, using Chernoff
bound and taking the logarithm, we have that, for all η > 0 and all integers K > ν,
log P (XK[α] < α) ≤ η +
ν + η
. (43)
Here the symbol [α] denotes the integer part of α. Next, observe that using the union bound we
P (M̃α > δ/2,XK[α] ≥ α) ≤ K[α] max
1≤n≤K[α]
(Sk − NCk−Xk(0,Xn − Xk]) ≥ αδ/2
Now we remark that, for n ≥ 1, (Xn − X1, . . . ,Xn − Xn−1) and (Xn−1, . . . ,X1) have the same
joint distribution. Moreover, given I, the centered processes Ck − Xk are iid and independent of
the {Xk}. Hence, letting An denote the random variable defined in the statement of Lemma 4.6,
we have
P (M̃α > δ/2,XK[α] ≥ α) ≤ K[α] max
1≤n≤K[α]
P (An ≥ αδ/2).
The random variables An are increasing in n, therefore
P (M̃α > δ,XK[α] ≥ α) ≤ K[α]P (AK[α] ≥ αδ/2),
and by Lemma 4.6 we have
log P (M̃α > δ/2,XK[α] ≥ α) = −∞. (44)
Now note that
P (M̃α > δ/2) ≤ P (M̃α > δ/2,XK[α] ≥ α) + P (XK[α] < α),
for arbitrary K > ν. Hence by (43) and(44) we have
lim sup
log P (M̃α > δ/2) ≤ inf
η + K log
ν + η
= K − ν − K log K
Then we obtain (42) by letting K tend to ∞.
Proof of Theorem 4.4. Throughout the proof we assume NC0((−∞, 0)) = 0 a.s.. The general case
is solved as in the proof of Theorem 4.2. Let {CT (t)} be the process defined in the proof of Theorem
4.5. The claim follows if we show that {CT (α·)/α} and {NX(0, α·]/α} are exponentially equivalent
with respect to the topology of uniform convergence. Note that the assumption NC0(−∞, 0) = 0
a.s. implies
NX(0, t] = NXt,T (0, t] +
Xk∈I|(−∞,−T )
NCk(0, t], t > 0 a.s..
Therefore, since we already proved that {CT (α·)/α} and {NXα·,T (0, α·]/α} are exponentially equiv-
alent with respect to the uniform topology (see the proof of Theorem 4.5), the claim follows if we
prove that
log P
Xk∈I|(−∞,−T )
NCk(0, α] > αδ
 = −∞, for any δ > 0. (45)
Using the Chernoff bound we have, for all θ > 0,
Xk∈I|(−∞,−T )
NCk(0, α] > αδ
 ≤ e−αθδE
Xk∈I|(−∞,−T )
θNCk(0, α]
= e−αθδ exp
E[eθNC0 (x,α+x] − 1] dx
≤ e−αθδ exp
νE[(eθS − 1)L]
Taking the logarithm, dividing by α, letting α tend to ∞ and using assumption (38) we get
lim sup
log P
Xk∈I|(−∞,−T )
NCk(0, α] > αδ
 ≤ −θδ, for all θ > 0.
Relation (45) follows letting θ tend to infinity in the above inequality.
5 Large deviations of spatial Poisson cluster processes
5.1 The large deviations principle
A spatial Poisson cluster process X is a Poisson cluster process in Rd, where d ≥ 1 is an integer. The
clusters centers are the points {Xi} of a homogeneous Poisson process I ⊂ Rd with intensity ν ∈
(0,∞). Each immigrant Xi ∈ I generates a cluster Ci = CXi , which is a finite point process. Given
I, the centered clusters {CXi −Xi} are iid and independent of I. X is the union of all clusters. As
in dimension 1, we denote by S the number of points in a cluster, with a little abuse of notation
by C0 the cluster generated by a point at 0, and by L the radius of C0. Moreover, we denote by
NX(b(0, r)) the number of points of X in the ball b(0, r), and by
ωd(r) =
rdπd/2
Γ(1 + d/2)
the volume of b(0, r). The following LDP holds:
Theorem 5.1 Assume (5) and
E[LdeθS ] < ∞, for all θ ∈ D◦S. (46)
Then {NX(b(0, r))/ωd(r)} satisfies a LDP on R with speed ωd(r) and good rate function (7).
Before proving Theorem 5.1, we show that the same LDP holds for the non-stationary Poisson
cluster process Xr,R with immigrant process empty in R
d \ b(0, R + r). As usual this LDP holds
under a weaker condition.
Theorem 5.2 Assume (5). Then {NXr,R(b(0, r))/ωd(r)} satisfies a LDP on R with speed ωd(r)
and good rate function (7).
Proof of Theorem 5.2 The proof is similar to that one for the non-stationary Poisson cluster
process on the line. Here we just sketch the main differences. As in the proof of Theorem 3.3,
the claim follows by the Gärtner-Ellis theorem. Indeed, letting I|b(0,r) denote the point process of
immigrants in b(0, r), and I|b(0,R+r)\b(0,r) the point process of immigrants in b(0, R + r) \ b(0, r) we
have, for each θ ∈ R,
θNXr,R (b(0,r))
Xi∈I|b(0,r)
NCi (b(0,r))
Xi∈I|b(0,R+r)\b(0,r)
NCi(b(0,r))
As usual, with a little abuse of notation denote by C0 the cluster generated by an immigrant at
0. It holds:
Xi∈I|b(0,r)
NCi (b(0,r))
= exp
b(0,r)
eθNC0 (b(−x,r)) − 1
for each θ ∈ R;
Xi∈I|b(0,R+r)\b(0,r)
NCi(b(0,r))
≤ exp
ν(ωd(R + r) − ωd(r))E
eθS − 1
for θ ∈ [0,∞) ∩ DS ;
1 ≥ E
Xi∈I|b(0,R+r)\b(0,r)
NCi (b(0,r))
≥ exp
ν(ωd(R + r) − ωd(r))E
eθS − 1
for θ ≤ 0. Therefore,
ωd(r)
log E
Xi∈I|b(0,r)
NCi(b(0,r))
= lim
ωd(r)
b(0,r)
E[eθNC0 (b(−x,r)) − 1] dx
ωd(1)
b(0,1)
E[eθNC0 (b(−ry,r)) − 1] dy
= νE[eθS − 1], for each θ ∈ R,
and, since limr→∞ ωd(R + r)/ωd(r) = 1, for each θ ∈ DS ,
ωd(r)
log E
Xi∈I|b(0,R+r)\b(0,r)
NCi (b(0,r))
The rest of the proof is exactly as in the one-dimensional case.
Proof of Theorem 5.1 The proof is similar to that one of Theorem 5.2 and is again based on the
Gärtner-Ellis theorem. We start showing that
ωd(r)
log E
θ(NX(b(0,r))−NXr,R (b(0,r)))
= 0, for all θ ∈ D◦S . (47)
This relation is easily verified for θ ≤ 0. Thus we only check (47) for θ ∈ D◦S ∩ (0,∞). We have, for
all θ ∈ D◦S ∩ (0,∞),
θ(NX(b(0,r))−NXr,R (b(0,r)))
Xi∈I|Rd\b(0,r+R)
NCi (b(0,r))
= exp
Rd\b(0,r+R)
eθNC0 (b(−x,r)) − 1
Now notice that since θ > 0 we have
eθNC0 (b(−x,r)) − 1 ≤ (eθNC0 (Rd) − 1)1{‖x‖ ≤ L + r}, for all x ∈ Rd.
The limit (47) follows by assumption (46) noticing that the above relations yield, for all θ ∈
S ∩ (0,∞), r > 0,
Xi∈I|Rd\b(0,r+R)
NCi(b(0,r))
≤ exp((νπd/2/Γ(1 + d/2))E[(L + r)d(eθS − 1)]).
Now notice that
eθNX(b(0,r))
θNXr,R (b(0,r))
θ(NX(b(0,r))−NXr,R (b(0,r)))
, for all θ ∈ R, r > 0.
Therefore, if DS = D
S then the claim is a consequence of the computation of the log-Laplace limit
of {NXr,R(b(0, r))} in the proof of Theorem 5.2, the Gärtner-Ellis theorem and assumption (5). It
remains to deal with the case DS 6= D◦S . Arguing exactly as in the proof of Theorem 3.2 it can
be proved that for any sequence {rn}n≥1 ⊂ (0,∞) diverging to +∞, as n → ∞, there exists a
subsequence {qn} ⊆ {rn} such that
lim sup
log P (NX(b(0, qn))/ωd(qn) ∈ F ) ≤ − inf
Λ∗(x), for all closed sets F
lim inf
log P (NX(b(0, qn))/ωd(qn) ∈ G) ≥ − inf
Λ∗(x), for all open sets G,
where Λ∗ is defined by (7). Then the large deviations upper and lower bounds hold for any sequence
{rn} and the claim follows.
5.2 The asymptotic behavior of the void probability function and the empty
space function
Apart some specific cases, the void probability function v(r) = P (NX(b(0, r)) = 0), r > 0, of
a spatial Poisson cluster process is not known in closed form. Comparing X with the immigrant
process I we easily obtain
v(r) ≤ P (NI(b(0, r)) = 0) = e−νωd(r), r > 0. (48)
A more precise information on the asymptotic behavior of v(·), as r → ∞, is provided by the
following proposition:
Proposition 5.3 Assume E[Ld] < ∞. Then
ωd(r)
log v(r) = −ν.
