A convergence criterion for systems of point processes from the convergence of their stochastic intensities

We define a metric d on the space N of locally finite simple point measures. By construction, the convergence of a sequence of point measures with respect to d means the convergence of their atoms. The probabilistic and topological properties of this metric allow us prove that the convergence in distribution of the stochastic intensities of a system of point processes implies the convergence of the system. MSC 2010 subject classifications: 60B05, 60G55, 60G57.


Introduction
In this paper we consider systems of point processes admitting stochastic intensities. Processes of these type arise naturally in the study of particle systems, such as modeling of neural activity (see e.g. [14] or [15]) or financial data (see e.g. [2] or [13]). Note that such processes can always be written as thinnings of Poisson measures.
The natural question is: when does the convergence in distribution in Skorohod space of systems of such processes follow from the convergence of their intensities? In this paper we study this question. Our main result, Theorem 1.1, roughly states that if the intensities of point processes converge in distribution and their limits are independent of the underlying Poisson measures, then the systems of point processes converge as well.
To give the formal statement of Theorem 1.1, denote by D(R + , R) Skorohod space, M the space of locally finite measures endowed with the topology of vague convergence and N the subspace of locally finite simple point measures. Assume that, for every n ≥ 1, (Y N,1 , π 1 , ..., Y N,n , π n ) converges in distribution to (Ȳ 1 , π 1 , ...,Ȳ n ,π n ) in (D(R + , R) × N ) n , and that, for each k ≥ 1,Ȳ k is independent ofπ k . Then, for any n ≥ 1, Z N,k 1≤k≤n converges to Z k 1≤k≤n in distribution in D(R + , R n ). In particular, Z N,k k≥1 converges to Z k k≥1 in distribution in D(R + , R) N * endowed with the product topology. Remark 1.2. In the statement of Theorem 1.1, we need to guarantee the following property: Poisson random measures are N −valued random variables. This is a direct consequence of Theorem 2.6.III.(ii) of [6] and of the definition of Poisson measures (see Definition 3.1).

Remark 1.3.
According to Lemma 4 of [4], a point process Z having stochastic intensity (Y s− ) s≥0 (where Y is a càdlàg process) can always be written in the form of Theorem 1.1.
Let us note that, in Theorem 1.1, the processes Y N,k are not assumed to be independent of the Poisson measures π k . Otherwise, the proof would be straightforward by conditioning by Y N,k . Let us also note that the condition of independence of the limiting intensitiesȲ k from Poisson measuresπ k is often satisfied and natural in many examples of application. It holds for example when the limiting intensities are deterministic, or when they are functionals of Brownian motions with respect to the same filtration asπ k .
If the processes Y N,k are semimartingales, the result of Theorem 1.1 can follow from Theorem IX.4.15 of [10], provided the convergence of the characteristics of the corresponding semimartingales holds. But we do not assume the semimartingale structure for the intensities Y N,k in Theorem 1.1.
When the limiting intensities are deterministic, Theorem 1.1 can be compared to Theorem 1 of [5] that states that the convergence of point processes is implied by the pointwise convergence in distribution of their compensators (i.e. for each t ≥ 0, the compensator at time t converges in distribution in R). In [5], Theorem 1 holds when the compensator of the limit point process is a deterministic function, whereas in Theorem 1.1, the limit point processes have stochastic intensities.
The idea of the proof of Theorem 1.1 consists in writing the point processes Z N,k andZ k as a function Φ of respectively Y N,k , π k andȲ k ,π k (k ≥ 1). Then, knowing that, (Y N,k , π k ) converges in distribution to (Ȳ k ,π k ), we can use Skorohod representation theorem to assume that this convergence is almost sure. Finally, proving that Φ is almost surely continuous at (Ȳ k , π k ) concludes the proof.
The organization of the paper is the following: in Section 2 we give some examples of applications of Theorem 1.1, then in Section 3 we recall some classical properties of Poisson measures and of the vague convergence. Finally, Section 4 is devoted to the proof of Theorem 1.1

