Functional central limit theorems for epidemic models with varying infectivity
aa r X i v : . [ m a t h . P R ] S e p Functional central limit theorems for epidemic modelswith varying infectivity
GUODONG PANG AND ´ETIENNE PARDOUX
Abstract.
In this paper, we prove functional central limit theorems (FCLTs) for a stochasticepidemic model with varying infectivity and general infectious periods recently introduced in [10].The infectivity process (total force of infection at each time) is composed of the independent in-fectivity random functions of each infectious individual at the elapsed time (that is, infection-agedependent). These infectivity random functions induce the infectious periods (as well as exposed,recovered or immune periods in full generality), whose probability distributions can be very general.The epidemic model includes the generalized non–Markovian SIR, SEIR, SIS, SIRS models withinfection-age dependent infectivity. In the FCLT for the generalized SEIR model (including SIRas a special case), the limits for the infectivity and susceptible processes are a unique solution toa two-dimensional Gaussian-driven stochastic Volterra integral equations, and then given these so-lutions, the limits for the exposed/latent, infected and recovered processes are Gaussian processesexpressed in terms of the solutions to those stochastic Volterra integral equations. We also presentthe FCLTs for the generalized SIS and SIRS models. Introduction
It has been observed in recent studies of the Covid-19 pandemic (for example, [13]) that theinfectivity of infectious individuals decreases from the epoch of symptom first appearing to fullrecovery. The varying infectivity characteristics also appears in many other epidemic diseases[16, 6]. We have presented a stochastic epidemic model with varying infectivity in [10], where eachindividual has an i.i.d. infectivity random function, and the total force of infection at each timeis the aggregate infectivity of all the individuals that are currently infectious. We have proved afunctional law of large numbers (FLLN) for the epidemic dynamics which results in a deterministicepidemic model, which is the model described as an “age-of-infection epidemic model” in [5, 6]. Inaddition, we have deduced the initial basic reproduction number R from the limit process andcomputed its value for the case of the early phase of the Covid-19 epidemic in France. We haveconcluded a decreased value of R induced by the decrease of the infectivity with age-infection.In this paper we establish functional central limit theorems (FCLTs) for this stochastic epidemicmodel. As discussed in [10], the model can be regarded as a generalization of the SIR and SEIRmodels. In particular, the infectivity random function can take a very general form (see Assumption2.2) and start with a value zero for a period of time, which then in turn determines the durationsof the exposed and infectious periods. Their distributions are also determined by the law of therandom function, and can be very general. As in the FLLN, we study the infectivity process jointlywith the counting processes for the susceptible, exposed, infectious and recovered individuals. Forthe generalized SEIR model, in the FLLN, the infectivity and susceptible functions in the limit areuniquely determined by a two-dimensional Volterra integral equation, and given these two functions,the exposed, infectious and recovered functions in the limit are given by Volterra integral equations.In the FCLT, we first show that the diffusion-scaled infectivity and susceptible processes convergejointly to a two-dimensional Gaussian-driven linear stochastic Volterra integral equation (Theorem Date : September 25, 2020.
Key words and phrases. epidemic model, varying infectivity, infection-age dependent infectivity, Gaussian-drivenstochastic Volterra integral equations, Poisson random measure. D , and takeadvantage of some useful properties of stochastic integrals with respect to the corresponding com-pensated PRMs. We first give a useful decomposition of this process, and construct two auxiliaryprocesses by replacing the random instantaneous infectivity rate process by its deterministic limitfunction in the FLLN. For these auxiliary processes, we employ the moment method to prove theirtightness (using a slightly reinforced criterion for tightness in [4, Theorem 13.5], as stated in theAppendix), which, together with the convergence of finite dimensional distributions, proves theirweak convergence. The martingale approaches employed for the infectious process in the SIR orSEIR models in [18] cannot be used for the aggregate infectivity process, since it is impossible toconstruct appropriate martingales for our purpose. Then the convergence of the susceptible, ex-posed, infectious and recovered individuals follows similarly as those in the classical non-MarkovianSEIR model in [18].We also state the FCLTs without proofs for the generalized SIS and SIRS models with varyinginfectivity (which follow from a slight modification). For the SIS model, the epidemic dynamics isdetermined by the aggregate infectivity process and the infectious process, whose limits are givenby a two-dimensional Gaussian-driven linear stochastic Volterra integral equation (Theorem 2.3).For the SIRS model, the epidemic dynamics is determined by the aggregate infectivity processand the infectious and recovered/immune processes, whose limits are given by a three-dimensionalGaussian-driven linear stochastic Volterra integral equation (Theorem 2.4).This work contributes to the literature of stochastic epidemic models in the aspects of infection-age dependent infectivity, and general infectious periods. The existing work in epidemic modelswith infection-age dependent infectivity has all been about the deterministic models, including themodels by Brauer [5] (see also [6, Chapter 4.5]), and the PDE models (see, e.g., [14, 22, 15, 17]).They are not established as the FLLNs of a well specified stochastic model either, except our workin [10]. Evidently, no FCLT has been yet established for infection-age dependent epidemic models.Our work is the very first for such models. On the other hand, for non-Markovian epidemic modelswith general infectious periods, although some deterministic models (including Volterra integralequations) appeared in the literature (see, e.g., [6, Chapter 4.5] and references therein), rigorouslyestablishing them as a FLLN from a stochastic model was done for the SIR model using Stein’smethod in [20] (using measure-valued processes), and is recently done for the general SIS, SIR,SEIR and SIRS models by the authors in [18] and for multipatch epidemic models in [19]. FCLTsfor these classical stochastic epidemic models and multi–patch models where the infectivity is aconstant, have also been recently established in [18, 19]. This work presents new techniques toestablish the FCLTs for the more realistic but challenging model with varying infectivity. Notethat the study of the final size of an epidemic with general infectious (and possibly latent) period(s)can be done using the Sellke construction [21], see in particular the recent survey [7], and [1, 2, 3].The paper may have insightful practical implications on pandemic crisis studies and management.We have illustrated how integral equations for the SEIR model can be used to estimate the stateof the Covid-19 pandemic using French data in [11], and how the integral equation for the general-ized SEIR model with varying infectivity can be used to better estimate initial basic reproduction number R in [10]. Another recent work by Fodor et al. [9] also uses integral equation (with deter-ministic infectious periods and constant infectivity) to provide a better estimate of R . However,all these papers use the deterministic integral equations arising from the FLLNs. It is clear thatthe FCLTs provides a characterization of stochastic fluctuations around the deterministic averagefunctions. It will be interesting to investigate how these FCLTs can be used to predict when thestate of an epidemic is likely to deviate significantly from the LLN deterministic model.The paper is organized as follows. In Section 2.1, we provide a detailed description of the modeland the assumptions, which is followed by the FCLTs for the generalized SEIR model in Section 2.2.The FCLTs for the generalized SIR and SIRS models are stated in Sections 2.3 and 2.4, respectively.The proofs are given in Section 3, with a tightness criterion stated and established in the Appendix.2. Main Results
Generalized SEIR model with varying infectivity.
In our epidemic model, each individ-ual is associated with an infectivity random function at the epoch of infection, which exerts the infec-tivity to the susceptible individuals. Let the population size be N , and S N ( t ) , E N ( t ) , I N ( t ) , R N ( t )be the numbers of susceptible, exposed, infectious and recovered individuals at each time t , respec-tively. We have the balance equation N = S N ( t ) + E N ( t ) + I N ( t ) + R N ( t ) for t ≥
0. Assume that R n (0) = 0, S N (0) > I N (0) > { λ j ( · ) } , { λ ,Ik ( · ) } and { λ i ( · ) } be the infectivity processes associated with each initially ex-posed, infectious and newly exposed individual, respectively. Assume that the sequence { λ j ( · ) } isi.i.d., and so are { λ ,Ij ( · ) } , and { λ i ( · ) } . These processes are only taking effect during the infectiousperiods, and generate the corresponding exposed and infectious periods. Assume that they all havec`adl`ag paths. In particular, the exposed and infectious periods ( ζ i , η i ) of a newly exposed individualare determined from λ i ( · ) as follows: ζ i = inf { t > , λ i ( t ) > } , and ζ i + η i = inf { t > , λ i ( r ) = 0 , ∀ r ≥ t } , a.s. (2.1)Similarly, the remaining exposed period and the infectious period ( ζ j , η j ) of an initially exposedindividual: ζ j = inf { t > , λ j ( t ) > } > , and ζ j + η j = inf { t > , λ j ( r ) = 0 , ∀ r ≥ t } , a.s. , (2.2)and the remaining infectious period η ,Ik of an initially infectious individual:inf { t > , λ ,Ik ( t ) > } = 0 , and η ,Ik = inf { t > , λ ,Ik ( r ) = 0 , ∀ r ≥ t } , a.s. (2.3)Under the i.i.d. assumptions of the corresponding infectivity processes, the random vectors { ( ζ i , η i ) : i ∈ N } and { ( ζ j , η j ) : j ∈ N } are i.i.d., and so is the sequence { η k : k ∈ N } . Let H ( du, dv ) denotethe law of ( ζ, η ), H ( du, dv ) that of ( ζ , η ) and F ,I the c.d.f. of η ,I . DefineΦ( t ) := Z t Z t − u H ( du, dv ) = P ( ζ + η ≤ t ) , Ψ( t ) := Z t Z ∞ t − u H ( du, dv ) = P ( ζ ≤ t < ζ + η ) , Φ ( t ) := Z t Z t − u H ( du, dv ) = P ( ζ + η ≤ t ) , Ψ ( t ) := Z t Z ∞ t − u H ( du, dv ) = P ( ζ ≤ t < ζ + η ) , and F ,I ( t ) := P ( η ,I ≤ t ) . We write H ( du, dv ) = G ( du ) F ( dv | u ) and H ( du, dv ) = G ( du ) F ( dv | u ),i.e., G is the c.d.f. of ζ and F ( ·| u ) is the conditional law of η , given that ζ = u , G is the c.d.f.of ζ and F ( ·| u ) is the conditional law of η , given that ζ = u . In the case of independent GUODONG PANG AND ´ETIENNE PARDOUX exposed and infectious periods, it is reasonable that the infectious periods of the initially exposedindividuals have the same distribution as the newly exposed ones, that is, F = F . Note that in theindependent case, Ψ( t ) = G ( t ) − Φ( t ) and Ψ ( t ) = G ( t ) − Φ ( t ). Also, let G c = 1 − G , G c = 1 − G , F c ,I = 1 − F ,I , and F c = 1 − F .Let A N ( t ) be the number of individuals that are exposed in (0 , t ], and τ Ni denote the time of the i th individual that gets exposed. Let I N ( t ) be the total force of infection which is exerted on thesusceptibles at time t . By definition, we have I N ( t ) = E N (0) X j =1 λ j ( t ) + I N (0) X k =1 λ ,Ik ( t ) + A N ( t ) X i =1 λ i ( t − τ Ni ) , t ≥ . (2.4)Thus, the infection process A N ( t ) can be expressed by A N ( t ) = Z t Z ∞ u ≤ Υ N ( s ) Q ( ds, du ) , t ≥ , (2.5)where Υ N ( t ) := S N ( t ) N I N ( t ) , (2.6)is the instantaneous infectivity rate function at time t , and Q is a standard Poisson random measure(PRM) on R .The epidemic dynamics of the model can be described by S N ( t ) = S N (0) − A N ( t ) ,E N ( t ) = E N (0) X j =1 ζ j >t + A N ( t ) X i =1 τ Ni + ζ i >t ,I N ( t ) = E N (0) X j =1 ζ j ≤ t<ζ j + η j + I N (0) X k =1 η ,Ik >t + A N ( t ) X i =1 τ Ni + ζ i ≤ t<τ Ni + ζ i + η i ,R N ( t ) = E N (0) X j =1 ζ j + η j ≤ t + I N (0) X k =1 η ,Ik ≤ t + A N ( t ) X i =1 τ Ni + ζ i + η i ≤ t . FCLTs for the generalized SEIR model.