Proof Note that
v(r) = P (NCi(b(0, r)) = 0, for all Xi ∈ I)
1{NI(b(0, r)) = 0}
Xi∈I|Rd\b(0,r)
1{NCi(b(0, r)) = 0}
= e−νωd(r)E
Xi∈I|Rd\b(0,r)
1{NCi(b(0, r)) = 0}
= e−νωd(r) exp
Rd\b(0,r)
P (NC0(b(−x, r)) > 0) dx
where in (49) we used Lemma 6.4.VI in Daley and Vere-Jones (2003). Thus the claim follows if we
prove
ωd(r)
Rd\b(0,r)
P (NC0(b(−x, r)) > 0) dx = 0.
For this note that
1{NC0(b(0, r)) > 0} ≤ 1{‖x‖ − r ≤ L}, for all x ∈ Rd, r > 0.
Therefore,
ωd(r)
Rd\b(0,r)
P (NC0(b(−x, r)) > 0) dx ≤ E[(1 + L/r)d − 1],
and the right-hand side in the above inequality goes to 0 as r → ∞ by the dominated convergence
theorem (note that E[Ld] < ∞ by assumption).
In spatial statistics, a widely used summary statistic is the so-called empty space function, which is
the distribution function of the distance from the origin to the nearest point in X (see, for instance,
Møller and Waagepetersen (2004)), that is
e(r) = 1 − v(r), r > 0.
Apart some specific cases, the empty space function of Poisson cluster processes seems to be in-
tractable. Next Corollary 5.4 concerns the asymptotic behavior of e(r), as r → ∞.
Corollary 5.4 Under the assumption of Proposition 5.3 it holds
ωd(r)
log log e(r)−1 = −ν.
Proof The proof is an easy consequence of Proposition 5.3. By the upper bound (48) we obtain
lim sup
ωd(r)
log log e(r)−1 ≤ lim
ωd(r)
log log(1 − e−νωd(r))−1 = −ν.
To get the matching lower bound we note that the inequality log(1 − x) ≤ −x, x ∈ [0, 1), gives
log e(r)−1 ≥ v(r), r > 0, and therefore by Proposition 5.3 we get
lim inf
ωd(r)
log log e(r)−1 ≥ lim inf
ωd(r)
log v(r) = −ν.
5.3 Spatial Hawkes processes
Spatial Hawkes processes have been introduced in Daley and Vere-Jones (2003). Brémaud, Mas-
soulié and Ridolfi (2005) considered spatial Hawkes processes with random fertility rate and not
necessarily Poisson immigrants, and computed the Bartlett spectrum; the reader is directed to
Daley and Vere-Jones (2003) for the notion of Bartlett spectrum. Møller and Torrisi (2005) de-
rived the pair correlation function of spatial Hawkes processes; we refer the reader to Møller and
Waagepetersen (2004) for the notion of pair correlation function.
For the sake of completeness, we briefly recall the definition of spatial Hawkes process. A spatial
Hawkes process is a Poisson cluster process X ⊂ Rd where d ≥ 1 is an integer. The clusters centers
are the points {Xi} of a homogeneous Poisson process I ⊂ Rd with intensity ν ∈ (0,∞). Each
immigrant Xi ∈ I generates a cluster Ci = CXi which is formed by the points of generations
n = 0, 1, . . . with the following branching structure: the immigrant Xi ∈ I is said to be of zero-
th generation. Given generations 0, 1, . . . , n in Ci, each point Y ∈ Ci of generation n generates
a Poisson process on Rd of offspring of generation n + 1 with intensity function h(· − Y ). Here
h : Rd → [0,∞) is a non-negative Borel function. In the model it is assumed that, given the
immigrants, the centered clusters {Ci −Xi} are iid, and independent of I. By definition the spatial
Hawkes process is X ≡
i Ci. As in the one-dimensional case, it is assumed
0 < µ ≡
h(ξ) dξ < 1. (50)
This assumption guarantees that the number of points in a cluster has a finite mean equal to
1/(1−µ), excludes the trivial case where there are no offspring, and ensures that X is ergodic, with
a finite and positive intensity given by ν/(1−µ). Due to the branching structure, the number S of
offspring in a cluster follows the distribution (3). Finally, we note that the classical Hawkes process
considered in the previous sections corresponds to the special case where d = 1 and h(t) = 0 for
t ≤ 0.
A LDP for spatial Hawkes processes can be obtained by Theorem 5.1. The precise statement is
as Theorem 5.1 with (50) and
‖ξ‖h(ξ) dξ < ∞,
in place of (5) and (46), moreover the rate function is Λ∗(·) defined by (20). Here the symbol ‖ · ‖
denotes the Euclidean norm.
Similarly, the asymptotic behavior of the void probability function and the empty space function
of spatial Hawkes processes can be obtained as immediate consequences of Proposition 5.3 and
Corollary 5.4, respectively. The precise statements are as Proposition 5.3 and Corollary 5.4, with
conditions (50) and
‖ξ‖dh(ξ) dξ < ∞
in place of E[Ld] < ∞.
6 Extensions and open problems
In this paper we studied large deviations of Poisson cluster processes. Applications of these results
to insurance and queueing models are presently under investigation by the authors.
The definition of Hawkes process extends immediately to the case of random fertility rate
h(·, Z), where Zk’s are iid unpredictable marks associated to the points Xk (see Daley and Vere-
Jones (2003) for the definition of unpredictable marks, and Brémaud, Massoulié and Ridolfi (2005)
for the construction of Hawkes processes with random fertility rate specified by an unpredictable
mark). Due to the form of the distribution of S in this case (see formula (6) in Møller and Rasmussen
(2005)) it is not clear if the LDPs for Hawkes processes proved in this paper are still valid for Hawkes
processes with random fertility rate.
The generalization of our results to non-linear Hawkes processes (Kerstan (1964); Brémaud and
Massoulié (1996); Massoulié (1998); Brémaud, Nappo and Torrisi (2002); Torrisi (2002)) would be
interesting. However, since a non-linear Hawkes process is not even a Poisson cluster process, a
different approach is needed.
Acknowledgements
We thank Kamil Szczegot for reporting two mistakes and for a careful reading of a first draft of
the paper.
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	Introduction
	Preliminaries
	Poisson cluster processes
	Hawkes processes
	Large deviation principles
	Scalar large deviations
	Scalar large deviations of Poisson cluster processes
	Scalar large deviations of Hawkes processes
	Sample path large deviations
	Sample path large deviations in the topology of point-wise convergence
	Sample path large deviations in the topology of uniform convergence
	Large deviations of spatial Poisson cluster processes
	The large deviations principle
	The asymptotic behavior of the void probability function and the empty space function
	Spatial Hawkes processes
	Extensions and open problems

  In this paper we prove scalar and sample path large deviation principles for
a large class of Poisson cluster processes. As a consequence, we provide a
large deviation principle for ergodic Hawkes point processes.

Introduction
Poisson cluster processes are one of the most important classes of point process models (see Daley
and Vere-Jones (2003) and Møller and Waagepetersen (2004)). They are natural models for the
location of objects in the space, and are widely used in point process studies whether theoretical
or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or
Hawkes processes (Hawkes (1971a), (1971b); Hawkes and Oakes (1974)). From a theoretical point
of view Hawkes processes combine both a Poisson cluster process representation and a simple
stochastic intensity representation.
Poisson cluster processes found applications in cosmology, ecology and epidemiology; see, respec-
tively, Neyman and Scott (1958), Brix and Chadoeuf (2002) and Møller (2003). Hawkes processes
are particularly appealing for seismological applications. Indeed, they are widely used as statistical
models for the standard activity of earthquake series; see the papers by Ogata and Akaike (1982),
Vere-Jones and Ozaki (1982), Ogata (1988) and Ogata (1998). Hawkes processes have also aspects
appealing to neuroscience applications; see the paper by Johnson (1996). More recently, Hawkes
processes found applications to finance, see Chavez-Demoulin, Davison and Mc Neil (2005), and to
DNA modeling, see Gusto and Schbath (2005).
In this paper we derive scalar and sample path large deviation principles for Poisson cluster pro-
cesses. The paper is organized as follows. In Section 2 we give some preliminaries on Poisson cluster
processes, Hawkes processes and large deviations. In Section 3 we provide scalar large deviation
principles for Poisson cluster processes, under a light-tailed assumption on the number of points per
cluster. As consequence, we provide scalar large deviations for ergodic Hawkes processes. Section
4 is devoted to sample path large deviations of Poisson cluster processes. First, we prove a sample
path large deviation principle on D[0, 1] equipped with the topology of point-wise convergence,
under a light-tailed assumption on the number of points per cluster. Second, we give a sample
path large deviation principle on D[0, 1] equipped with the topology of uniform convergence, under
a super-exponential assumption on the number of points per cluster. In Section 5 we prove large
deviations for spatial Poisson cluster processes, and we provide the asymptotic behavior of the void
probability function and the empty space function. We conclude the paper with a short discussion.
∗INRIA/ENS, Départment d’Informatique, 45 rue d’Ulm, F-75230 Paris Cedex 05, France. e-mail:
charles.bordenave@ens.fr
†Istituto per le Applicazioni del Calcolo ”Mauro Picone” (IAC), Consiglio Nazionale delle Ricerche (CNR), Viale
del Policlinico 137, I-00161 Roma, Italia. e-mail: torrisi@iac.rm.cnr.it
http://arXiv.org/abs/0704.1463v1
2 Preliminaries
In this section we recall the definition of Poisson cluster process, Hawkes process, and the notion
of large deviation principle.