Examples
We start this section with an example of system of point processes based on [8]. In this example the stochastic intensities are in fact semimartingales. We illustrate both methods of proof of the convergence of this system: using Theorem IX.4.15 of [10] about semimartingale's convergence and using our Theorem 1.1.
Example 2.1. Let α > 0 and π k (k ≥ 1) be independent Poisson measures on R + ×R + ×R with intensity dt · dz · dν(u) where ν is the centered normal distribution with variance one. Define X N as solution of It was shown in Theorem 1.4 of [8] that X N converges in distribution in D(R + , R) toX, whereX is solution of dX t = −αX t dt + 1 +X 2 t dW t , with W some one-dimensional standard Brownian motion. Define alsō whereπ i (i ≥ 1) are independent Poisson measures on R 2 + having Lebesgue intensity.
In order to show the convergence of (Z N,i ) 1≤i≤n in distribution in D(R + , R n ) using Theorem IX.4.15 of [10], we have to consider the semimartingale (X N , Z N,1 , ..., Z N,n ) and show that its characteristics converge to those of (X,Z 1 , ...,Z n ).
An alternative proof of the convergence of (Z N,i ) 1≤i≤n to (Z i ) 1≤i≤n relies on the Theorem 1.1. Indeed, following Theorem 1.4 of [8], X N converges in distribution in D(R + , R) toX, and, being adapted to the same filtration, the Brownian motion W is independent of the Poisson measuresπ i (i ≥ 1) (see Theorem II.6.3 of [9]). Theorem 1.1 allows us to consider processes that are not semimartingales, such as Hawkes processes and Volterra processes. Since the stochastic intensities of Hawkes processes are not, in general, semimartingales, Theorem 1.1 can be interesting to show the convergence of Hawkes processes, provided that one can show the convergence of their stochastic intensities. Let us give an example of application of Theorem 1.1 in this case. The example is based on Examples 7.3 and 7.4 of [1]. Example 2.2. Let us consider K(t) := t γ for some γ > 0, K N (t) := K(t/N ) and some Poisson random measure π on R 2 + having Lebesgue intensity. Let X N satisfy Theorem 7.2 of [1] implies that the sequence of processes (X N t ) t≥0 = (N −1 X N N t ) t≥0 has converging subsequences (in distribution in the topology L 2 loc ), and that every limit for some standard Brownian motion B. Besides, one can prove with standard arguments, the tightness of (X N ) N in Skorohod topology. Then, we can consider a subsequence of (X N ) N that converges in distribution in the topology of L 2 loc and in Skorohod topology. The limit for both topologies is necessarily the same on Skorohod space. Indeed, letx be the limit for L 2 loc topology andx be the limit for Skorohod topology of a sequence of càdlàg functions (x n ) n . The L 2 loc convergence implies that x n As it was mentioned in Introduction, if the stochastic intensities of the limit point processes are deterministic, then the hypotheses of Theorem 1.1 are satisfied. Let us give a practical example based on [7]. Example 2.3. Consider any locally bounded K : R + → R and any Lipschitz continuous function f : R → R + . Define X N as solution of where π j (j ≥ 1) are independent Poisson measures on R 2 + with Lebesgue intensity, and X as the deterministic solution of Then, Theorem 8 of [7] states that, for some constant C T . In other words, Z N,i converges toZ in a L 1 −sense. AsX is a deterministic function, Theorem 1 of [5], and a fortiori Theorem 1.1, can be used to