Let D = D ([0 , + ∞ ) , R ) denote the space of R –valued c`adl`ag functions defined on [0 , + ∞ ). Let C denote its subspace of continuous functions.Throughout the paper, convergence in D means convergence in the Skorohod J topology, seeChapter 3 of [4]. Also, D k stands for the k -fold product equipped with the product topology.We first make the following assumptions on the distribution functions and the random infectivityfunctions. Assumption 2.1.
The c.d.f. G satisfies the following assumption: G can be written as G = G + G , where G ( t ) = P i a i t ≥ t i for a finite or countable number of positive numbers a i and thecorresponding t i such that P i a i ≤ and t < t < · · · < t k < . . . , and G is H¨older continuouswith exponent + θ for some θ > , that is, G ( t + δ ) − G ( t ) ≤ cδ / θ for some c > . Moreover,the conditional c.d.f. F ( ·| u ) satisfies the same assumption, uniformly in u . We now state our assumptions on λ , λ ,I and λ . Let ¯ λ ( t ) = E [ λ ( t )], ¯ λ ,I ( t ) = E [ λ ,I ( t )] and¯ λ ( t ) = E [ λ ( t )] for t ≥ Assumption 2.2.
The random functions λ ( t ) (resp. λ ( t ) and resp. λ ,I ( t ) ), of which λ ( t ) , λ ( t ) , . . . (resp. λ ( t ) , λ ( t ) , . . . and resp. λ ,I ( t ) , λ ,I ( t ) , . . . ) are i.i.d. copies, satisfying the following prop-erties. There exists a constant λ ∗ < ∞ such that sup t ∈ [0 ,T ] max { λ ( t ) , λ ,I ( t ) , λ ( t ) } ≤ λ ∗ almostsurely. (i) There exist nondecreasing functions φ and ψ in C and α > / and β > such that forall ≤ r ≤ s ≤ t , denoting ˘ λ ( t ) = λ ( t ) − ¯ λ ( t ) , ( a ) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( s ) (cid:1) (cid:3) ≤ ( φ ( t ) − φ ( s )) α , ( b ) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( s ) (cid:1) (cid:0) ˘ λ ( s ) − ˘ λ ( r ) (cid:1) (cid:3) ≤ ( ψ ( t ) − ψ ( r )) β . Similarly for the infectivity processes { λ ,Ik } k ≥ . (ii) There exist a given number k ≥ , a random sequence ξ ≤ ξ ≤ · · · ≤ ξ k = η andrandom functions λ j ∈ C , ≤ j ≤ k such that λ ( t ) = k X j =1 λ j ( t ) [ ξ j − ,ξ j ) ( t ) . (2.7) Moreover, denoting by F j the c.d.f. of ξ j , we assume that each F j satisfies the conditionson G in Assumption 2.1, and that there exists a nondecreasing function ϕ ∈ C satisfying ϕ ( r ) ≤ Cr α , with α > / and C > arbitrary , (2.8) such that | λ j ( t ) − λ j ( s ) | ≤ ϕ ( | t − s | ) , a.s., (2.9) for all t, s ≥ , ≤ j ≤ k . We remark that the conditions in Assumption 2.2 (i) and (2.8) are not required to establish theFLLN [10]. It is not surprising that the FCLT requires additional assumptions, compared with theFLLN.Let ¯ X N = N − X N for any process X N . Then under Assumptions 2.1 and 2.2, assuming thatthere exist deterministic constants ¯ E (0) , ¯ I (0) ∈ (0 ,
1) such that ¯ E (0)+ ¯ I (0) <
1, and ( ¯ E N (0) , ¯ I N (0)) → ( ¯ E (0) , ¯ I (0)) ∈ R in probability as N → ∞ , it is shown in [10, Theorem 2.1] that (cid:0) ¯ S N , ¯ I N , ¯ I N , ¯ E N , ¯ R N (cid:1) → (cid:0) ¯ S, ¯ I , ¯ I, ¯ E, ¯ R (cid:1) in D as N → ∞ , (2.10)in probability, locally uniformly in t . The limit ( ¯ S, ¯ I ) is the unique solution of the following systemof integral equations: ¯ S ( t ) = 1 − ¯ I (0) − Z t ¯ S ( s )¯ I ( s ) ds , (2.11)¯ I ( t ) = ¯ E (0)¯ λ ( t ) + ¯ I (0)¯ λ ,I ( t ) + Z t ¯ λ ( t − s ) ¯ S ( s )¯ I ( s ) ds , (2.12)and the limits ( ¯ E, ¯ I, ¯ R ) are given by the following formulas:¯ E ( t ) = ¯ E (0) G c ( t ) + Z t G c ( t − s ) ¯ S ( s )¯ I ( s ) ds , (2.13)¯ I ( t ) = ¯ I (0) F c ,I ( t ) + ¯ E (0)Ψ ( t ) + Z t Ψ( t − s ) ¯ S ( s )¯ I ( s ) ds , (2.14)¯ R ( t ) = ¯ I (0) F ,I ( t ) + ¯ E (0)Φ ( t ) + Z t Φ( t − s ) ¯ S ( s )¯ I ( s ) ds . (2.15)We also have ¯Υ N → ¯Υ in D in probability as N → ∞ , where¯Υ( t ) := ¯ S ( t )¯ I ( t ) , t ≥ . (2.16) GUODONG PANG AND ´ETIENNE PARDOUX
Let ˆ X N := √ N ( ¯ X N − ¯ X ) for any process X N with its fluid-scaled process ¯ X N and limit ¯ X . Wemake the following assumption on the initial quantities. Assumption 2.3.
There exist deterministic constants ¯ E (0) , ¯ I (0) ∈ (0 , and random variables ˆ E (0) , ˆ I (0) such that ( ˆ E N (0) , ˆ I N (0)) := √ N ( ¯ E N (0) − ¯ E (0) , ¯ I N (0) − ¯ I (0)) ⇒ ( ˆ E (0) , ˆ I (0)) as N → ∞ and sup N E [ ˆ E N (0) ] < ∞ and sup N E [ ˆ I N (0) ] < ∞ , and thus, E [ ˆ E (0) ] < ∞ and E [ ˆ I (0) ] < ∞ . Theorem 2.1.
Under Assumptions 2.1, 2.2 and 2.3, (cid:0) ˆ S N , ˆ I N (cid:1) ⇒ (cid:0) ˆ S, ˆ I (cid:1) in D as N → ∞ . (2.17) The limit process ( ˆ S, ˆ I ) is the unique solution to the following system of stochastic integral equations: ˆ S ( t ) = − ˆ I (0) − ˆ M A ( t ) + Z t ˆΥ( s ) ds, (2.18)ˆ I ( t ) = ˆ I (0)¯ λ ,I ( t ) + ˆ E (0)¯ λ ( t ) + ˆ I , ( t ) + ˆ I , ( t ) + ˆ I ( t ) + ˆ I ( t ) + Z t ¯ λ ( t − s ) ˆΥ( s ) ds, (2.19) where ˆΥ( t ) = ˆ S ( t )¯ I ( t ) + ¯ S ( t )ˆ I ( t ) , (2.20) and ¯ S ( t ) and ¯ I ( t ) are given by the unique solutions to the integral equations (2.11) and (2.12) , ˆ M A , ˆ I , , ˆ I , , ˆ I and ˆ I are Gaussian processes with ˆ M A ( t ) = B (cid:18)Z t ¯ S ( s )¯ I ( s ) ds (cid:19) , t ≥ , (2.21) with B being a standard Brownian motion, and ˆ I , , ˆ I , , ˆ I and ˆ I have covariance functions: for t, t ′ ≥ , Cov(ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ I (0)Cov( λ ,I ( t ) , λ ,I ( t ′ )) , Cov(ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ E (0)Cov( λ ( t ) , λ ( t ′ )) , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ Cov( λ ( t − s ) , λ ( t ′ − s )) ¯ S ( s )¯ I ( s ) ds, Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s )¯ λ ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds. ˆ I , and ˆ I , are independent, and also independent of ˆ M A , ˆ I and ˆ I . ˆ M A , ˆ I and ˆ I havecovariance functions: Cov( ˆ M A ( t ) , ˆ I ( t ′ )) = 0 , Cov( ˆ M A ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ¯ λ ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds, Cov(ˆ I ( t ) , ˆ I ( t ′ )) = 0 . ˆ S has continuous paths, and if ¯ λ and ¯ λ ,I are in C , then ˆ I is also continuous. We next state the FCLT for the processes (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) , which extends Theorem 3.2 in [18]. Theorem 2.2.