2.1 Poisson cluster processes
A Poisson cluster process X ⊂ R is a point process. The clusters centers of X are given by particular
points called immigrants; the other points of the process are called offspring. The formal definition
of the process is the following:
(a) The immigrants are distributed according to a homogeneous Poisson process I with points
Xi ∈ R and intensity ν > 0.
(b) Each immigrant Xi generates a cluster Ci = CXi , which is a finite point process containing
(c) Given the immigrants, the centered clusters
Ci − Xi = {Y − Xi : Y ∈ Ci}, Xi ∈ I
are independent, identically distributed (iid for short), and independent of I.
(d) X consists of the union of all clusters.
The number of points in a cluster is denoted by S. We will assume that E[S] < ∞. Let Y be a point
process on R and NY(0, t] the number of points of Y in the interval (0, t]. Y is said stationary if
its law is translations invariant, is said ergodic if it is stationary, with a finite intensity E[NY(0, 1]],
NY(0, t]
= E[NY(0, 1]], a.s..
By the above definition of Poisson cluster process it is clear that X is ergodic with finite intensity
νE[S]. In particular,
NX(0, t]
= νE[S], a.s.. (1)
2.2 Hawkes processes
We say that X ⊂ R is a Hawkes process if it is a Poisson cluster process with (b) in the definition
above replaced by:
(b)’ Each immigrant Xi generates a cluster Ci = CXi , which is the random set formed by the
points of generations n = 0, 1, . . . with the following branching structure: the immigrant
Xi is said to be of generation 0. Given generations 0, 1, . . . , n in Ci, each point Y ∈ Ci of
generation n generates a Poisson process on (Y,∞), say Φ, of offspring of generation n + 1
with intensity function h(· − Y ). Here h : (0,∞) → [0,∞) is a non-negative Borel function
called fertility rate.
We refer the reader to Section 2 in Møller and Rasmussen (2005) for more insight into the branch-
ing structure and self-similarity property of clusters. Consider the mean number of points in any
offspring process Φ:
h(t) dt.
As usual in the literature on Hawkes processes, throughout this paper we assume
0 < µ < 1. (2)
Condition µ > 0 excludes the trivial case in which there are almost surely no offspring. Recalling
that the total number of points in a cluster is equivalent to the total progeny of the Galton-Watson
process with one ancestor and number of offspring per individual following a Poisson distribution
with mean µ (see p. 496 of Hawkes and Oakes (1974)), the other condition µ < 1 is equivalent to
assuming that E[S] = 1/(1 − µ) < ∞. For our purposes it is important to recall that for Hawkes
processes the distribution of S is given by
P (S = k) =
e−kµ(kµ)k−1
, k = 1, 2, . . . (3)
This follows by Theorem 2.11.2 in the book by Jagers (1975). Finally, since X is ergodic with a
finite and positive intensity equal to ν/(1 − µ) it holds:
NX(0, t]
1 − µ, a.s.. (4)
2.3 Large deviation principles
We recall here some basic definitions in large deviations theory (see, for instance, the book by
Dembo and Zeitouni (1998)). A family of probability measures {µα}α∈(0,∞) on a topological space
(M,TM ) satisfies the large deviations principle (LDP for short) with rate function J(·) and speed
v(·) if J : M → [0,∞] is a lower semi-continuous function, v : [0,∞) → [0,∞) is a measurable
function which increases to infinity, and the following inequalities hold for every Borel set B:
− inf
J(x) ≤ lim inf
log µα(B) ≤ lim sup
log µα(B) ≤ − inf
J(x),
where B◦ denotes the interior of B and B denotes the closure of B. Similarly, we say that a family
of M -valued random variables {Vα}α∈(0,∞) satisfies the LDP if {µα}α∈(0,∞) satisfies the LDP and
µα(·) = P (Vα ∈ ·). We point out that the lower semi-continuity of J(·) means that its level sets:
{x ∈ M : J(x) ≤ a}, a ≥ 0,
are closed; when the level sets are compact the rate function J(·) is said to be good.
3 Scalar large deviations
3.1 Scalar large deviations of Poisson cluster processes
Consider the ergodic Poisson cluster process X described above. In this section we prove that the
process {NX(0, t]/t} satisfies a LDP on R. Define the set
DS = {θ ∈ R : E[eθS ] < ∞}.
With a little abuse of notation, denote by C0 the cluster generated by an immigrant at 0 and let
L = supY ∈C0 |Y | be the radius of C0. We shall consider the following conditions:
the function θ 7→ E[eθS ] is essentially smooth and 0 ∈ D◦S (5)
E[LeθS ] < ∞ for all θ ∈ D◦S . (6)
For the definition of essentially smooth function, we refer the reader to Definition 2.3.5. in Dembo
and Zeitouni (1998).
Remark 3.1 Since S ≥ 1 we have that the function ϕ(θ) = E[eθS ] is increasing. It follows that
S = (−∞, θ0) with θ0 ∈ [0,∞]. By the dominated convergence theorem we have that ϕ′(θ) =
E[SeθS ] and ϕ′′(θ) = E[S2eθS ], for all θ ∈ D◦S . Hence, if θ0 < ∞, to prove that ϕ is essentially
smooth it suffices to show that E[Seθ0S] = ∞. On the other hand, if θ0 = +∞, the function ϕ is
always essentially smooth.
It holds:
Theorem 3.2 Assume (5) and (6). Then {NX(0, t]/t} satisfies a LDP on R with speed t and good
rate function
Λ∗(x) = sup
(θx − Λ(θ)). (7)
where Λ(θ) = ν(E[eθS ] − 1).
It is easily verified that Λ∗(νE[S]) = 0. Moreover, this is the unique zero of Λ∗(·). Therefore the
probability law of NX(0, t]/t concentrates in arbitrarily small neighborhoods of νE[S] as t → ∞,
as stated by the law of large numbers (1). The LDP is a refinement of the law of large numbers in
that it gives us the probability of fluctuations away the most probable value.
Before proving Theorem 3.2 we show that the same LDP holds for the non-stationary Poisson
cluster process Xt,T with immigrant process empty on (−∞,−T ) ∪ (t + T,∞), where T > 0 is a
fixed constant. Furthermore, the LDP for Xt,T holds under a weaker condition.
Theorem 3.3 Assume (5). Then {NXt,T (0, t]/t} satisfies a LDP on R with speed t and good rate
function (7).
Proof . The proof is based on the Gärtner-Ellis theorem (see, for instance, Theorem 2.3.6 in Dembo
and Zeitouni (1998)). We start proving that
log E[e
θNXt,T (0,t]] =
ν(E[eθS ] − 1) if θ ∈ DS
+∞ if θ /∈ DS
For a Borel set A ⊂ R, let I|A = I∩A be the point process of immigrants in A. Clearly I|(0,t], I|[−T,0]
and I|(t,t+T ] are independent Poisson processes with intensity ν, respectively on (0, t], [−T, 0] and
(t, T + t]. Since I|(0,t], I|[−T,0] , and I|(t,t+T ] are independent, by the definition of Poisson cluster
process it follows that the random sets {Ci : Xi ∈ I|(0,t]}, {Ci : Xi ∈ I|[−T,0]} and {Ci : Xi ∈ I|(t,t+T ]}
are independent. Therefore, for all θ ∈ R,
θNXt,T (0,t]
Xi∈I|(0,t]
NCi (0,t]+
Xi∈I|[−T,0]
NCi (0,t]+
Xi∈I|(t,t+T ]
NCi(0,t]
Xi∈I|(0,t]
NCi (0,t]
Xi∈I|[−T,0]
NCi (0,t]
Xi∈I|(t,t+T ]
NCi(0,t]
We shall show
log E
Xi∈I|(0,t]
NCi(0,t]
ν(E[eθS ] − 1) if θ ∈ DS
+∞ if θ /∈ DS
log E
Xi∈I|[−T,0]
NCi(0,t]
= lim
log E
Xi∈I|(t,t+T ]
NCi (0,t]
= 0, for θ ∈ DS . (10)
Note that (8) is a consequence of (9) and (10). We first prove (9). With a little abuse of notation,
denote by C0 the cluster generated by an immigrant at 0. Since {(Xi, Ci) : Xi ∈ I|(0,t]} is an
independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have
Xi∈I|(0,t]
NCi(0,t]
Xi∈I|(0,t]
NCi−Xi (−Xi,t−Xi]
= exp
eθNC0 (−x,t−x] − 1
= exp
eθNC0 (−tz,(1−z)t] − 1
. (11)
Therefore if θ ∈ DS , the expectation in (11) goes to E[eθS − 1] as t → ∞ by the monotone
convergence theorem. Hence, for θ ∈ DS the limit (9) follows from the dominated convergence
theorem. For θ /∈ DS the expectation in (11) goes to +∞ as t → ∞ by the monotone convergence
theorem, and the limit (9) follows by Fatou’s lemma. We now show (10). Here again, since {(Xi, Ci) :
Xi ∈ I|[−T,0]} is an independently marked Poisson process, by Lemma 6.4.VI in Daley and Vere-
Jones (2003) we have
Xi∈I|[−T,0]
NCi (0,t]
Xi∈I|[−T,0]
NCi−Xi(−Xi,t−Xi]
= exp
eθNC0 (x,x+t] − 1
. (12)
Now note that, for θ ∈ DS ∩ [0,∞), we have
0 ≤ 1
log E
Xi∈I|[−T,0]
NCi(0,t]
eθS − 1
dx < ∞
and, for each θ ≤ 0,
eθS − 1
eθNC0 (x,x+t] − 1
dx ≤ 0.