Classical properties of Poisson measures and the vague convergence
Let us begin with the usual definition of random measures and Poisson measures. We restrict this definition to the space R 2 + since we only need this space in the paper, but Definition 3.1 can be generalized to any measurable space. In the rest of the paper, R 2 + is always endowed with the Borel sigma algebra B(R 2 + ). Definition 3.1. A locally finite random measure on R 2 + is a M−valued random variable, where M is endowed with the σ−algebra generated by the functions π ∈ M → π(B) (B ∈ B(R 2 + )). A Poisson measure on R 2 + is a locally finite random measure π satisfying: Proof. Let us write Then, introducing the random measure π (2) (called the second factorial measure of π) defined as According to the multivariate Mecke equation (see Theorem 4.4 of [12]) and Fubini-Tonelli's theorem, the expectation of the integral above is This allows to conclude that P (∃t, π({t} × R + ) ≥ 2) = 0.
We end this section with a result claiming that the vague convergence of locally finite point measures implies the convergence of their atoms. ). Then, for k large enough, n k = n, and there exists a sequence of permutation (σ k ) k of 1, n such that, for all 1 ≤ i ≤ n, t k σ k (i) (resp. z k σ k (i) ) converges to t i (resp. z i ) as k goes to infinity.
Proof. To begin with, Proposition A2.6.II.(iv) of [6] implies that n k converges to n as k goes to infinity. As n k (k ∈ N) and n are integers, this implies that n k = n for k large enough.
To show the convergence of t k i and z k i , (1 ≤ i ≤ n) let us fix some ε > 0. Then, for each 1 ≤ i ≤ n, consider an open ball B i centered on (t i , z i ) of radius smaller than ε (for the supremum norm) such that, for all i = j, B i ∩ B j = ∅.
Thanks to Proposition A2.6.II.(iv) of [6], we know that P k (B i ) converges to P (B i ) = 1 (1 ≤ i ≤ n). Since P k are point measures, this implies that for all 1 ≤ i ≤ n, P k (B i ) = 1 for k large enough. As the sets B i (1 ≤ i ≤ n) are disjoint, there exists a permutation σ k such that (t k σ k (i) , z k σ k (i) ) ∈ B i . Hence, for k large enough, for all 1 ≤ i ≤ n, |t k σ k (i) − t i | ≤ ε and |z k σ k (i) − z i | ≤ ε.

Proof of Theorem 1.1
This section is dedicated to prove Theorem 1.1. Let us begin with an important result.
Let (x, π) ∈ D(R + , R + ) m × N m . A sufficient condition for Φ to be continuous at (x, π) is: Before proving Theorem 4.1, let us point out that, in general, Φ is not continuous at every point of D(R + , R + ) m × N m . This is shown in Example 4.2, where hypothesis (d) is not satisfied.