Under Assumptions 2.1, 2.2 and 2.3, (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) ⇒ (cid:0) ˆ E, ˆ I, ˆ R (cid:1) in D as N → ∞ , (2.22) jointly with (cid:0) ˆ S N , ˆ I N ) (i.e., (cid:0) ˆ S N , ˆ I N , ˆ E N , ˆ I N , ˆ R N ) ⇒ (cid:0) ˆ S, ˆ J , ˆ E, ˆ I, ˆ R (cid:1) in D ). The limit processes ˆ E , ˆ I and ˆ R are given by the expressions: ˆ E ( t ) = ˆ E (0) G c ( t ) + ˆ E ( t ) + ˆ E ( t ) + Z t G c ( t − s ) ˆΥ( s ) ds, ˆ I ( t ) = ˆ I (0) F c ,I ( t ) + ˆ E (0)Ψ ( t ) + ˆ I , ( t ) + ˆ I , ( t ) + ˆ I ( t ) + Z t Ψ( t − s ) ˆΥ( s ) ds, ˆ R ( t ) = ˆ I (0) F ,I ( t ) + ˆ E (0)Φ ( t ) + ˆ R , ( t ) + ˆ R , ( t ) + ˆ R ( t ) + Z t Φ( t − s ) ˆΥ( s ) ds, where ˆΥ is given in (2.20) , ˆ E , ˆ I , , ˆ I , , ˆ R , and ˆ R , are Gaussian processes, independent of B ,with covariance functions, for t, t ′ ≥ , Cov( ˆ E ( t ) , ˆ E ( t ′ )) = ¯ E (0)( G c ( t ∨ t ′ ) − G c ( t ) G c ( t ′ )) , Cov( ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ I (0)( F c ,I ( t ∨ t ′ ) − F c ,I ( t ) F c ,I ( t ′ )) , Cov( ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ E (0) (cid:0) Ψ ( t ∧ t ′ ) − Ψ ( t )Ψ ( t ′ ) (cid:1) , Cov( ˆ R , ( t ) , ˆ R , ( t ′ )) = ¯ I (0)( F ,I ( t ∧ t ′ ) − F ,I ( t ) F ,I ( t ′ )) , Cov( ˆ R , ( t ) , ˆ R , ( t ′ )) = ¯ E (0) (cid:0) Φ ( t ∧ t ′ ) − Φ ( t )Φ ( t ′ ) (cid:1) , Cov( ˆ I , ( t ) , ˆ R , ( t ′ )) = ¯ I (0) (cid:16) ( F ,I ( t ′ ) − F ,I ( t )) ( t ′ ≥ t ) − F c ,I ( t ) F ,I ( t ′ ) (cid:17) , Cov( ˆ E ( t ) , ˆ I , ( t ′ )) = ¯ E (0) Z t ′ t ( t ′ ≥ t ) F c ( t ′ − s | s ) dG ( s ) − G c ( t )Ψ ( t ′ ) ! , Cov( ˆ E ( t ) , ˆ R , ( t ′ )) = ¯ E (0) Z t ′ t F ( t ′ − s | s ) dG ( s ) − G c ( t )Φ ( t ′ ) ! , Cov( ˆ I , ( t ) , ˆ R , ( t ′ )) = ¯ E (0) Z t ∧ t ′ ( F ( t ′ − s | s ) − F ( t − s | s )) dG ( s ) − Ψ ( t )Φ ( t ′ ) ! . The other pairs of limit processes for the initial quantities ( ˆ E , ˆ I , ) , ( ˆ E , ˆ R , ) , ( ˆ I , , ˆ I , ) , ( ˆ I , , ˆ R , ) are independent. The limits ( ˆ E , ˆ I , ˆ R ) are three-dimensional continuous Gaussian processes, in-dependent of ˆ E , ˆ I , , ˆ I , , ˆ R , , ˆ R , and ˆ I (0) , and can be written as ˆ E ( t ) = W H ([0 , t ] × [ t, ∞ ) × [0 , ∞ )) , ˆ I ( t ) = W H ([0 , t ] × [0 , t ) × [ t, ∞ )) , ˆ R ( t ) = W H ([0 , t ] × [0 , t ) × [0 , t )) , where W H is a continuous Gaussian white noise process on R with mean zero and E (cid:2) W H ([ s, t ) × [ a, b ) × [ a ′ , b ′ )) (cid:3) = Z ts (cid:18)Z b − sa − s ( F ( b ′ − y − s | y ) − F ( a ′ − y − s | y )) G ( dy ) (cid:19) ¯ S ( s )¯ I ( s ) ds, for ≤ s ≤ t , ≤ a ≤ b and ≤ a ′ ≤ b ′ . They have the covariance functions: for t, t ′ ≥ , Cov( ˆ E ( t ) , ˆ E ( t ′ )) = Z t ∧ t ′ G c ( t ∨ t ′ − s ) ¯ S ( s )¯ I ( s ) ds, Cov( ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ Ψ( t ∨ t ′ − s ) ¯ S ( s )¯ I ( s ) ds, GUODONG PANG AND ´ETIENNE PARDOUX
Cov( ˆ R ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ Φ( t ∧ t ′ − s ) ¯ S ( s )¯ I ( s ) ds, Cov( ˆ E ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ( G c ( t − s ) − Ψ( t ′ − s )) ( t ′ ≥ t ) ¯ S ( s )¯ I ( s ) ds, Cov( ˆ E ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ ( G c ( t − s ) − Φ( t ′ − s )) ( t ′ ≥ t ) ¯ S ( s )¯ I ( s )) ds, Cov( ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ Z t ′ − s ( F ( t ′ − s − y | y ) − F ( t − s − y | y )) ( t ′ ≥ t ) dG ( y ) ¯ S ( s )¯ I ( s ) ds. If G and F ,I are continuous, then ˆ E , ˆ I , , ˆ I , , ˆ R , and ˆ R , are continuous, and thus ˆ E , ˆ I and ˆ R are continuous.In addition, the processes ˆ I , and ˆ I , in the integral expression of ˆ I in (2.19) have the followingcovariance functions with ˆ E , ˆ I , , ˆ I , , ˆ R , and ˆ R , : for t, t ′ ≥ , Cov(ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ I (0) (cid:0) E (cid:2) λ ,I ( t ) η ,I >t ′ (cid:3) − ¯ λ ,I ( t ) F c ,I ( t ′ ) (cid:1) , Cov(ˆ I , ( t ) , ˆ R , ( t ′ )) = ¯ I (0) (cid:0) E (cid:2) λ ,I ( t ) η ,I ≤ t ′ (cid:3) − ¯ λ ,I ( t ) F ,I ( t ′ ) (cid:1) , Cov(ˆ I , ( t ) , ˆ E ( t ′ )) = ¯ E (0) (cid:0) E (cid:2) λ ( t ) ζ >t ′ (cid:3) − ¯ λ ( t ) (cid:1) G c ( t ′ ) (cid:1) , Cov(ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ E (0) (cid:0) E (cid:2) λ ( t ) ζ + η >t ′ (cid:3) − ¯ λ ( t )Ψ ( t ′ ) (cid:1) , Cov(ˆ I , ( t ) , ˆ R , ( t ′ )) = ¯ E (0) (cid:0) E (cid:2) λ ( t ) ζ + η ≤ t ′ (cid:3) − ¯ λ ( t )Φ ( t ′ ) (cid:1) , ˆ I , is independent of ˆ E , ˆ I , and ˆ R , , and ˆ I , is independent of ˆ I , and ˆ R , . The processes ˆ I and ˆ I , independent of ˆ E , ˆ I , , ˆ I , , ˆ R , and ˆ R , , have the following covariance functions with ˆ E , ˆ I and ˆ R : for t, t ′ ≥ , Cov(ˆ I ( t ) , ˆ E ( t ′ )) = Z t ∧ t ′ (cid:16) E (cid:2) λ ( t − s ) ζ>t ′ − s (cid:3) − ¯ λ ( t − s ) G c ( t ′ − s ) (cid:17) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ (cid:16) E (cid:2) λ ( t − s ) ζ ≤ t ′ − s<ζ + η (cid:3) − ¯ λ ( t − s )Ψ( t ′ − s ) (cid:17) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ (cid:16) E (cid:2) λ ( t − s ) ζ + η ≤ t ′ − s (cid:3) − ¯ λ ( t − s )Φ( t ′ − s ) (cid:17) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ E ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s ) G c ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s )Ψ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s )Φ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds . ˆ M A is independent of ˆ E , ˆ I , , ˆ I , , ˆ R , and ˆ R , , and has covariance functions with ˆ E , ˆ I and ˆ R : for t, t ′ ≥ , Cov( ˆ M A ( t ) , ˆ E ( t ′ )) = Z t ∧ t ′ G c ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds , Cov( ˆ M A ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ Ψ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds , Cov( ˆ M A ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ Φ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds . Remark 2.1.
It is clear that the model includes the generalized SIR as a special case, where therandom infectivity function λ i ( t ) does not equal to zero at time , that is, an infected individual isimmediately infectious, so ζ = ζ = 0 a.s., and there are no exposed individuals, E N ( t ) = 0 for all t ≥ . Let F be the c.d.f. of the infectious duration η of newly infected individuals and F ,I be thec.d.f. of the infectious duration η ,I of initially infectious individuals. The FLLN gives the limits ( ¯ S, ¯ I ) determined by the following two-dimensional integral equation: ¯ S ( t ) = 1 − ¯ I (0) − Z t ¯ S ( s )¯ I ( s ) ds, ¯ I ( t ) = ¯ I (0)¯ λ ,I ( t ) + Z t ¯ λ ( t − s ) ¯ S ( s )¯ I ( s ) ds, and the limits ( ¯ I, ¯ R ) given by the following integral equations: ¯ I ( t ) = ¯ I (0) F c ,I ( t ) + Z t F c ( t − s ) ¯ S ( s )¯ I ( s ) ds, ¯ R ( t ) = ¯ I (0) F ,I ( t ) + Z t F ( t − s ) ¯ S ( s )¯ I ( s ) ds. The FCLT gives the limits ( ˆ S, ˆ I ) determined by the solutions to the following two-dimensionalstochastic integral equation: ˆ S ( t ) = − ˆ I (0) − ˆ M A ( t ) + Z t ˆΥ( s ) ds, ˆ I ( t ) = ˆ I (0)¯ λ ,I ( t ) + ˆ I , ( t ) + ˆ I ( t ) + ˆ I ( t ) + Z t ¯ λ ( t − s ) ˆΥ( s ) ds, where the processes ˆ M A , ˆ I , , ˆ I , ˆ I and ˆΥ are as given in Theorem 2.1. And the limits ( ˆ I, ˆ R ) aregiven by the following stochastic integral equations: ˆ I ( t ) = ˆ I (0) F c ,I ( t ) + ˆ I , ( t ) + ˆ I ( t ) + Z t F c ( t − s ) ˆΥ( s ) ds, ˆ R ( t ) = ˆ I (0) F ,I ( t ) + ˆ R , ( t ) + ˆ R ( t ) + Z t F ( t − s ) ˆΥ( s ) ds, where ˆ I , and ˆ R , are as given in Theorem 2.2, and the limits ( ˆ I , ˆ R ) are two-dimensional con-tinuous Gaussian processes, independent of ˆ I , , ˆ R , and ˆ I (0) , and can be written as ˆ I ( t ) = W H ([0 , t ] × [ t, ∞ )) , ˆ R ( t ) = W H ([0 , t ] × [0 , t )) , where W H is a continuous Gaussian white noise process on R with mean zero and E (cid:2) W H ([ s, t ) × [ a, b )) (cid:3) = Z ts ( F ( b − y − s ) − F ( a − y − s )) ¯ S ( s )¯ I ( s ) ds, for ≤ s ≤ t and ≤ a ≤ b . They have the covariance functions: for t, t ′ ≥ , Cov( ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ F c ( t ∨ t ′ − s ) ¯ S ( s )¯ I ( s ) ds, Cov( ˆ R ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ F ( t ∧ t ′ − s ) ¯ S ( s )¯ I ( s ) ds, Cov( ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ ( F ( t ′ − s − y | y ) − F ( t − s − y | y )) ( t ′ ≥ t ) ¯ S ( s )¯ I ( s ) ds. If F ,I is continuous, then ˆ I and ˆ R are continuous. In addition, the process ˆ I , has covariance functions with the processes ˆ I , and ˆ R , as given inTheorem 2.2, and the process ˆ I and ˆ I , independent of ˆ I , and ˆ R , , have the following covariancefunctions with ˆ I and ˆ R : for t, t ′ ≥ , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ (cid:16) E (cid:2) λ ( t − s ) η>t ′ − s (cid:3) − ¯ λ ( t − s ) F c ( t ′ − s ) (cid:17) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ (cid:16) E (cid:2) λ ( t − s ) η ≤ t ′ − s (cid:3) − ¯ λ ( t − s ) F ( t ′ − s ) (cid:17) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s ) F c ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds , Cov(ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s ) F ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds . ˆ M A is independent of ˆ I , and ˆ R , , and has covariance functions with ˆ I and ˆ R : for t, t ′ ≥ , Cov( ˆ M A ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ F c ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds , Cov( ˆ M A ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ F ( t ′ − s ) ¯ S ( s )¯ I ( s ) ds . Now consider the SEIR model, but suppose that we do not care to follow the numbers or propor-tions of exposed and infectious individuals, but only the number or proportion of infected individual(where infected means either exposed or infectious). Formally, the SEIR model then reduces to aSIR model, with the function λ being allowed to be zero on the interval [0 , ζ ) , with ζ > . For sucha model, the FLLN and the FCLT are exactly those described in this remark. Generalized SIS models with varying infectivity.