By passing to the limit as t → ∞ we get that the first limit in (10) is equal to 0. The proof for
the second limit in (10) is rigorously the same. Hence we proved (8). Using assumption (5), the
conclusion is a consequence of the Gärtner-Ellis theorem.
Proof of Theorem 3.2. The proof is similar to that one of Theorem 3.3 and is again based on the
Gärtner-Ellis theorem. We start showing that
log E[eθNX(0,t]] =
ν(E[eθS ] − 1) if θ ∈ D◦S
+∞ if θ /∈ DS
By similar arguments as in the proof of Theorem 3.3, using the definition of X, we have
eθNX(0,t]
θNXt,T (0,t]
θ(NX(0,t]−NXt,T (0,t])
, for all θ ∈ R, t > 0.
By the computations in the proof of Theorem 3.3, in order to prove (13) we only need to check
log E
θ(NX(0,t]−NXt,T (0,t])
= 0, for all θ ∈ D◦S . (14)
It is easily verified for θ ≤ 0 (the argument of the expectation is bounded below by eθS and above
by 1). We only check (14) for θ ∈ D◦S ∩ (0,∞). Here again, for a Borel set A ⊂ R, let I|A = I ∩ A
denote the point process of immigrants in A. Note that
NX(0, t] − NXt,T (0, t] =
Xi∈I|(−∞,−T )
NCi(0, t] +
Xi∈I|(t+T,∞)
NCi(0, t], t > 0.
Clearly I|(−∞,−T ) and I|(t+T,∞) are independent Poisson processes with intensity ν, respectively on
(−∞,−T ) and (t + T,∞). Thus, by the definition of Poisson cluster process it follows that the
random sets {Ci : Xi ∈ I|(−∞,−T )} and {Ci : Xi ∈ I|(t+T,∞)} are independent. Therefore, for all
θ ∈ D◦S ∩ (0,∞),
θ(NX(0,t]−NXt,T (0,t])
Xi∈I|(−∞,−T )
NCi(0,t]
Xi∈I|(t+T,∞)
NCi(0,t]
Since {(Xi, Ci) : Xi ∈ I|(−∞,−T )} and {(Xi, Ci) : Xi ∈ I|(t+T,∞)} are independently marked Poisson
processes, by Lemma 6.4.VI in Daley and Vere-Jones (2003) we have
Xi∈I|(−∞,−T )
NCi(0,t]
Xi∈I|(−∞,−T )
NCi−Xi(−Xi,t−Xi]
= exp
eθNC0 (x,t+x] − 1
Xi∈I|(t+T,∞)
NCi (0,t]
= exp
eθNC0 (−x,t−x] − 1
= exp
eθNC0 (−t−z,−z] − 1
Now notice that since θ > 0 we have
eθNC0 (x,x+t] − 1 ≤ (eθNC0 (R) − 1)1{x ≤ L}, for all x ≥ T
eθNC0 (−t−z,−z] − 1 ≤ (eθNC0 (R) − 1)1{z ≤ L}, for all z ≥ T .
Relation (14) follows by assumption (6) noticing that the above relations yield
Xi∈I|(−∞,−T )
NCi(0,t]
≤ exp(νE[L(eθS − 1)]), for all θ ∈ D◦S ∩ (0,∞), t > 0.
Xi∈I|(t+T,∞)
NCi (0,t]
≤ exp(νE[L(eθS − 1)]), for all θ ∈ D◦S ∩ (0,∞), t > 0.
Therefore, (13) is proved. Now, if DS = D
S then the claim is a consequence of the Gärtner-Ellis
theorem and assumption (5). It remains to deal with the case DS 6= D◦S . We shall show the large
deviations upper and lower bounds proving that for any sequence {tn}n≥1 ⊂ (0,∞) diverging to
+∞, as n → ∞, there exists a subsequence {sn} ⊆ {tn} such that
lim sup
log P (NX(0, sn]/sn ∈ F ) ≤ − inf
Λ∗(x), for all closed sets F (15)
lim inf
log P (NX(0, sn]/sn ∈ G) ≥ − inf
Λ∗(x), for all open sets G, (16)
where Λ∗ is defined by (7) (then the large deviations upper and lower bounds hold for any sequence
{tn} and the claim follows). By assumption (5), there exists θ0 > 0 such that DS = (∞, θ0]. Let
{tn}n≥1 ⊂ (0,∞) be a sequence diverging to +∞, as n → ∞, and define the extended non-negative
real number l ∈ [0,∞] by
l ≡ lim sup
log E[eθ0NX(0,tn]].
Clearly, there exists a subsequence {sn} ⊆ {tn} which realizes this lim sup, i.e.
log E[eθ0NX(0,sn]] = l.
By (13) it follows that
log E[eθNX(0,sn]] = Λ̃(θ), θ ∈ R
where
Λ̃(θ) =
Λ(θ) if θ < θ0
l if θ = θ0
+∞ if θ > θ0.
Note that, irrespective to the value of l, Λ̃ is essentially smooth (however it may be not lower
semi-continuous). We now show that the Legendre transform of Λ and Λ̃ coincide, i.e.
Λ̃∗(x) = Λ∗(x), x ∈ R. (17)
A straightforward computation gives Λ̃∗(x) = Λ∗(x) = +∞, for x < 0, and Λ̃∗(0) = Λ∗(0) = ν. Now,
note that since θ0 < ∞, Λ̃∗(x) and Λ∗(x) are both finite, for x > 0. Moreover, since Λ and Λ̃ are
essentially smooth, if x > 0 we have that Λ∗(x) = θxx − Λ(θx) and Λ̃∗(x) = θ̃xx − Λ̃(θ̃x), where
θx (respectively θ̃x) is the unique solution of Λ
′(θ) = x (respectively Λ̃(θ) = x) on (−∞, θ0). The
claim (17) follows recalling that Λ̃(θ) = Λ(θ) = ν(E[eθS ] − 1) on D◦S . Now, applying part (a) of
Theorem 2.3.6 in Dembo and Zeitouni (1998) we have (15). Applying part (b) of Theorem 2.3.6 in
Dembo and Zeitouni (1998) we get
lim inf
log P (NX(0, sn]/sn ∈ G) ≥ − inf
x∈G∩F
Λ∗(x), for any open set G, (18)
where F is the set of exposed points of Λ∗ whose exposing hyperplane belongs to (−∞, θ0), i.e.
F = {y ∈ R : ∃ θ ∈ D◦S such that for all x 6= y, θy − Λ∗(y) > θx − Λ∗(x)}.
We now prove that F = (0,+∞). For y < 0, Λ∗(y) = ∞, therefore an exposing hyperplane satisfying
the corresponding inequality does not exist. For y > 0 consider the exposing hyperplane θ = θy,
where θy is the unique positive solution on (−∞, θ0) of E[SeθS ] = y/ν. Note that Λ′(θ) = E[SeθS ]
and Λ′′(θ) = E[S2eθS ] for all θ < θ0. In particular, since S ≥ 1, we have that Λ is strictly convex
on (−∞, θ0). Therefore, for all x 6= y, it follows
θyy − Λ∗(y) = Λ(θy) > Λ(θx) + Λ′(θx)(θy − θx)
= θyx − Λ∗(x).
It remains to check that 0 /∈ F. Notice that since E[SeθxS ] = x/ν, limx↓0 θx = −∞. Also, by the
implicit function theorem, x 7→ θx is a continuous mapping on (0,∞). Now assume that 0 ∈ F, then
there would exist θ < θ0, such that for all x > 0, −Λ∗(0) > θx−Λ∗(x). However, by the intermediate
values theorem, there exists y > 0 such that θ = θy, and we obtain a contradiction. This implies
F = (0,+∞) as claimed. Now recall that Λ∗(x) = +∞ for x < 0; moreover, limx↓0 Λ∗(x) = Λ∗(0) =
ν (indeed, limx↓0 θx = −∞). Therefore
x∈G∩F
Λ∗(x) ≤ inf
Λ∗(x), for any open set G.
Finally, by (18) and the above inequality we obtain (16).
3.2 Scalar large deviations of Hawkes processes
Consider the ergodic Hawkes process X described before. In this section we prove that the process
{NX(0, t]/t} satisfies a LDP, and we give the explicit expression of the rate function. Our result is
a refinement of the law of large numbers (4). The following theorem holds:
Theorem 3.4 Assume (2) and
th(t) dt < ∞. (19)
Then {NX(0, t]/t} satisfies a LDP on R with speed t and good rate function
Λ∗(x) =
xθx + ν − νxν+µx if x ∈ (0,∞)
ν if x = 0
+∞ if x ∈ (−∞, 0)
, (20)
where θ = θx is the unique solution in (−∞, µ − 1 − log µ) of
= x/ν, x > 0, (21)
or equivalently of
E[eθS ] =
ν + xµ
, x > 0.
Proof . The proof is a consequence of Theorem 3.2. We start noticing that by (3) we have
E[eθS ] =
(eθ−µ)k(kµ)k−1
and this sum is infinity for θ > µ − 1 − log µ and finite for θ < µ − 1 − log µ (apply, for instance,
the ratio criterion). If θ = µ − 1 − log µ the sum above is finite. Indeed, in this case
E[eθS ] = (1/µ)
e−kkk−1
= 1/µ.