Example 4.2.
Let us consider the point measure π = δ (1,1) and the constant function x : t ∈ R + → 1. In addition, we consider the functions x n defined as in Figure 1 below. Obviously, ||x − x n || ∞ = 1/n, but Φ(x, π) t = 1 {t≥1} and Φ(x n , π) = 0. In other words, x n converges uniformly to x, but Φ(x n , π) does not converge to Φ(x, π) for non-trivial topologies.    ] vanishes as N goes to infinity since This concludes the proof.
We assume that there exists a dense subset A ⊆ [0, T ] containing T and such that, for all t ∈ A, g N (t) converges to g(t). Moreover, we assume that for all i 1 = i 2 , for all j 1 ∈ 1, n i1−1 and j 2 ∈ 1, n i2−1 , t i1,j1 = t i2,j2 . Then g N converges to g in D [0, T ], R k .
Proof. Since g N (T ) = (n N i − 1) 1≤i≤k converges to g(T ) = (n i − 1) 1≤i≤k , we know that n N i = n i for all N (large enough) and all 1 ≤ i ≤ k. Now, we show that for each 1 ≤ i ≤ k, 1 ≤ j ≤ n i − 1, t N i,j converges to t i,j . As the sequence (t N i,j ) N is bounded, it is sufficient to show that t i,j is its only limit point. Let s be a limit of a subsequence (t We show that s = t i,j . If s > t i,j there would exist some r ∈ A∩]t i,j , s[ satisfying that g ϕ(N ) (r) converges to g(r). This is not possible because, as r < s = lim N t ϕ(N ) i,j , g ϕ(N ) (r) i ≤ j − 1 for N large enough, and as r > t i,j , g(r) i ≥ j. For the same reason, it is not possible to have s < t i,j . As a consequence, t i,j is the only limit point of the bounded sequence t N i,j N . This implies the convergence of t N i,j to t i,j . In the rest of the proof, let us re-index the set {t i,j : 1 ≤ i ≤ k, 1 ≤ j ≤ n i − 1} as {s i : 1 ≤ i ≤ n} where n = k i=1 (n i − 1), such that s 1 < s 2 < ... < s n . And we consider the same indexes for the points t N i,j (1 ≤ i ≤ k, 1 ≤ j ≤ n i − 1). To prove the convergence of g N to g in D([0, T ], R), we just have to define the sequence of functions (λ N ) N such that each λ N is the function that is linear on each interval [s N i , s N i+1 ] and that satisfies λ N (s N i ) = s i . These functions verify g N = g • λ N and This implies that (g N ) N converges to g in Skorohod topology.
Let us consider T ≥ 0 such that for all 1 ≤ j ≤ m, π j ({T } × R + ) = 0 and for all k ∈ N * , π j k ({T } × R + ) = 0. In particular T is a point of continuity of Z and of each Z k , and, as the set of all the atoms of the measures π j and π j k (1 ≤ j ≤ m, k ≥ 1) is countable, the set of points T satisfying the previous conditions is dense.
According to the proof of Theorem 16.2 of [3], in order to prove the convergence of Z k to Z in D(R + , R m ), it is sufficient to prove this convergence in D([0, T ], R m ) for the points T satisfying the conditions of the previous paragraph. Indeed, these points T are continuity points of Z and there exists an increasing sequence of such points T going to infinity. Then, by Lemma 4.4 (whose hypothesis is satisfied thanks to hypothesis (a) and (b)), the convergence of Z k to Z in D([0, T ], R m ) will follow from the convergence of (Z k ) t to Z t for every point t satisfying the same conditions as T . Let us show the convergence of (Z j k ) t to Z j t for every 1 ≤ j ≤ m. In the rest of the proof, we work with fixed j, t, T.
To Notice that To end the proof, one has to note that In the first case, we consider ε > 0 such that ζ i + ε < x j τ i − . Then, noticing that hypothesis (c) guarantees that x j is continuous at τ i , Lemma 4.3 and the convergence of τ i k and ζ i k respectively to τ i and ζ i imply that, for The second case is handled in the same way, as well as the convergence of 1 {τ i k ≤t} to 1 {τ i ≤t} , recalling that π j ({t} × R + ) = 0, and so τ i < t.
Let us finally prove our main result.
Proof of Theorem 1.1.
Then Theorem 4.1 implies the almost sure convergence of the multivariate point processes ( Z N,k ) 1≤k≤n := Φ(( Y N,k , π N,k ) 1≤k≤n ) to ( Z k ) 1≤k≤n := Φ(( Y k , π k ) 1≤k≤n ) in D(R + , R n ). Let us show that the hypothesis of Theorem 4.1 are satisfied almost surely. Hypothesis (a) is a classical property of Poisson measures (see Lemma 3.4). Hypothesis (b) is satisfied since the Poisson measuresπ i (i ≥ 1) are independent, whence, considering i = j, denoting A(π j ) the set of points t ≥ 0 such thatπ j ({t} × R + ) = 0, we have that t∈A(π j ) {t} × R + is a null set (since A(π j ) is finite or countable) independent ofπ i , and consequently Hypothesis (c) and (d) are satisfied for a similar reason. For (c), one has to observe that P ∃t > 0, π j ({t}×R + ) = 1 andȲ j is not continuous at t = P  π j ( t∈D(Ȳ j ) where D(Ȳ j ) is the set of discontinuity points ofȲ j . As D(Ȳ j ) is a.s. finite or countable (see e.g. the discussion after Lemma 1 of Section 12 of [3]), whence t∈D(Ȳ j ) {t} × R + is a null set independent ofπ j , and Lemma 3.3 gives the result: And hypothesis (d) holds true, because the set {(t,Ȳ j t− ) : t ≥ 0} is also a null set independent ofπ j .
Then, the almost sure convergence of ( Z N,k ) 1≤k≤n to ( Z k ) 1≤k≤n implies the convergence in distribution of (Z N,k ) 1≤k≤n to (Z k ) 1≤k≤n in D(R + , R n ).
Step 2: We have shown that, for every n ∈ N * , (Z N,k ) 1≤k≤n converges to (Z k ) 1≤k≤n in distribution in D(R + , R n ). This implies the convergence in the weaker topology D(R + , R) n . Then, the convergence of (Z N,k ) k≥1 to (Z k ) k≥1 as N goes to infinity in D(R + , R) N * is classical (see e.g. Theorem 3.29 of [11]).