In the SIS model, individuals becomesusceptible immediately after going through the infectious periods. Since S N ( t ) = N − I N ( t ), theepidemic dynamics is determined by the two dimensional processes I N ( t ) and I N ( t ). As statedin Remark 2.3 of [10], the FLLN limit (¯ I ( t ) , ¯ I ( t )) is determined by the two–dimensional integralequations: ¯ I ( t ) = ¯ I (0)¯ λ ( t ) + Z t ¯ λ ( t − s )(1 − ¯ I ( s ))¯ I ( s ) ds , (2.23)¯ I ( t ) = ¯ I (0) F c ,I ( t ) + Z t F c ( t − s )(1 − ¯ I ( s ))¯ I ( s ) ds . (2.24)Here the c.d.f.’s F and F ,I denote the distributions of the infectious periods of newly infectedindividuals and those of initially infectious ones. Theorem 2.3.
In the generalized SIS model, under Assumptions 2.1, 2.2 and 2.3 (with E N ( t ) ≡ and only infectious periods), (cid:0) ˆ I N , ˆ I N (cid:1) → (cid:0) ˆ I , ˆ I (cid:1) in D as N → ∞ . The limit processes ˆ I and ˆ I are the unique solution to the following stochastic integral equations: ˆ I ( t ) = ˆ I (0)¯ λ ,I ( t ) + ˆ I ( t ) + ˆ I ( t ) + ˆ I ( t ) + Z t ¯ λ ( t − s ) ˆΥ( s ) ds, ˆ I ( t ) = ˆ I (0) F c ,I ( t ) + ˆ I ( t ) + ˆ I ( t ) + Z t F c ( t − s ) ˆΥ( s ) ds, where ˆΥ( t ) = (1 − ¯ I ( t ))ˆ I ( t ) − ¯ I ( t ) ˆ I ( t ) , and ¯ I ( t ) and ¯ I ( t ) are given by the unique solutions to the integral equations (2.23) and (2.24) . ˆ I , ˆ I and ˆ I are Gaussian processes with covariance functions: for t, t ′ ≥ , Cov(ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ I (0)Cov( λ ,I ( t ) , λ ,I ( t ′ )) , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ Cov( λ ( t − s ) , λ ( t ′ − s ))(1 − ¯ I ( s ))¯ I ( s ) ds, Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s )¯ λ ( t ′ − s )(1 − ¯ I ( s ))¯ I ( s ) ds. ˆ I , is independent of ˆ I and ˆ I . ˆ I and ˆ I have covariance function Cov(ˆ I ( t ) , ˆ I ( t ′ )) = 0 for t, t ′ ≥ . ˆ I and ˆ I are independent Gaussian processes with covariance functions: for t, t ′ ≥ , Cov( ˆ I ( t ) , ˆ I ( t ′ )) = ¯ I (0)( F c ,I ( t ∨ t ′ ) − F c ,I ( t ) F c ,I ( t ′ )) , Cov( ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ F c ( t ∨ t ′ − s )(1 − ¯ I ( s ))¯ I ( s ) ds. ˆ I and ˆ I have covariance function Cov(ˆ I ( t ) , ˆ I ( t ′ )) = ¯ I (0) (cid:0) E (cid:2) λ ,I ( t ) η ,I >t ′ (cid:3) − ¯ λ ,I ( t ) F c ( t ′ ) (cid:1) , and ˆ I , ˆ I and ˆ I have covariance functions Cov(ˆ I ( t ) , ˆ I ( t ′ )) = 0 , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ (cid:0) E (cid:2) λ ( t − s ) η>t ′ − s (cid:3) − ¯ λ ( t − s ) F c ( t ′ − s ) (cid:1) (1 − ¯ I ( s ))¯ I ( s ) ds. If ¯ λ ,I ( t ) and F ,I are continuous, then the limits ˆ I and ˆ I are continuous. Generalized SIRS models with varying infectivity.
In the SIRS model, individuals ex-perience the infectious and recovered/immune periods, and then become susceptible. We use I N ( t )and R N ( t ) to denote the numbers of infectious and recovered/immune individuals, respectively,corresponding to the processes E N ( t ) and I N ( t ) in the SEIR model. Similarly, we use { λ j } j ≥ and { λ i } i ≥ to denote the infectivity processes of the initially and newly infectious individuals,respectively, and also use { ( ξ j , η j ) } j ≥ and { ( ξ i , η i ) } i ≥ for the infectious and recovered/immuneperiods for the initially and newly infected individuals, respectively. Denote the remaining immunetime of the initially recovered/immune individuals by η ,Rk (changing notation η ,I to η ,R accord-ingly). Of course, { λ j } j ≥ and { λ i } i ≥ only take positive values over the intervals [0 , ζ j ) and [0 , ζ i ),respectively. The definitions of the variables ( ξ i , η i ), ( ξ j , η j ) and η ,Rk in (2.1), (2.2) and (2.3) alsoneed to change accordingly in a natural manner. The c.d.f.’s G , G denote the distributions of in-fectious periods of the initially and newly infectious individuals, and the c.d.f.’s F ,R and F denotethe distributions of the recovered/immune periods of the initially and newly recovered individuals.Similarly for the notation Ψ , Ψ , Φ , Φ.Since S N ( t ) = N − I N ( t ) − R N ( t ), the epidemic dynamics is described by the three processes( I N , I N , R N ). As stated in Remark 2.3 in [10], the FLLN limit (cid:0) ¯ I , ¯ I, ¯ R (cid:1) is determined by thethree–dimensional integral equations:¯ I ( t ) = ¯ I (0)¯ λ ( t ) + Z t ¯ λ ( t − s ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds , (2.25)¯ I ( t ) = ¯ I (0) G c ( t ) + Z t G c ( t − s ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds , (2.26)¯ R ( t ) = ¯ R (0) F c ,R ( t ) + ¯ I (0)Ψ ( t ) + Z t Ψ( t − s ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds . (2.27) We next state the FCLT for these processes.
Theorem 2.4.
In the generalized SIRS model, under Assumptions 2.1, 2.2 and 2.3, (cid:0) ˆ I N , ˆ I N , ˆ R N (cid:1) ⇒ (cid:0) ˆ I , ˆ I, ˆ R (cid:1) in D as N → ∞ . The limit process (ˆ I , ˆ I, ˆ R ) is the unique solution to the following system of stochastic integral equa-tions: ˆ I ( t ) = ˆ I (0)¯ λ ( t ) + ˆ I ( t ) + ˆ I ( t ) + ˆ I ( t ) + Z t ¯ λ ( t − s ) ˆΥ( s ) ds, ˆ I ( t ) = ˆ I (0) G c ( t ) + ˆ I ( t ) + ˆ I ( t ) + Z t G c ( t − s ) ˆΥ( s ) ds, ˆ R ( t ) = ˆ R (0) F c ,R ( t ) + ˆ I (0)Ψ ( t ) + ˆ R , ( t ) + ˆ R , ( t ) + ˆ R ( t ) + Z t Ψ( t − s ) ˆΥ( s ) ds, where ˆΥ( t ) = (1 − ¯ I ( t ) − ¯ R ( t ))ˆ I ( t ) − ¯ I ( t )( ˆ I ( t ) + ˆ R ( t )) , and ¯ I ( t ) , ¯ I ( t ) and ¯ R ( t ) are given by the unique solution to the integral equations (2.25) – (2.27) . ˆ I , ˆ I and ˆ I are Gaussian processes with covariance functions: for t, t ′ ≥ , Cov(ˆ I , ( t ) , ˆ I , ( t ′ )) = ¯ I (0)Cov( λ ( t ) , λ ( t ′ )) , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ Cov( λ ( t − s ) , λ ( t ′ − s )) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds, Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ ¯ λ ( t − s )¯ λ ( t ′ − s ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds. ˆ I , is independent of ˆ I and ˆ I . ˆ I and ˆ I have covariance function Cov(ˆ I ( t ) , ˆ I ( t ′ )) = 0 for t, t ′ ≥ . ˆ I and ˆ I are independent Gaussian processes with covariance functions: for t, t ′ ≥ , Cov( ˆ I ( t ) , ˆ I ( t ′ )) = ¯ I (0)( G c ( t ∨ t ′ ) − G c ( t ) G c ( t ′ )) , Cov( ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ G c ( t ∨ t ′ − s ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds. ˆ R , , ˆ R , and ˆ R are mutually independent Gaussian processes with covariance functions: for t, t ′ ≥ , Cov( ˆ R , ( t ) , ˆ R , ( t ′ )) = ¯ R (0)( F c ,R ( t ∨ t ′ ) − F c ,R ( t ) F c ,R ( t ′ )) , Cov( ˆ R , ( t ) , ˆ R , ( t ′ )) = ¯ I (0) (cid:0) Ψ ( t ∧ t ′ ) − Ψ ( t )Ψ ( t ′ ) (cid:1) , Cov( ˆ R ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ Ψ( t ∨ t ′ − s ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds. The processes ˆ I , ˆ I , ˆ R , , ˆ R , are independent of ˆ I , ˆ I , ˆ I , ˆ R . ˆ I , ˆ I and ˆ R , are independent of ˆ R , , and have covariance functions Cov(ˆ I ( t ) , ˆ I ( t ′ )) = ¯ I (0) (cid:0) E (cid:2) λ ( t ) ζ >t ′ (cid:3) − ¯ λ ( t ) G c ( t ′ ) (cid:1) , Cov(ˆ I ( t ) , ˆ R , ( t ′ )) = ¯ I (0) (cid:0) E (cid:2) λ ( t ) ζ + η >t ′ (cid:3) − ¯ λ ( t )Ψ( t ′ ) (cid:1) , Cov( ˆ I ( t ) , ˆ R , ( t ′ )) = ¯ I (0) Z t ′ t ( t ′ ≥ t ) F c ( t ′ − s | s ) dG ( s ) − G c ( t )Ψ ( t ′ ) ! . The processes ˆ I , ˆ I , ˆ I and ˆ R have covariance functions Cov(ˆ I ( t ) , ˆ I ( t ′ )) = 0 , Cov(ˆ I ( t ) , ˆ R ( t ′ )) = 0 , Cov(ˆ I ( t ) , ˆ I ( t ′ )) = Z t ∧ t ′ (cid:0) E (cid:2) λ ( t − s ) ζ>t ′ − s (cid:3) − ¯ λ ( t − s ) F c ( t ′ − s ) (cid:1)(cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds, Cov(ˆ I ( t ) , ˆ R ( t ′ )) = Z t ∧ t ′ (cid:0) E (cid:2) λ ( t − s ) ζ + η>t ′ − s (cid:3) − ¯ λ ( t − s )Ψ( t ′ − s ) (cid:1)(cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds, Cov( ˆ I ( t ′ ) , ˆ R ( t ′ )) = Z t ∧ t ′ ( G c ( t − s ) − Ψ( t ′ − s )) ( t ′ ≥ t ) (cid:0) − ¯ I ( s ) − ¯ R ( s ) (cid:1) ¯ I ( s ) ds. If ¯ λ ( t ) , G and F ,R are continuous, the limits ˆ I , ˆ I and ˆ R have continuous paths. Proofs