Therefore DS = (−∞, µ − 1 − log µ]. The origin belongs to D◦S in that by (2) and the inequality
ex > x + 1, x 6= 0, we have eµ−1
> 1. The function θ 7→ E[eθS ] is essentially smooth. Indeed, it is
differentiable in the interior of DS and
E[Se(µ−1−log µ)S ] = ∞
because
E[Se(µ−1−log µ)S ] = (1/µ)
e−kkk
and this sum is infinity since by Stirling’s formula e
2πk. We now check assumption
(6). By the structure of the clusters, it follows that there exists a sequence of independent non-
negative random variables {Vn}n≥1, independent of S, such that V1 has probability density h(·)/µ
and the following stochastic domination holds:
Vn, a.s.
(see Reynaud-Bouret and Roy (2007)). Therefore, for all θ < µ − 1 − log µ, we have
E[LeθS ] ≤ E
= E[V1]E[Se
θS ].
Since θ < µ − 1 − log µ, we have E[SeθS ] < ∞; moreover, assumption (19) yields
E[V1] =
th(t) dt < ∞.
Hence, condition (6) holds, and by Theorem 3.2, {NX(0, t]/t} satisfies a LDP on R with speed t
and good rate function
Λ∗(x) = sup
(θx − Λ(θ)) = sup
θ≤µ−1−log µ
(θx − Λ(θ)).
Now Λ∗(x) = ∞ if x < 0, in that in such a case limθ→−∞(θx − Λ(θ)) = ∞. If x > 0, letting
θx ∈ (−∞, µ − 1 − log µ) denote the unique solution of the equation (21) easily follows that
Λ∗(x) = xθx − Λ(θx). (22)
It is well-known (see, for instance, p. 39 in Jagers (1975)) that, for all θ ∈ (−∞, µ − 1 − log µ),
E[eθS ] satisfies
E[eθS ] = eθ exp{µ(E[eθS ] − 1)},
therefore differentiating with respect to θ we get
E[SeθS ] =
eθ exp{µ(E[eθS ] − 1)}
1 − µeθ exp{µ(E[eθS ] − 1)} =
E[eθS ]
1 − µE[eθS ] . (23)
Setting θ = θx in the above equality and using (21) we have
E[eθxS ]
1 − µE[eθxS ] ,
which yields
E[eθxS ] =
ν + xµ
Thus, by (22) we have for x > 0
Λ∗(x) = xθx + ν −
ν + µx
The conclusion follows noticing that a direct computation gives Λ∗(0) = ν.
4 Sample path large deviations
Let X be the ergodic Poisson cluster process described at the beginning. The results proved in this
section are sample path LDP for X.
4.1 Sample path large deviations in the topology of point-wise convergence
Let D[0, 1] be the space of càdlàg functions on the interval [0, 1]. Here we prove that {NX(0,α·]
satisfies a LDP on D[0, 1] equipped with the topology of point-wise convergence on D[0, 1]. The
LDP we give is a refinement of the following functional law of large numbers:
NX(0, α·]
= χ(·) a.s., (24)
where χ(t) = νE[S]t. As this is a corollary of the LDP we establish, we do not include a separate
proof of this result. Letting Λ∗(·) denote the rate function of the scalar LDP, we have:
Theorem 4.1 Assume (5) and (6). If moreover DS is open, then {NX(0,α·]α } satisfies a LDP on
D[0, 1], equipped with the topology of point-wise convergence, with speed α and good rate function
J(f) =
{ ∫ 1
Λ∗(ḟ(t))dt iff ∈ AC0[0, 1]
∞ otherwise , (25)
where AC0[0, 1] is the family of absolutely continuous functions f(·) defined on [0, 1], with f(0) = 0.
While it is tempting to conjecture that the result above holds even if the effective domain of S is not
open, we do not have a proof of this claim. If we take χ(t) = νE[S]t, then J(χ) = 0. Moreover this is
the unique zero of J(·). Thus the law of NX(0, α·]/α concentrates in arbitrarily small neighborhoods
of χ(·) as α → ∞, as ensured by the functional law of large numbers (24). The sample path LDP is
a refinement of the functional law of large numbers in that it gives the probability of fluctuations
away the most likely path.
As in Section 3.1, denote by Xt,T the non-stationary Poisson cluster process with immigrant
process empty on (−∞,−T )∪ (t+T,∞), where T > 0 is a fixed constant. Before proving Theorem
4.1 we show that the same LDP holds for Xt,T . Furthermore, the LDP for Xt,T holds under a
weaker condition.
Theorem 4.2 Assume (5). Then {NXα·,T (0, α·]/α} satisfies a LDP on D[0, 1], equipped with the
topology of point-wise convergence, with speed α and good rate function (25).
To prove this theorem we need Lemma 4.3 below, whose proof can be found in Ganesh, Macci and
Torrisi (2005) (see Lemma 2.3 therein).
Lemma 4.3 Let (θ1, . . . , θn) ∈ Rn and let w1, . . . , wn ≥ 0 be such that w1 ≤ . . . ≤ wn. Then
i=k θiwi ≤ θ∗w∗ for all k ∈ {1, . . . , n}, for any θ∗ ≥ max{max{
i=k θi : k ∈ {1, . . . , n}}, 0} and
any w∗ ≥ wn.
Proof of Theorem 4.2. With a little abuse of notation denote by C0 the cluster generated by an
immigrant at 0. We first show the theorem under the additional condition
NC0((−∞, 0)) = 0, a.s.. (26)
The idea in proving Theorem 4.2 is to apply the Dawson-Gärtner theorem to ”lift” a LDP for the
finite-dimensional distributions of {NXαt,T (0, αt]/α} to a LDP for the process. Therefore, we first
show the following claim:
(C) For all n ≥ 1 and 0 ≤ t1 < . . . < tn ≤ 1,
NXαt1,T (0, αt1]/α, . . . ,NXαtn,T (0, αtn]/α
satisfies
the LDP in Rn with speed α and good rate function
Jt1,...,tn(x1, . . . , xn) =
(tj − tj−1)Λ∗
xj − xj−1
tj − tj−1
, (27)
where x0 = 0 and t0 = 0.
Claim (C) is a consequence of the Gärtner-Ellis theorem in Rn, and will be shown in three steps:
(a) For each (θ1, . . . , θn) ∈ Rn, we prove that
Λt1,...,tn(θ1, . . . , θn) ≡ lim
log E
θiNXαti,T
(0, αti]
(tj−tj−1)Λ
where the existence of the limit (as an extended real number) is part of the claim, and Λ(·)
is defined in the statement of Theorem 3.2.
(b) The function Λt1,...,tn(·) satisfies the hypotheses of the Gärtner-Ellis theorem.
(c) The rate function
Jt1,...,tn(x1, . . . , xn) ≡ sup
(θ1,...,θn)∈Rn
θixi − Λt1,...,tn(θ1, . . . , θn)
coincides with the rate function defined in (27).
Proof of (a). For a Borel set A ⊂ R, denote by I|A = I ∩ A the Poisson process of immigrants in
A. Since, for each t, I|(0,t] and I|[−T,0] are independent, it follows from the definition of Poisson
cluster process that, for each i, the random sets {Ck : Xk ∈ I|(0,αti]} and {Ck : Xk ∈ I|[−T,0]} are
independent. Therefore,
θiNXαti,T
(0, αti]
Xk∈I|(0,αti]
NCk(0, αti]
Xk∈I|[−T,0]
NCk(0, αti]
where we used the independence and the assumption that NC0(−∞, 0) = 0 a.s.. In order to prove
(28), we treat successively the two terms in (29). Viewing I|(0,αti] as the superposition of the i
independent Poisson processes: I|(αtj−1,αtj ] on (αtj−1, αtj ] (j = 1, . . . , i) with intensity ν we get
Xk∈I|(0,αti]
NCk(0, αti]
 = E
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
, (30)
where in the latter equality we used the independence of {Ck : Xk ∈ I|(αtj−1,αtj ]} (j = 1, . . . , n). Since,
for each j, {(Xk, Ck) : Xk ∈ I|(αtj−1,αtj ]} is an independently marked Poisson process, by Lemma
6.4.VI in Daley and Vere-Jones (2003) we have
Xk∈I|(αtj−1,αtj ]
θiNCk(0, αti]
∫ α(tj−tj−1)
θiNC0(−αtj−1 − s, α(ti − tj−1) − s]
 . (31)
We now show
log E
Xk∈I|(0,αti]
NCk(0, αti]
(tj − tj−1)Λ
 (32)
for each (θ1, . . . , θn) ∈ Rn. We first notice that by (30) and (31) we have
log E
Xk∈I|(0,αti]
NCk(0, αti]
(tj − tj−1)Jj(α),
where
Jj(α) =
α(tj − tj−1)
∫ α(tj−tj−1)
θiNC0(−αtj−1 − s, α(ti − tj−1) − s]
 ds.
Now suppose that (θ1, . . . , θn) ∈ Rn is such that
i=j θi ∈ DS for each j ∈ {1, . . . , n}. Then by
Lemma 4.3 it follows that there exists θ∗ ∈ DS such that θ∗ ≥ 0,
i=j θi ≤ θ∗ for all j ∈ {1, . . . , n},
θiNC0(−αtj−1 − s, α(ti − tj−1) − s] ≤ θ∗NC0(R), a.s..