In this section we prove Theorems 2.1 and 2.2. We focus on the proof of the convergence of( ˆ S N , ˆ I N ) stated in Theorem 2.1. The proof of Theorem 2.2 on the convergence of ( ˆ E N , ˆ I N , ˆ R N )follows essentially the same as that of Theorem 3.2 in [18], for which we only highlight the differ-ences.3.1. Convergence of ( ˆ S N , ˆ I N ) . For the process A N ( t ), we have the decomposition A N ( t ) = M NA ( t ) + Z t Υ N ( s ) ds , (3.1)where M NA ( t ) = Z t Z ∞ u ≤ Υ N ( s ) Q ( ds, du ) , with Q ( ds, du ) := Q ( ds, du ) − dsdu being the compensated PRM. It is clear that the process { M NA ( t ) : t ≥ } is a square-integrable martingale with respect to the filtration {F Nt : t ≥ } defined by F Nt := σ n I N (0) , E N (0) , λ j ( · ) j ≥ , λ ,Ik ( · ) k ≥ , λ i ( · ) i ≥ , Z t ′ Z ∞ u ≤ Υ N ( s ) Q ( ds, du ) : t ′ ≤ t o . It has the following finite quadratic variation (see e.g. [8, Chapter VI]): h M NA i ( t ) = Z t Υ N ( s ) ds , t ≥ . Under Assumption 2.2, we have0 ≤ N − Z ts Υ N ( u ) du ≤ λ ∗ ( t − s ) , w.p. 1 for 0 ≤ s ≤ t. (3.2)It is shown in Section 4.1 of [10] that Z · ¯Υ N ( s ) ds = Z · ¯ S N ( s )¯ I N ( s ) ds ⇒ Z · ¯ S ( s )¯ I ( s ) ds in D as N → ∞ . (3.3)and ¯ A N ⇒ ¯ A = Z · ¯ S ( s )¯ I ( s ) ds in D as N → ∞ . (3.4)By (3.1), we have ˆ A N ( t ) = √ N ( ¯ A N ( t ) − ¯ A ( t )) = ˆ M NA ( t ) + Z t ˆΥ N ( s ) ds, (3.5) where ˆ M NA ( t ) = 1 √ N Z t Z ∞ u ≤ Υ N ( s ) Q ( ds, du ) , and ˆΥ N ( t ) = √ N ( ¯ S N ( t )¯ I N ( t ) − ¯ S ( t )¯ I ( t )) = ˆ S N ( t )¯ I N ( t ) + ¯ S ( t )ˆ I N ( t ) . (3.6)The process { ˆ M NA ( t ) : t ≥ } is a square-integrable martingale with respect to the filtration F N and has the quadratic variation h ˆ M NA i ( t ) = N − Z t Υ N ( s ) ds, t ≥ . By (3.3), we obtain h ˆ M NA i ( t ) ⇒ Z t ¯ S ( s )¯ I ( s ) ds in D as N → ∞ . Thus by the FCLT for martingales, see, e.g., [23], we obtain the following lemma.
Lemma 3.1.
Under Assumptions 2.1, 2.2 and 2.3, ˆ M NA ⇒ ˆ M A in D as N → ∞ , (3.7) where ˆ M A is given in (2.21) . It then follows thatˆ S N ( t ) = ˆ S N (0) − ˆ A N ( t ) = − ˆ I N (0) − ˆ M NA ( t ) − Z t ˆΥ N ( s ) ds. (3.8)By (2.4), we haveˆ I N ( t ) = ˆ I N (0)¯ λ ,I ( t ) + ˆ E N (0)¯ λ ( t ) + ˆ I N , ( t ) + ˆ I N , ( t ) + ˆ I N ( t ) + ˆ I N ( t )+ Z t ¯ λ ( t − s ) ˆΥ N ( s ) ds, (3.9)where ˆ I N , ( t ) := 1 √ N I N (0) X k =1 (cid:0) λ ,Ik ( t ) − ¯ λ ,I ( t ) (cid:1) , ˆ I N , ( t ) := 1 √ N E N (0) X j =1 (cid:0) λ j ( t ) − ¯ λ ( t ) (cid:1) , ˆ I N ( t ) := 1 √ N A N ( t ) X i =1 (cid:0) λ i ( t − τ Ni ) − ¯ λ ( t − τ Ni ) (cid:1) , (3.10)and ˆ I N ( t ) := 1 √ N A N ( t ) X i =1 ¯ λ ( t − τ Ni ) − Z t ¯ λ ( t − s )Υ N ( s ) ds . (3.11) Lemma 3.2.
Under Assumptions 2.2(i) and 2.3, (cid:0) ˆ I N , , ˆ I N , (cid:1) ⇒ (cid:0) ˆ I , , ˆ I , (cid:1) in D as N → ∞ , (3.12) where (cid:0) ˆ I , , ˆ I , (cid:1) is given in Theorem 2.1. Proof.
Define ˜ I N , ( t ) := 1 √ N N ¯ I (0) X k =1 (cid:0) λ ,Ik ( t ) − ¯ λ ,I ( t ) (cid:1) , (3.13)˜ I N , ( t ) := 1 √ N N ¯ E (0) X j =1 (cid:0) λ j ( t ) − ¯ λ ( t ) (cid:1) . (3.14)By the FCLT for the random elements in D (see Theorem 2 in [12], whose conditions (i) and (ii)are satisfied thanks to Assumption 2.2 (i) (a) and (b), respectively) and by the independence ofthe sequences { λ j } j ≥ and { λ ,Ik } k ≥ , we obtain (cid:0) ˜ I N , , ˜ I N , (cid:1) ⇒ (cid:0) ˆ I , , ˆ I N , (cid:1) in D as N → ∞ . (3.15)It then suffices to show that (cid:0) ˜ I N , − ˆ I N , , ˜ I N , − ˆ I N , (cid:1) ⇒ D as N → ∞ . (3.16)We focus on ˜ I N , − ˆ I N , ⇒
0. It is clear from the definition in (3.14) and the i.i.d. property of λ j ( · )that for each t ≥ E (cid:2) ˜ I N , ( t ) − ˆ I N , ( t ) (cid:3) = 0, and E (cid:2)(cid:0) ˜ I N , ( t ) − ˆ I N , ( t ) (cid:1) (cid:3) = ν ( t ) E (cid:2) | ¯ E (0) − ¯ E N (0) | (cid:3) → N → ∞ , where ν ( t ) = E (cid:2)(cid:0) λ j ( t ) − ¯ λ ( t ) (cid:1) (cid:3) < ∞ under Assumption 2.2, and the convergence follows fromAssumption 2.3 and the dominated convergence theorem. It then remains to show that { ˜ I N , − ˆ I N , : N ∈ N } is tight in D . We have˜ I N , ( t ) − ˆ I N , ( t ) = sign( ¯ E (0) − ¯ E N (0)) 1 √ N N ( ¯ E (0) ∨ ¯ E N (0)) X j = N ( ¯ E (0) ∧ ¯ E N (0))+1 (cid:0) λ j ( t ) − ¯ λ ( t ) (cid:1) . We use the moment criterion in Theorem 13.5 of [4], and consider the moment: for t ′ ≤ t ≤ t ′′ , E (cid:2)(cid:12)(cid:12)(cid:0) ˜ I N , ( t ) − ˆ I N , ( t ) (cid:1) − (cid:0) ˜ I N , ( t ′ ) − ˆ I N , ( t ′ ) (cid:1)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:0) ˜ I N , ( t ) − ˆ I N , ( t ) (cid:1) − (cid:0) ˜ I N , ( t ′′ ) − ˆ I N , ( t ′′ ) (cid:1)(cid:12)(cid:12) (cid:3) . Recall ˘ λ j ( t ) = λ j ( t ) − ¯ λ ( t ), and we drop the subscript j for the generic variable ˘ λ ( t ). Then bythe i.i.d. and mean zero properties of ˘ λ j ( t ), and by the independence between ¯ E N (0) and ˘ λ j ( t ),we obtain that the moment above is equal to1 N E N ( ¯ E (0) ∨ ¯ E N (0)) X j = N ( ¯ E (0) ∧ ¯ E N (0))+1 (cid:0) ˘ λ j ( t ) − ˘ λ j ( t ′ ) (cid:1) N ( ¯ E (0) ∨ ¯ E N (0)) X j = N ( ¯ E (0) ∧ ¯ E N (0))+1 (cid:0) ˘ λ j ( t ) − ˘ λ j ( t ′′ ) (cid:1) = 1 N (cid:18) E (cid:2) N | ¯ E N (0) − ¯ E (0) | (cid:3) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′ ) (cid:1) (cid:0) ˘ λ ( t ) − ˘ λ ( t ′′ ) (cid:1) (cid:3) + E (cid:2) N | ¯ E N (0) − ¯ E (0) | ( N | ¯ E N (0) − ¯ E (0) | − (cid:3) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′ ) (cid:1) (cid:3) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′′ ) (cid:1) (cid:3) + 2 E (cid:2) N | ¯ E N (0) − ¯ E (0) | ( N | ¯ E N (0) − ¯ E (0) | − (cid:3)(cid:16) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′ ) (cid:1)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′′ ) (cid:1)(cid:3)(cid:17) (cid:19) ≤ N (cid:18) E (cid:2) N | ¯ E N (0) − ¯ E (0) | (cid:3) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′ ) (cid:1) (cid:0) ˘ λ ( t ) − ˘ λ ( t ′′ ) (cid:1) (cid:3) + 3 E (cid:2) N | ¯ E N (0) − ¯ E (0) | ( N | ¯ E N (0) − ¯ E (0) | − (cid:3) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′ ) (cid:1) (cid:3) E (cid:2)(cid:0) ˘ λ ( t ) − ˘ λ ( t ′′ ) (cid:1) (cid:3)(cid:19) ≤ N E (cid:2) | ¯ E N (0) − ¯ E (0) | (cid:3) ( ψ ( t ′′ ) − ψ ( t ′ )) β + 3 E (cid:2) | ¯ E N (0) − ¯ E (0) | (cid:3) ( φ ( t ′′ ) − φ ( t ′ )) α , where we have used Cauchy-Schwarz inequality in the first inequality, and conditions (a) and (b)in Assumption 2.2 (i) in the second inequality. By Assumption 2.3, E (cid:2) | ¯ E N (0) − ¯ E (0) | (cid:3) → E (cid:2) | ¯ E N (0) − ¯ E (0) | (cid:3) → N → ∞ . Thus we can conclude the tightness of { ˜ I N , − ˆ I N , : N ∈ N } follows from Theorem 4.1 in the Appendix below. This completes the proof. (cid:3) We next prove the convergence of ˆ I N . We introduce a PRM ˘ Q on R + × R + × D , which to thepoint τ Ni associates the copy λ i of the random function λ , so that the mean measure of the PRM is ds × du × Law of λ . ˆ I N can be writtenˆ I N ( t ) = N − / Z t Z ∞ Z D [ λ − ¯ λ ]( t − s ) u ≤ Υ N ( s ) ˘ Q ( ds, du, dλ ) . (3.17)We note that if we replace in the above ˘ Q by its mean measure, then the resulting integral vanishes.Consequently we also haveˆ I N ( t ) = N − / Z t Z ∞ Z D [ λ − ¯ λ ]( t − s ) u ≤ Υ N ( s ) e Q ( ds, du, dλ ) , (3.18)where e Q is the compensated PRM of ˘ Q . Hence E [ˆ I N ( t )] = 0 and E (cid:2) (ˆ I N ( t )) (cid:3) = E (cid:20)Z t [ λ − ¯ λ ] ( t − s ) ¯Υ N ( s ) ds (cid:21) . Before we establish the next result, let us recall a well–known formula for the exponential momentof an integral with respect to a compensated Poisson random measure.
Lemma 3.3.
Let Q be a PRM on some measurable space ( E, E ) , with mean measure ν , and ¯ Q the associated compensated measure. Let f : E C be measurable and such that e f − − f is ν integrable. Then E (cid:20) exp (cid:18)Z E f ( x ) ¯ Q ( dx ) (cid:19)(cid:21) = exp (cid:18)Z E h e f ( x ) − − f ( x ) i ν ( dx ) (cid:19) . If ν ( f + f ) < ∞ , then E "(cid:18)Z E f ( x ) ¯ Q ( dx ) (cid:19) = Z E f ( x ) ν ( dx ) , E "(cid:18)Z E f ( x ) ¯ Q ( dx ) (cid:19) = Z E f ( x ) ν ( dx ) + 3 (cid:18)Z E f ( x ) ν ( dx ) (cid:19) . We also have the following bounds on the increments of the infectivity functions.
Lemma 3.4.
For t ≥ s ≥ , (cid:12)(cid:12) λ i ( t ) − λ i ( s ) (cid:12)(cid:12) ≤ ϕ ( t − s ) + λ ∗ k X j =1 s<ξ ji ≤ t , and | ¯ λ ( t ) − ¯ λ ( s ) | ≤ ϕ ( t − s ) + λ ∗ k X j =1 ( F j ( t ) − F j ( s )) . Moreover, if ˜ λ i := λ i − ¯ λ , then E [ | ˜ λ i ( t ) − ˜ λ i ( s ) | ] ≤ ϕ ( t − s ) + 2 λ ∗ k X j =1 ( F j ( t ) − F j ( s )) . Proof.
We have λ i ( t ) − λ i ( s ) = k X j =1 (cid:0) λ ji ( t ) − λ ji ( s ) (cid:1) ξ j − ≤ s, t<ξ j + (cid:0) λ i ( t ) − λ i ( s ) (cid:1) k X j =1 s<ξ ji ≤ t . Thus the first statement follows from Assumption 2.2 (ii), and the next statements follow readilyfrom the first one. (cid:3)
Lemma 3.5.
Under Assumptions 2.1, 2.2(ii) and 2.3, ˆ I N ⇒ ˆ I in D as N → ∞ , (3.19) where ˆ I is given in Theorem 2.1.Proof. Let us first prove the convergence of finite dimensional distributions. We consider only anarbitrary two–dimensional distribution, the general result being obtained exactly in the same way.Let e I N ( t ) := N − / Z t Z ∞ Z D [ λ − ¯ λ ]( t − s ) u ≤ N ¯Υ( s ) ˘ Q ( ds, du, dλ ) . (3.20)It is not hard to se that for any t >
0, ˆ I N ( t ) − e I N ( t ) → N → ∞ . Therefore itis enough to compute the limit as N → ∞ of E h exp (cid:16) iθ e I N ( t ) + iθ e I N ( t ) (cid:17)i , where θ , θ ∈ R and t , t > E = R × D , Q = ˘ Q and f ( s, u, λ ) = iN − / { θ ( λ − ¯ λ )( t − s ) s ≤ t + θ ( λ − ¯ λ )( t − s ) s ≤ t } u ≤ N ¯Υ( s ) , from which we easily deduce thatlim N →∞ E h exp (cid:16) iθ e I N ( t ) + iθ e I N ( t ) (cid:17)i = exp (cid:18) − θ Z t E (cid:2)(cid:0) λ ( t − s ) − ¯ λ ( t − s ) (cid:1) (cid:3) ¯Υ( s ) ds − θ Z t E (cid:2)(cid:0) λ ( t − s ) − ¯ λ ( t − s ) (cid:1) (cid:3) ¯Υ( s ) ds − θ θ Z t ∧ t E (cid:2)(cid:0) λ ( t − s ) − ¯ λ ( t − s ) (cid:1)(cid:0) λ ( t − s ) − ¯ λ ( t − s ) (cid:1)(cid:3) ¯Υ( s ) ds (cid:19) . (3.21)We next prove tightness. The moment criterion requires to calculate for t ′ < t < t ′′ , E h(cid:12)(cid:12)e I N ( t ) − e I N ( t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)e I N ( t ) − e I N ( t ′′ ) (cid:12)(cid:12) i and find a bound of the form ( ψ ( t ′′ ) − ψ ( t ′ )) α for some α > / ψ ( · ). By Cauchy-Schwartz inequality, we calculate the fourth moment. Wehave E (cid:2)(cid:12)(cid:12)e I N ( t ) − e I N ( r ) (cid:12)(cid:12) (cid:3) = E "(cid:12)(cid:12)(cid:12)(cid:12) N − / Z tr Z ∞ Z D [ λ − ¯ λ ]( t − s ) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) (cid:12)(cid:12)(cid:12)(cid:12) + E "(cid:12)(cid:12)(cid:12)(cid:12) N − / Z r Z ∞ Z D (cid:16) [ λ − ¯ λ ]( t − s ) − [ λ − ¯ λ ]( r − s ) (cid:17) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) (cid:12)(cid:12)(cid:12)(cid:12) + 6 E "(cid:12)(cid:12)(cid:12)(cid:12) N − / Z tr Z ∞ Z D [ λ − ¯ λ ]( t − s ) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) (cid:12)(cid:12)(cid:12)(cid:12) × E "(cid:12)(cid:12)(cid:12)(cid:12) N − / Z r Z ∞ Z D (cid:16) [ λ − ¯ λ ]( t − s ) − [ λ − ¯ λ ]( r − s ) (cid:17) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) (cid:12)(cid:12)(cid:12)(cid:12) . (3.22)For the first two terms, we have E "(cid:12)(cid:12)(cid:12)(cid:12) N − / Z tr Z ∞ Z D [ λ − ¯ λ ]( t − s ) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 N Z tr E [[ λ − ¯ λ ]( t − s ) ] ¯Υ( s ) ds + 3 (cid:18) E Z tr (cid:0) [ λ − ¯ λ ]( t − s ) (cid:1) ¯Υ( s ) ds (cid:19) ≤ N (2 λ ∗ ) Z tr ¯Υ( s ) ds + 3(2 λ ∗ ) (cid:18)Z tr ¯Υ( s ) ds (cid:19) ≤ N λ ∗ ) ( t − r ) + 48( λ ∗ ) ( t − r ) , (3.23)and E "(cid:12)(cid:12)(cid:12)(cid:12) N − / Z r Z ∞ Z D (cid:16) [ λ − ¯ λ ]( t − s ) − [ λ − ¯ λ ]( r − s ) (cid:17) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 N Z r E (cid:2)(cid:0) [ λ − ¯ λ ]( t − s ) − [ λ − ¯ λ ]( r − s ) (cid:1) (cid:3) ¯Υ( s ) ds + 3 (cid:18) E Z r (cid:16) [ λ − ¯ λ ]( t − s ) − [ λ − ¯ λ ]( r − s ) (cid:17) ¯Υ( s ) ds (cid:19) . (3.24)Here for the first term on the right hand side of (3.24), we use E [ | ˜ λ ( t ) − ˜ λ ( s ) | ] ≤ E [ | λ ( t ) − λ ( s ) | ] + 8 | ¯ λ ( t ) − ¯ λ ( s ) | , | ¯ λ ( t ) − ¯ λ ( s ) | ≤ ϕ ( t − s ) + 8( λ ∗ ) k X j =1 ( F j ( t ) − F j ( s )) , and | λ ( t ) − λ ( s ) | ≤ ϕ ( t − s ) + 8( λ ∗ ) k X j =1 s<ξ j ≤ t ≤ ϕ ( t − s ) + 8 k ( λ ∗ ) k X j =1 s<ξ j ≤ t . Thus we obtain the bound for the first term on the right hand side of (3.24):1 N ϕ ( t − r ) λ ∗ T + 1 N λ ∗ ) Z r (cid:18) k X j =1 ( F j ( t − s ) − F j ( r − s )) (cid:19) ¯Υ( s ) ds + 1 N k ( λ ∗ ) Z r k X j =1 ( F j ( t − s ) − F j ( r − s )) ¯Υ( s ) ds . (3.25)For the second term on the right hand side of (3.24), we have3 (cid:18) E Z r (cid:16) [ λ − ¯ λ ]( t − s ) − [ λ − ¯ λ ]( r − s ) (cid:17) ¯Υ( s ) ds (cid:19) ≤ (cid:18) E Z r (cid:16) λ ( t − s ) − λ ( r − s )) + 2(¯ λ ( t − s ) − ¯ λ ( r − s )) (cid:17) ¯Υ( s ) ds (cid:19) ≤ (cid:18) Z r (cid:18) ϕ ( t − r ) + 4( λ ∗ ) k k X j =1 (cid:18) F j ( t − s ) − F j ( r − s ) (cid:19) + 4( λ ∗ ) (cid:18) k X j =1 ( F j ( t − s ) − F j ( r − s )) (cid:19) (cid:19) ¯Υ( s ) ds (cid:19) ≤ (cid:18) ϕ ( t − r ) Z r ¯Υ( s ) ds (cid:19) + 9 Z r λ ∗ ) k k X j =1 (cid:18) F j ( t − s ) − F j ( r − s ) (cid:19) ¯Υ( s ) ds + 9 Z r λ ∗ ) (cid:18) k X j =1 ( F j ( t − s ) − F j ( r − s )) (cid:19) ¯Υ( s ) ds ≤ × ϕ ( t − r ) ( λ ∗ ) T + 9 × k ( λ ∗ ) k X j =1 (cid:18)Z r ( F j ( t − s ) − F j ( r − s )) ds (cid:19) + 9 × λ ∗ ) k k X j =1 (cid:18)Z r [ F j ( t − s ) − F j ( r − s )] ds (cid:19) . (3.26)Finally, the third term on the right hand side of (3.22) equals6 × E Z tr ˜ λ ( t − s ) ¯Υ( s ) ds × E Z r [˜ λ ( t − s ) − ˜ λ ( r − s )] ¯Υ( s ) ds ≤ λ ∗ ) ( t − r ) T ϕ ( t − r ) + k ( λ ∗ ) k X j =1 Z r [ F j ( t − s ) − F j ( r − s )] ds , where we have used the obvious inequality ( F j ( t ) − F j ( s )) ≤ F j ( t ) − F j ( s ) for 0 ≤ s ≤ t .Combining the bounds for the three terms in the right hand side of (3.22), we obtain the followingbound or the left hand side of (3.22):1 N C (cid:18) ( t − r ) + ϕ ( t − r ) + k X j =1 Z r (cid:0) F j ( t − s ) − F j ( r − s ) (cid:1) ds (cid:19) + C ( t − r ) + ϕ ( t − r ) + k X j =1 (cid:18)Z r (cid:0) F j ( t − s ) − F j ( r − s ) (cid:1) ds (cid:19) ! (3.27)+ C ( t − r ) ϕ ( t − r ) + k X j =1 Z r [ F j ( t − s ) − F j ( r − s )] ds ! , for some positive constants C , C , C >
0, which depend only upon λ ∗ , T and k . Under Assumption 2.2(ii), supposing that F j satisfies the H¨older continuity condition in Assump-tion 2.1, we have Z r ( F j ( t − s ) − F j ( r − s )) ds ≤ C ( t − r ) / θ . (3.28)On the other hand, if F j satisfies the discrete condition in Assumption 2.1, say F j ( t ) = P i a ji t ≥ t ji for P i a ji = 1 and t ji ≤ t ji +1 , then Z r ( F j ( t − s ) − F j ( r − s )) ds = Z r X i a ji ( r − t ji ) +
0. Taking into account our assumption (2.8), we deduce from (3.27), (3.28)and (3.29) that there exist δ > C ′ and C ′ > t ′ < t < t ′′ E h(cid:12)(cid:12)e I N ( t ) − e I N ( t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)e I N ( t ) − e I N ( t ′′ ) (cid:12)(cid:12) i ≤ C ′ N ( t ′′ − t ′ ) + C ′ ( t ′′ − t ′ ) δ , which allows us to deduce from Theorem 4.1, a reinforced version of Theorem 13.5 in [4] which isestablished in the Appendix below, that the sequence { e J N : N ∈ N } is tight.Then we conclude the convergence e I N ⇒ ˆ I in D . The lemma follows from the asymptoticequivalence between ˆ I N and e I N , see Lemma 3.8 below. (cid:3) We next prove the convergence ofˆ I N ( t ) := 1 √ N Z t Z ∞ ¯ λ ( t − s ) u ≤ Υ N ( s ) Q ( ds, du ) . (3.30) Lemma 3.6.
Under Assumptions 2.1, 2.2(ii) and 2.3, ˆ I N ⇒ ˆ I in D as N → ∞ , (3.31) where ˆ I is given in Theorem 2.1.Proof. We define e I N ( t ) := 1 √ N Z t Z ∞ ¯ λ ( t − s ) u ≤ N ¯Υ( s ) Q ( ds, du ) . (3.32)and apply again Lemma 3.3, this time with E = R and f ( s, u ) = iN − / { θ ¯ λ ( t − s ) s ≤ t + θ ¯ λ ( t − s ) s ≤ t } u ≤ N ¯Υ( s ) . We obtain lim N →∞ E h exp (cid:16) iθ e I N ( t ) + iθ e I N ( t ) (cid:17)i = exp (cid:18) − θ Z t ¯ λ ( t − s ) ¯Υ( s ) ds − θ Z t ¯ λ ( t − s ) ¯Υ( s ) ds − θ θ Z t ∧ t ¯ λ ( t − s )¯ λ ( t − s ) ¯Υ( s ) ds (cid:19) . (3.33)We next establish tightness. We have for t > r ≥ e I N ( t ) − e I N ( r ) = 1 √ N Z tr Z ∞ ¯ λ ( t − s ) u ≤ N ¯Υ( s ) Q ( ds, du )+ 1 √ N Z r Z ∞ (cid:0) ¯ λ ( t − s ) − ¯ λ ( r − s ) (cid:1) u ≤ N ¯Υ( s ) Q ( ds, du ) . (3.34) The moment criterion requires to calculate E h(cid:12)(cid:12)e I N ( t ) − e I N ( t ′ ) (cid:12)(cid:12) (cid:12)(cid:12)e I N ( t ) − e I N ( t ′′ ) (cid:12)(cid:12) i . By Cauchy-Schwartz inequality, we calculate the fourth moment. By (3.34), noting that Q iscompensated PRM, we have for t ≥ r ≥ E h(cid:12)(cid:12)e I N ( t ) − e I N ( r ) (cid:12)(cid:12) i = E "(cid:12)(cid:12)(cid:12)(cid:12) √ N Z tr Z ∞ ¯ λ ( t − s ) u ≤ N ¯Υ( s ) Q ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12) + E "(cid:12)(cid:12)(cid:12)(cid:12) √ N Z r Z ∞ (cid:0) ¯ λ ( t − s ) − ¯ λ ( r − s ) (cid:1) u ≤ N ¯Υ( s ) Q ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12) + 6 E "(cid:12)(cid:12)(cid:12)(cid:12) √ N Z tr Z ∞ ¯ λ ( t − s ) u ≤ N ¯Υ( s ) Q ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12) × E "(cid:12)(cid:12)(cid:12)(cid:12) √ N Z r Z ∞ (cid:0) ¯ λ ( t − s ) − ¯ λ ( r − s ) (cid:1) u ≤ N ¯Υ( s ) Q ( ds, du ) (cid:12)(cid:12)(cid:12)(cid:12) . (3.35)The first term on the right hand side is equal to1 N Z tr ¯ λ ( t − s ) ¯Υ( s ) ds + 3 (cid:18)Z tr ¯ λ ( t − s ) ¯Υ( s ) ds (cid:19) ≤ N ( λ ∗ ) ( t − r ) + 3( λ ∗ ) ( t − r ) . The second term on the right hand side of (3.35) is equal to1 N Z r (cid:0) ¯ λ ( t − s ) − ¯ λ ( r − s ) (cid:1) ¯Υ( s ) ds + 3 (cid:18)Z r (cid:0) ¯ λ ( t − s ) − ¯ λ ( r − s ) (cid:1) ¯Υ( s ) ds (cid:19) ≤ N ϕ ( t − r ) Z r ¯Υ( s ) ds + 8( λ ∗ ) N Z r (cid:18) k X j =1 ( F j ( t − s ) − F j ( r − s )) (cid:19) ¯Υ( s ) ds + 24 ϕ ( t − r ) (cid:18)Z r ¯Υ( s ) ds (cid:19) + 24( λ ∗ ) Z r (cid:18) k X j =1 ( F j ( t − s ) − F j ( r − s )) (cid:19) ¯Υ( s ) ds ≤ λ ∗ TN ϕ ( t − r ) + 24( λ ∗ T ) ϕ ( t − r ) + 8 k ( λ ∗ ) N k X j =1 Z r ( F j ( t − s ) − F j ( r − s )) ds + 24 k ( λ ∗ ) k X j =1 (cid:18)Z r ( F j ( t − s ) − F j ( r − s )) ds (cid:19) . Finally the third term on the right hand side of (3.35) is bounded by12( λ ∗ ) ( t − r ) ϕ ( t − r ) T + ( kλ ∗ ) k X j =1 Z r ( F j ( t − s ) − F j ( r − s )) ds . Combining the above three bounds, we conclude as in the proof of Lemma 3.5. (cid:3)
We also need the following technical lemma (which is a direct consequence of the inequality (4.21)in [18]).
Lemma 3.7.
Let { X N } N ≥ be a sequence of random elements in D . If for all ǫ > , the twoconditions (i) sup ≤ t ≤ T P (cid:0) | X N ( t ) | > ǫ (cid:1) → , as N → ∞ , and (ii) lim sup N sup ≤ t ≤ T δ P (cid:0) sup ≤ u ≤ δ | X N ( t + u ) − X N ( t ) | > ǫ (cid:1) → , as δ → are satisfied, then X N ( t ) → in probability uniformly in t . Lemma 3.8.
Under Assumptions 2.1, 2.2(ii) and 2.3, ˆ I N − e I N ⇒ in D as N → ∞ . (3.36) Proof.