By (33) and the dominated convergence theorem, we have
Jj(α) = ν
i=j θiS
Hence we proved (32) whenever (θ1, . . . , θn) ∈ Rn satisfies
i=j θi ∈ DS for every j ∈ {1, . . . , n}. Now
suppose that (θ1, . . . , θn) ∈ Rn is such that
i=j θi /∈ DS for some j ∈ {1, . . . , n}. We have that
Jj(α) is bigger than or equal to
α(tj − tj−1)
∫ α(tj−tj−1)
1{θi < 0}θiS +
1{θi > 0}θiNC0 [0, α(tj − tj−1) − s]
1{θi < 0}θiS +
1{θi > 0}θiNC0 [0, α(tj − tj−1)(1 − z)]
 − 1
 dz.
The expectation in the latter formula goes to E[exp(
i=j θiS) − 1] as α → ∞ by the monotone
convergence theorem. Therefore, by Fatou’s lemma we have
Jj(α) ≥ νE
 = ∞.
Thus, since the quantities J1(α), . . . , Jn(α) are bounded below by −ν, we get (32) also in this
case. We now show
log E
Xk∈I|[−T,0]
NCk(0, αti]
 = 0 (34)
for all (θ1, . . . , θn) ∈ Rn such that
i=j θi ∈ DS for each j ∈ {1, . . . , n}. By Lemma 4.3 we have
that there exists θ∗ ∈ DS such that θ∗ ≥ 0,
i=j θi ≤ θ∗ for all j ∈ {1, . . . , n} and
Xk∈I|[−T,0]
NCk(R) ≤
Xk∈I|[−T,0]
NCk(0, αti] ≤ θ∗
Xk∈I|[−T,0]
NCk(R), a.s.,
where θ− ≡
i:θi<0
θi and θ− ≡ 0 if {i : θi < 0} = ∅. Therefore, using again Lemma 6.4 VI in
Daley and Vere-Jones (2003), we have
νT (E[eθ−S ] − 1)
Xk∈I|[−T,0]
NCk(0, αti]
 ≤ exp
νT (E[eθ
∗S ] − 1)
Equation (34) follows taking the logarithms in the above inequalities and passing to the limit. The
conclusion follows putting together (29), (32) and (34).
Proof of (b) and Proof of (c). Part (b) can be shown using assumption (5) and following the lines
of the proof of part (b) of Proposition 2.2 in Ganesh, Macci and Torrisi (2005). The proof of part
(c) is identical to the proof of part (c) of Proposition 2.2 in Ganesh, Macci and Torrisi (2005).
End of the proof under condition (26). By claim (C) and the Dawson-Gärtner theorem, {NXα·,T (0, α·]/α}
satisfies the LDP on D[0, 1], equipped with the topology of point-wise convergence, with speed α
and good rate function
J̃(f) = sup
(tk − tk−1)Λ∗
(f(tk) − f(tk−1)
tk − tk−1
: n ≥ 1, 0 ≤ t1 < . . . < tn ≤ 1
The conclusion follows noticing that J̃(·) coincides with J(·) in (25), as can be checked following
the same lines as in the proof of Lemma 5.1.6 in Dembo and Zeitouni (1998).
Removing the additional condition (26). The general case is solved as follows. Since Ck is almost
surely finite, there exists a left-most extremal point Yk ∈ Ck such that NCk(−∞, Yk) = 0 a.s.. Note
that, given the immigrants, Yk − Xk is an iid sequence. Therefore, by a classical result on Poisson
processes we have that {Yk} is a Poisson process with intensity ν. Viewing Xt,T as a Poisson cluster
process with cluster centers Yk and clusters Ck, the conclusion follows by the first part of the proof.
Proof of Theorem 4.1. The proof uses similar steps as in the proof of Theorem 4.2. Here we sketch
the main difference. Assume the additional condition NC0((−∞, 0)) = 0 a.s. (the general case can
be treated as in the proof of Theorem 4.2). Define the following subsets of Rn:
(θ1, . . . , θn) ∈ Rn :
θi ∈ DS for all j ∈ {1, . . . , n}
(θ1, . . . , θn) ∈ Rn :
θi /∈ DS for some j ∈ {1, . . . , n}
We start showing that for all n ≥ 1 and 0 ≤ t1 < . . . < tn ≤ 1
Λt1,...,tn(θ1, . . . , θn) =
j=1(tj − tj−1)Λ
i=j θi
for (θ1, . . . , θn) ∈ A1
+∞ for (θ1, . . . , θn) ∈ A2,
where
Λt1,...,tn(θ1, . . . , θn) ≡ lim
log E
θiNX(0, αti]
and Λ(·) is defined in the statement of Theorem 3.2. Using the definition of X and the assumption
NC0((−∞, 0)) = 0 a.s., we have
θiNX(0, αti]
Xk∈I|(0,αti]
NCk(0, αti]
Xk∈I|[−T,0]
NCk(0, αti]
Xk∈I|(−∞,−T )
NCk(0, αti]
As noticed in the proof of Theorem 4.2 we have
θiNXαti,T
(0, αti]
Xk∈I|(0,αti]
NCk(0, αti]
Xk∈I|[−T,0]
NCk(0, αti]
Therefore, by the computations in the proof of Theorem 4.2, to prove (35) we only need to check
log E
Xk∈I|(−∞,−T )
NCk(0, αti]
 = 0, for all (θ1, . . . , θn) ∈ A1. (36)
Since {(Xi, Ci) : Xi ∈ I|(−∞,−T )} is an independently marked Poisson process, by Lemma 6.4.VI in
Daley and Vere-Jones (2003) we have
Xk∈I|(−∞,−T )
NCk(0, αti]
 = E
Xk∈I|(−∞,−T )
θiNCk−Xk(−Xk, αti − Xk]
= exp
i=1 θiNC0 (x,αti+x] − 1
Take (θ1, . . . , θn) ∈ A1. By Lemma 4.3 we have that there exists θ∗ ∈ DS such that θ∗ ≥ 0,
i=j θi ≤ θ∗ for all j ∈ {1, . . . , n} and
θ−NC0(R) ≤
θiNC0(x, αti + x] ≤ θ∗NC0(R), a.s.
where θ− ≡
i:θi<0
θi and θ− ≡ 0 if {i : θi < 0} = ∅. Thus,
i=1 θiNC0 (x,αti+x] − 1 ≤ (eθ∗NC0 (R) − 1)1{x ≤ L}, for all x ≥ T
i=1 θiNC0 (x,αti+x] − 1 ≥ (eθ−NC0(R) − 1)1{x ≤ L}, for all x ≥ T
The limit (36) follows by assumption (6) noticing that the above relations yield, for all (θ1, . . . , θn) ∈
Xk∈I|(−∞,−T )
NCk(0, αti]
 ≤ exp(νE[L(eθ∗S − 1)])
Xk∈I|(−∞,−T )
NCk(0, αti]
 ≥ exp(νE[(eθ−S − 1)(L − T )1{L ≥ T}]).
Now since DS is open, the claim follows by applying first the Gärtner-Ellis theorem in R
n to get
the LDP for the finite-dimensional distributions, and then the Dawson-Gärtner theorem to have
the LDP for the process (argue as in the proof of Theorem 4.2 for the remaining steps).
4.2 Sample path large deviations in the topology of uniform convergence
In the applications, one usually derives LDPs for continuous functions of sample paths of stochastic
processes by using the contraction principle. Since the topology of uniform convergence is finer than
the topology of point-wise convergence, it has a larger class of continuous functions. Thus, it is of
interest to understand if {NX(0, α·]/α} satisfies a LDP on D[0, 1] equipped with the topology of
uniform convergence. In this section we give an answer to this question assuming that the tails of
S decay super-exponentially.
Theorem 4.4 Assume
E[eθS ] < ∞ for each θ ∈ R (37)
E[LeθS ] < ∞ for each θ ∈ R. (38)
Then {NX(0,α·]
} satisfies a LDP on D[0, 1], equipped with the topology of uniform convergence, with
speed α and good rate function (25).
In this section, without loss of generality we assume that the points of I are {Xi}i∈Z∗ , where
∗ = Z\{0}, Xi < Xi+1, and we set X0 = 0. As usual, we denote by Xt,T the non-stationary
Poisson cluster process with immigrant process empty on (−∞,−T ) ∪ (t + T,∞), where T > 0 is
a fixed constant, and by C0 the cluster generated by an immigrant at 0.
Before proving Theorem 4.4 we show that the same LDP holds for Xt,T , under a weaker con-
dition.
Theorem 4.5 Assume (37). Then {NXα·,T (0, α·]/α} satisfies a LDP on D[0, 1], equipped with the
topology of uniform convergence, with speed α and good rate function (25).
To prove Theorem 4.5 above we use the following Lemma 4.6, whose proof is omitted since it is
similar to the proof of Lemma 3.3 in Ganesh, Macci and Torrisi (2005). Let {Sk}k∈Z be the iid
sequence of random variables (distributed as S) defined by Sk = NCk(R).
Lemma 4.6 Assume (37), NC0(−∞, 0) = 0 a.s., and define
(Sk − NCk−Xk(0,Xk]), n ≥ 1.
It holds
log P (An ≥ nδ) = −∞ for each δ > 0.
Proof of Theorem 4.5. We prove the theorem assuming that NC0(−∞, 0) = 0 a.s.. The general
case is solved as in the proof of Theorem 4.2. As usual denote by I|A the restriction of I on the
Borel set A ⊂ R. Define
C(t) =
Xk∈I|(0,t]
Sk, t > 0.
We prove that {NXα·,T (0, α·]/α} and {C(α·)/α} are exponentially equivalent (see, for instance,
Definition 4.2.10 in the book of Dembo and Zeitouni, (1998)) with respect to the topology of
uniform convergence. Therefore the conclusion follows by a well-known result on sample path large
deviations, with respect to the uniform topology, of compound Poisson processes (see, for instance,
Borovkov (1967); see also de Acosta (1994) and the references cited therein) and Theorem 4.2.13
in Dembo and Zeitouni (1998). Define
CT (t) =
Xk∈I|[−T,t]
Sk, t > 0.
Using Chernoff bound and condition (37) can be easily realized that the processes {C(α·)/α}
and {CT (α·)/α} are exponentially equivalent with respect to the topology of uniform conver-
gence. Therefore, it suffices to show that {CT (α·)/α} and {NXα·,T (0, α·]/α} are exponentially equiv-
alent with respect to the topology of uniform convergence. Note that the assumption NC0(−∞, 0) =
0 a.s. gives
NXt,T (0, t] =
Xk∈I|(0,t]
NCk(0, t] +
Xk∈I|[−T,0]
NCk(0, t] t > 0, a.s..
Therefore, we need to show that
log P (Mα > δ) = −∞, for any δ > 0, (39)
where
t∈[0,1]
CT (αt) −
Xk∈I|(0,αt]
NCk(0, αt] −
Xk∈I|[−T,0]
NCk(0, αt]
Since
Mα ≤ M (1)α + M (2)α , a.s.,
where
M (1)α =
Xk∈I|[−T,0]
Sk and M
t∈[0,1]
Xk∈I|(0,αt]
(Sk − NCk(0, αt]),
the limit (39) follows if we prove
log P (M (1)α > δ/2) = −∞, for any δ > 0 (40)
log P (M (2)α > δ/2) = −∞, for any δ > 0. (41)
The limit (40) easily follows by the Chernoff bound and condition (37). It remains to show
(41). Since the random function t 7→ NCk(0, αt] is non-decreasing, it is clear that the supremum
over t is attained at one of the points Xn, n ≥ 1. Thus
M (2)α =
n≥1:Xn≤α
(Sk − NCk(0,Xn]).
Note that
M (2)α ≤ M̃α where M̃α =
n≥1:Xn≤α
(Sk − NCk−Xk(0,Xn − Xk]) a.s..
Therefore (41) follows if we show
log P (M̃α > δ/2) = −∞, for any δ > 0. (42)
Since Xn, n ≥ 1, is the sum of n exponential random variables with mean 1/ν, using Chernoff
bound and taking the logarithm, we have that, for all η > 0 and all integers K > ν,
log P (XK[α] < α) ≤ η +
ν + η
. (43)
Here the symbol [α] denotes the integer part of α. Next, observe that using the union bound we
P (M̃α > δ/2,XK[α] ≥ α) ≤ K[α] max
1≤n≤K[α]
(Sk − NCk−Xk(0,Xn − Xk]) ≥ αδ/2
Now we remark that, for n ≥ 1, (Xn − X1, . . . ,Xn − Xn−1) and (Xn−1, . . . ,X1) have the same
joint distribution. Moreover, given I, the centered processes Ck − Xk are iid and independent of
the {Xk}. Hence, letting An denote the random variable defined in the statement of Lemma 4.6,
we have
P (M̃α > δ/2,XK[α] ≥ α) ≤ K[α] max
1≤n≤K[α]
P (An ≥ αδ/2).
The random variables An are increasing in n, therefore
P (M̃α > δ,XK[α] ≥ α) ≤ K[α]P (AK[α] ≥ αδ/2),
and by Lemma 4.6 we have
log P (M̃α > δ/2,XK[α] ≥ α) = −∞. (44)
Now note that
P (M̃α > δ/2) ≤ P (M̃α > δ/2,XK[α] ≥ α) + P (XK[α] < α),
for arbitrary K > ν. Hence by (43) and(44) we have
lim sup
log P (M̃α > δ/2) ≤ inf
η + K log
ν + η
= K − ν − K log K
Then we obtain (42) by letting K tend to ∞.
Proof of Theorem 4.4. Throughout the proof we assume NC0((−∞, 0)) = 0 a.s.. The general case
is solved as in the proof of Theorem 4.2. Let {CT (t)} be the process defined in the proof of Theorem
4.5. The claim follows if we show that {CT (α·)/α} and {NX(0, α·]/α} are exponentially equivalent
with respect to the topology of uniform convergence. Note that the assumption NC0(−∞, 0) = 0
a.s. implies
NX(0, t] = NXt,T (0, t] +
Xk∈I|(−∞,−T )
NCk(0, t], t > 0 a.s..
Therefore, since we already proved that {CT (α·)/α} and {NXα·,T (0, α·]/α} are exponentially equiv-
alent with respect to the uniform topology (see the proof of Theorem 4.5), the claim follows if we
prove that
log P
Xk∈I|(−∞,−T )
NCk(0, α] > αδ
 = −∞, for any δ > 0. (45)
Using the Chernoff bound we have, for all θ > 0,
Xk∈I|(−∞,−T )
NCk(0, α] > αδ
 ≤ e−αθδE
Xk∈I|(−∞,−T )
θNCk(0, α]
= e−αθδ exp
E[eθNC0 (x,α+x] − 1] dx
≤ e−αθδ exp
νE[(eθS − 1)L]
Taking the logarithm, dividing by α, letting α tend to ∞ and using assumption (38) we get
lim sup
log P
Xk∈I|(−∞,−T )
NCk(0, α] > αδ
 ≤ −θδ, for all θ > 0.
Relation (45) follows letting θ tend to infinity in the above inequality.
5 Large deviations of spatial Poisson cluster processes
5.1 The large deviations principle
A spatial Poisson cluster process X is a Poisson cluster process in Rd, where d ≥ 1 is an integer. The
clusters centers are the points {Xi} of a homogeneous Poisson process I ⊂ Rd with intensity ν ∈
(0,∞). Each immigrant Xi ∈ I generates a cluster Ci = CXi , which is a finite point process. Given
I, the centered clusters {CXi −Xi} are iid and independent of I. X is the union of all clusters. As
in dimension 1, we denote by S the number of points in a cluster, with a little abuse of notation
by C0 the cluster generated by a point at 0, and by L the radius of C0. Moreover, we denote by
NX(b(0, r)) the number of points of X in the ball b(0, r), and by
ωd(r) =
rdπd/2
Γ(1 + d/2)
the volume of b(0, r). The following LDP holds:
Theorem 5.1 Assume (5) and
E[LdeθS ] < ∞, for all θ ∈ D◦S. (46)
Then {NX(b(0, r))/ωd(r)} satisfies a LDP on R with speed ωd(r) and good rate function (7).
Before proving Theorem 5.1, we show that the same LDP holds for the non-stationary Poisson
cluster process Xr,R with immigrant process empty in R
d \ b(0, R + r). As usual this LDP holds
under a weaker condition.
Theorem 5.2 Assume (5). Then {NXr,R(b(0, r))/ωd(r)} satisfies a LDP on R with speed ωd(r)
and good rate function (7).
Proof of Theorem 5.2 The proof is similar to that one for the non-stationary Poisson cluster
process on the line. Here we just sketch the main differences. As in the proof of Theorem 3.3,
the claim follows by the Gärtner-Ellis theorem. Indeed, letting I|b(0,r) denote the point process of
immigrants in b(0, r), and I|b(0,R+r)\b(0,r) the point process of immigrants in b(0, R + r) \ b(0, r) we
have, for each θ ∈ R,
θNXr,R (b(0,r))
Xi∈I|b(0,r)
NCi (b(0,r))
Xi∈I|b(0,R+r)\b(0,r)
NCi(b(0,r))
As usual, with a little abuse of notation denote by C0 the cluster generated by an immigrant at
0. It holds:
Xi∈I|b(0,r)
NCi (b(0,r))
= exp
b(0,r)
eθNC0 (b(−x,r)) − 1
for each θ ∈ R;
Xi∈I|b(0,R+r)\b(0,r)
NCi(b(0,r))
≤ exp
ν(ωd(R + r) − ωd(r))E
eθS − 1
for θ ∈ [0,∞) ∩ DS ;
1 ≥ E
Xi∈I|b(0,R+r)\b(0,r)
NCi (b(0,r))
≥ exp
ν(ωd(R + r) − ωd(r))E
eθS − 1
for θ ≤ 0. Therefore,
ωd(r)
log E
Xi∈I|b(0,r)
NCi(b(0,r))
= lim
ωd(r)
b(0,r)
E[eθNC0 (b(−x,r)) − 1] dx
ωd(1)
b(0,1)
E[eθNC0 (b(−ry,r)) − 1] dy
= νE[eθS − 1], for each θ ∈ R,
and, since limr→∞ ωd(R + r)/ωd(r) = 1, for each θ ∈ DS ,
ωd(r)
log E
Xi∈I|b(0,R+r)\b(0,r)
NCi (b(0,r))
The rest of the proof is exactly as in the one-dimensional case.
Proof of Theorem 5.1 The proof is similar to that one of Theorem 5.2 and is again based on the
Gärtner-Ellis theorem. We start showing that
ωd(r)
log E
θ(NX(b(0,r))−NXr,R (b(0,r)))
= 0, for all θ ∈ D◦S . (47)
This relation is easily verified for θ ≤ 0. Thus we only check (47) for θ ∈ D◦S ∩ (0,∞). We have, for
all θ ∈ D◦S ∩ (0,∞),
θ(NX(b(0,r))−NXr,R (b(0,r)))
Xi∈I|Rd\b(0,r+R)
NCi (b(0,r))
= exp
Rd\b(0,r+R)
eθNC0 (b(−x,r)) − 1
Now notice that since θ > 0 we have
eθNC0 (b(−x,r)) − 1 ≤ (eθNC0 (Rd) − 1)1{‖x‖ ≤ L + r}, for all x ∈ Rd.
The limit (47) follows by assumption (46) noticing that the above relations yield, for all θ ∈
S ∩ (0,∞), r > 0,
Xi∈I|Rd\b(0,r+R)
NCi(b(0,r))
≤ exp((νπd/2/Γ(1 + d/2))E[(L + r)d(eθS − 1)]).
Now notice that
eθNX(b(0,r))
θNXr,R (b(0,r))
θ(NX(b(0,r))−NXr,R (b(0,r)))
, for all θ ∈ R, r > 0.
Therefore, if DS = D
S then the claim is a consequence of the computation of the log-Laplace limit
of {NXr,R(b(0, r))} in the proof of Theorem 5.2, the Gärtner-Ellis theorem and assumption (5). It
remains to deal with the case DS 6= D◦S . Arguing exactly as in the proof of Theorem 3.2 it can
be proved that for any sequence {rn}n≥1 ⊂ (0,∞) diverging to +∞, as n → ∞, there exists a
subsequence {qn} ⊆ {rn} such that
lim sup
log P (NX(b(0, qn))/ωd(qn) ∈ F ) ≤ − inf
Λ∗(x), for all closed sets F
lim inf
log P (NX(b(0, qn))/ωd(qn) ∈ G) ≥ − inf
Λ∗(x), for all open sets G,
where Λ∗ is defined by (7). Then the large deviations upper and lower bounds hold for any sequence
{rn} and the claim follows.
5.2 The asymptotic behavior of the void probability function and the empty
space function
Apart some specific cases, the void probability function v(r) = P (NX(b(0, r)) = 0), r > 0, of
a spatial Poisson cluster process is not known in closed form. Comparing X with the immigrant
process I we easily obtain
v(r) ≤ P (NI(b(0, r)) = 0) = e−νωd(r), r > 0. (48)
A more precise information on the asymptotic behavior of v(·), as r → ∞, is provided by the
following proposition:
Proposition 5.3 Assume E[Ld] < ∞. Then
ωd(r)
log v(r) = −ν.
Proof Note that
v(r) = P (NCi(b(0, r)) = 0, for all Xi ∈ I)
1{NI(b(0, r)) = 0}
Xi∈I|Rd\b(0,r)
1{NCi(b(0, r)) = 0}
= e−νωd(r)E
Xi∈I|Rd\b(0,r)
1{NCi(b(0, r)) = 0}
= e−νωd(r) exp
Rd\b(0,r)
P (NC0(b(−x, r)) > 0) dx
where in (49) we used Lemma 6.4.VI in Daley and Vere-Jones (2003). Thus the claim follows if we
prove
ωd(r)
Rd\b(0,r)
P (NC0(b(−x, r)) > 0) dx = 0.
For this note that
1{NC0(b(0, r)) > 0} ≤ 1{‖x‖ − r ≤ L}, for all x ∈ Rd, r > 0.
Therefore,
ωd(r)
Rd\b(0,r)
P (NC0(b(−x, r)) > 0) dx ≤ E[(1 + L/r)d − 1],
and the right-hand side in the above inequality goes to 0 as r → ∞ by the dominated convergence
theorem (note that E[Ld] < ∞ by assumption).
In spatial statistics, a widely used summary statistic is the so-called empty space function, which is
the distribution function of the distance from the origin to the nearest point in X (see, for instance,
Møller and Waagepetersen (2004)), that is
e(r) = 1 − v(r), r > 0.
Apart some specific cases, the empty space function of Poisson cluster processes seems to be in-
tractable. Next Corollary 5.4 concerns the asymptotic behavior of e(r), as r → ∞.
Corollary 5.4 Under the assumption of Proposition 5.3 it holds
ωd(r)
log log e(r)−1 = −ν.
Proof The proof is an easy consequence of Proposition 5.3. By the upper bound (48) we obtain
lim sup
ωd(r)
log log e(r)−1 ≤ lim
ωd(r)
log log(1 − e−νωd(r))−1 = −ν.
To get the matching lower bound we note that the inequality log(1 − x) ≤ −x, x ∈ [0, 1), gives
log e(r)−1 ≥ v(r), r > 0, and therefore by Proposition 5.3 we get
lim inf
ωd(r)
log log e(r)−1 ≥ lim inf
ωd(r)
log v(r) = −ν.
5.3 Spatial Hawkes processes
Spatial Hawkes processes have been introduced in Daley and Vere-Jones (2003). Brémaud, Mas-
soulié and Ridolfi (2005) considered spatial Hawkes processes with random fertility rate and not
necessarily Poisson immigrants, and computed the Bartlett spectrum; the reader is directed to
Daley and Vere-Jones (2003) for the notion of Bartlett spectrum. Møller and Torrisi (2005) de-
rived the pair correlation function of spatial Hawkes processes; we refer the reader to Møller and
Waagepetersen (2004) for the notion of pair correlation function.
For the sake of completeness, we briefly recall the definition of spatial Hawkes process. A spatial
Hawkes process is a Poisson cluster process X ⊂ Rd where d ≥ 1 is an integer. The clusters centers
are the points {Xi} of a homogeneous Poisson process I ⊂ Rd with intensity ν ∈ (0,∞). Each
immigrant Xi ∈ I generates a cluster Ci = CXi which is formed by the points of generations
n = 0, 1, . . . with the following branching structure: the immigrant Xi ∈ I is said to be of zero-
th generation. Given generations 0, 1, . . . , n in Ci, each point Y ∈ Ci of generation n generates
a Poisson process on Rd of offspring of generation n + 1 with intensity function h(· − Y ). Here
h : Rd → [0,∞) is a non-negative Borel function. In the model it is assumed that, given the
immigrants, the centered clusters {Ci −Xi} are iid, and independent of I. By definition the spatial
Hawkes process is X ≡
i Ci. As in the one-dimensional case, it is assumed
0 < µ ≡
h(ξ) dξ < 1. (50)
This assumption guarantees that the number of points in a cluster has a finite mean equal to
1/(1−µ), excludes the trivial case where there are no offspring, and ensures that X is ergodic, with
a finite and positive intensity given by ν/(1−µ). Due to the branching structure, the number S of
offspring in a cluster follows the distribution (3). Finally, we note that the classical Hawkes process
considered in the previous sections corresponds to the special case where d = 1 and h(t) = 0 for
t ≤ 0.
A LDP for spatial Hawkes processes can be obtained by Theorem 5.1. The precise statement is
as Theorem 5.1 with (50) and
‖ξ‖h(ξ) dξ < ∞,
in place of (5) and (46), moreover the rate function is Λ∗(·) defined by (20). Here the symbol ‖ · ‖
denotes the Euclidean norm.
Similarly, the asymptotic behavior of the void probability function and the empty space function
of spatial Hawkes processes can be obtained as immediate consequences of Proposition 5.3 and
Corollary 5.4, respectively. The precise statements are as Proposition 5.3 and Corollary 5.4, with
conditions (50) and
‖ξ‖dh(ξ) dξ < ∞
in place of E[Ld] < ∞.
6 Extensions and open problems
In this paper we studied large deviations of Poisson cluster processes. Applications of these results
to insurance and queueing models are presently under investigation by the authors.
The definition of Hawkes process extends immediately to the case of random fertility rate
h(·, Z), where Zk’s are iid unpredictable marks associated to the points Xk (see Daley and Vere-
Jones (2003) for the definition of unpredictable marks, and Brémaud, Massoulié and Ridolfi (2005)
for the construction of Hawkes processes with random fertility rate specified by an unpredictable
mark). Due to the form of the distribution of S in this case (see formula (6) in Møller and Rasmussen
(2005)) it is not clear if the LDPs for Hawkes processes proved in this paper are still valid for Hawkes
processes with random fertility rate.
The generalization of our results to non-linear Hawkes processes (Kerstan (1964); Brémaud and
Massoulié (1996); Massoulié (1998); Brémaud, Nappo and Torrisi (2002); Torrisi (2002)) would be
interesting. However, since a non-linear Hawkes process is not even a Poisson cluster process, a
different approach is needed.
Acknowledgements
We thank Kamil Szczegot for reporting two mistakes and for a careful reading of a first draft of
the paper.
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	Introduction
	Preliminaries
	Poisson cluster processes
	Hawkes processes
	Large deviation principles
	Scalar large deviations
	Scalar large deviations of Poisson cluster processes
	Scalar large deviations of Hawkes processes
	Sample path large deviations
	Sample path large deviations in the topology of point-wise convergence
	Sample path large deviations in the topology of uniform convergence
	Large deviations of spatial Poisson cluster processes
	The large deviations principle
	The asymptotic behavior of the void probability function and the empty space function
	Spatial Hawkes processes
	Extensions and open problems