We first have E (cid:2) ˆ I N ( t ) − e I N ( t ) (cid:3) = 0 , and E (cid:2)(cid:0) ˆ I N ( t ) − e I N ( t ) (cid:1) (cid:3) = E (cid:20)Z t [ λ − ¯ λ ]( t − s ) (cid:12)(cid:12) ¯Υ N ( s ) − ¯Υ( s ) (cid:12)(cid:12) ds (cid:21) ≤ (2 λ ∗ ) E (cid:2)(cid:12)(cid:12) ¯Υ N ( s ) − ¯Υ( s ) (cid:12)(cid:12)(cid:3) → N → ∞ . Here the convergence follows from E [ | ¯Υ N ( s ) − ¯Υ( s ) | ] → N → ∞ , (3.37)which holds by (3.3) and the dominated convergence theorem. It then suffices to show the tightnessof { ˆ I N − e I N : N ∈ N } . We haveˆ I N ( t ) − e I N ( t ) = 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D [ λ − ¯ λ ]( t − s )sign( ¯Υ N ( s ) − ¯Υ( s )) e Q ( ds, du, dλ ) . (Note that the equality also holds with e Q replaced by ˘ Q .) It then suffices to show the tightness ofthe processes { Ξ N : N ∈ N } defined byΞ N ( t ) := 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D [ λ − ¯ λ ]( t − s ) e Q ( ds, du, dλ ) . By Lemma 3.7, it suffices to show thatlim sup N →∞ δ P sup v ∈ [0 ,δ ] | Ξ N ( t + v ) − Ξ N ( t ) | > ǫ ! → δ → . (3.38)We have | Ξ N ( t + v ) − Ξ N ( t ) |≤ √ N Z t + vt Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D [ λ − ¯ λ ]( t + v − s ) ˘ Q ( ds, du, dλ )+ 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D | [ λ − ¯ λ ]( t + v − s ) − [ λ − ¯ λ ]( t − s ) | ˘ Q ( ds, du, dλ ) ≤ λ ∗ √ N Z t + vt Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du )+ 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D (cid:18) ϕ ( v ) + λ ∗ k X j =1 t − s<ξ j ≤ t + v − s + λ ∗ k X j =1 ( F j ( t + v − s ) − F j ( t − s )) (cid:19) ˘ Q ( ds, du, dλ ) , where the second inequality follows from Lemma 3.4. It is clear that the above upper bound isincreasing in v . Thus, we obtain that for any ǫ > P sup v ∈ [0 ,δ ] | Ξ N ( t + v ) − Ξ N ( t ) | > ǫ ! ≤ P λ ∗ √ N Z t + vt Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) > ǫ/ ! + P ϕ ( δ ) √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) > ǫ/ ! + P √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D (cid:18) λ ∗ k X j =1 t − s<ξ j ≤ t + v − s (cid:19) ˘ Q ( ds, du, dλ ) > ǫ/ + P √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:18) λ ∗ k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) Q ( ds, du ) > ǫ/ . (3.39)The first term is bounded by16 ǫ E λ ∗ √ N Z t + vt Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) ! ≤ λ ∗ ) ǫ ( E Z t + δt | ¯Υ N ( s ) − ¯Υ( s ) | ds + E "(cid:18)Z t + δt | ˆΥ N ( s ) | ds (cid:19) ≤ λ ∗ ) ǫ ( δ sup s ≤ T E | ¯Υ N ( s ) − ¯Υ( s ) | + δ sup s ≤ T E (cid:16) | ˆΥ N ( s ) | (cid:17)) , (3.40)where the first inequality follows from the decomposition Q ( ds, du ) = ¯ Q ( ds, du ) + dsdu . We notethat the first term on the right of (3.40) tends to 0 as N → ∞ , while the lim sup N of the secondterm multiplied by δ − tends to 0, as δ →
0, which is exactly what we want.The second term is bounded by16 ǫ E √ N ϕ ( δ ) Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) ! ≤ ǫ E √ N ϕ ( δ ) Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) ! + 32 ǫ E "(cid:18) ϕ ( δ ) Z t | ˆΥ N ( s ) | ds (cid:19) ≤ ǫ (2 ϕ ( δ )) Z t E | ¯Υ N ( s ) − ¯Υ( s ) | ds + 32 ǫ (2 ϕ ( δ )) T sup s ∈ [0 ,T ] E (cid:2)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) (cid:3) . By (3.37), the first term converges to zero as N → ∞ , while, thanks to our assumption (2.8), thelim sup N of the second term multiplied by δ − tends to 0, as δ →
0, which again is exactly whatwe want.The third term on the right hand side of (3.39) is bounded by16 ǫ E √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D (cid:18) λ ∗ k X j =1 t − s<ξ j ≤ t + v − s (cid:19) ˘ Q ( ds, du, dλ ) ≤ ǫ E √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Z D (cid:18) λ ∗ k X j =1 t − s<ξ j ≤ t + v − s (cid:19) e Q ( ds, du, dλ ) + 32 ǫ ( λ ∗ ) E Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds . Here the first term is equal to32 ǫ Z t E (cid:18) λ ∗ k X j =1 t − s<ξ j ≤ t + v − s (cid:19) | ¯Υ N ( s ) − ¯Υ( s ) | ds ≤ ǫ ( λ ∗ ) k Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) E [ | ¯Υ N ( s ) − ¯Υ( s ) | ] ds, which converges to zero as N → ∞ by (3.37). The second term satisfies1 δ E Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds → δ → , (3.41)which follows from Lemma (3.10) and a similar argument as in the proof of (5.13) in [18] under theconditions on F j in Assumption 2.2 (ii).The fourth and last term on the right hand side of (3.39) is bounded by16 ǫ E √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:18) λ ∗ k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) Q ( ds, du ) ≤ ǫ ( λ ∗ ) E √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) Q ( ds, du ) + 32 ǫ ( λ ∗ ) E Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds ≤ ǫ ( λ ∗ ) Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) E [ | ¯Υ N ( s ) − ¯Υ( s ) | ] ds + 32 ǫ ( λ ∗ ) E Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds . Here the first term converges to zero as N → ∞ by (3.37). The second term also satisfies (3.41).It is then clear that (3.38) holds for Ξ N . This completes the proof. (cid:3) Lemma 3.9.
Under Assumptions 2.1, 2.2(ii) and 2.3, ˆ I N − e I N ⇒ in D as N → ∞ . (3.42) Proof.
It is clear that E (cid:2) ˆ I N ( t ) − e I N ( t ) (cid:3) = 0 , and E (cid:2)(cid:0) ˆ I N ( t ) − e I N ( t ) (cid:1) (cid:3) = Z t ¯ λ ( t − s ) E (cid:2)(cid:12)(cid:12) ¯Υ N ( s ) − ¯Υ( s ) (cid:12)(cid:12)(cid:3) ds → N → ∞ , where the convergence follows from the bounded convergence theorem and (3.37). It then sufficesto show tightness of the sequence { ˆ I N − e I N : N ∈ N } . We writeˆ I N ( t ) − e I N ( t ) = 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) ¯ λ ( t − s )sign( ¯Υ N ( s ) − ¯Υ( s )) Q ( ds, du ) − √ N Z t ¯ λ ( t − s ) (cid:0) ¯Υ N ( s ) − ¯Υ( s ) (cid:1) ds . Notice that the tightness of the processes { ˆ I N − e I N : N ∈ N } can be deduced from the tightnessof the following two processesΞ N ( t ) := 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) ¯ λ ( t − s ) Q ( ds, du ) , Ξ N ( t ) := Z t ¯ λ ( t − s ) (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds. By Lemma 3.7, it suffices to show that for each ℓ = 1 , N →∞ δ P sup v ∈ [0 ,δ ] | Ξ Nℓ ( t + v ) − Ξ Nℓ ( t ) | > ǫ ! → δ → . (3.43)For the process Ξ N ( t ), we can writeΞ N ( t ) := 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) ¯ λ ( t − s ) Q ( ds, du ) , and thus, we have | Ξ N ( t + v ) − Ξ N ( t ) |≤ λ ∗ √ N Z t + vt Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du )+ 1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) | ¯ λ ( t + v − s ) − ¯ λ ( t − s ) | Q ( ds, du ) . We already know how to treat the first term, see (3.40). By Lemma 3.4, the second term on theright hand side is bounded by1 √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:16) ϕ ( v ) + λ ∗ k X j =1 ( F j ( t + v − s ) − F j ( t − s )) (cid:17) Q ( ds, du ) , which is nondecreasing in v . Thus, we obtain that for any ǫ > P sup v ∈ [0 ,δ ] | Ξ N ( t + v ) − Ξ N ( t ) | > ǫ ! ≤ P λ ∗ √ N Z t + δt Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) > ǫ/ ! + P √ N ϕ ( δ ) Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) > ǫ/ ! + P √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:18) λ ∗ k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) Q ( ds, du ) > ǫ/ (3.44)The first term is bounded as in (3.40). Let us bound the second term.9 ǫ E √ N ϕ ( δ ) Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) ! ≤ ǫ E √ N ϕ ( δ ) Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) Q ( ds, du ) ! + 18 ǫ E "(cid:18) ϕ ( δ ) Z t | ˆΥ N ( s ) | ds (cid:19) ≤ ǫ ϕ ( δ ) Z t E | ¯Υ N ( s ) − ¯Υ( s ) | ds + 18 ǫ ϕ ( δ ) T sup s ∈ [0 ,T ] E (cid:2)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) (cid:3) . This upper bound satisfies the proper bound (3.43), by the same argument as already used in theproof of the previous lemma.The third term on the right hand side of (3.44) is bounded by9 ǫ E √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:18) λ ∗ k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) Q ( ds, du ) ≤ ǫ ( λ ∗ ) E √ N Z t Z N ( ¯Υ N ( s ) ∨ ¯Υ( s )) N ( ¯Υ N ( s ) ∧ ¯Υ( s )) (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) Q ( ds, du ) + 18 ǫ ( λ ∗ ) E Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds ≤ ǫ ( λ ∗ ) Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) E [ | ¯Υ N ( s ) − ¯Υ( s ) | ] ds + 18 ǫ ( λ ∗ ) E Z t (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds . Here the first term converges to zero as N → ∞ by (3.37). The second term satisfies1 δ E √ N Z t + δ (cid:18) k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:19) | ˆΥ N ( s ) | ds → δ → , (3.45)which follows from a similar argument in the proof of (5.13) in [18] under the conditions on F j inAssumption 2.2 (ii), and using Lemma 3.10.Next for the process Ξ N ( t ), we have (cid:12)(cid:12) Ξ N ( t + v ) − Ξ N ( t ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z t + v ¯ λ ( t + v − s ) (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds − Z t ¯ λ ( t − s ) (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t + vt ¯ λ ( t + v − s ) (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds + Z t (cid:12)(cid:12) ¯ λ ( t + v − s ) − ¯ λ ( t − s ) (cid:12)(cid:12)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds. (3.46) The first term is bounded from above by λ ∗ Z t + vt (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds, which is increasing in v . By Lemma 3.4, the second term on the right hand side can be boundedby Z t (cid:16) ϕ ( v ) + λ ∗ k X j =1 ( F j ( t + v − s ) − F j ( t − s )) (cid:17)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds, which is nondecreasing in v . Thus, for any ǫ > P sup v ∈ [0 ,δ ] | Ξ N ( t + v ) − Ξ N ( t ) | > ǫ ! ≤ P (cid:18) λ ∗ Z t + δt (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds > ǫ/ (cid:19) + P (cid:18) ϕ ( δ ) Z t (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds > ǫ/ (cid:19) + P Z t (cid:16) λ ∗ k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:17)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds > ǫ/ ≤ λ ∗ ) ǫ E "(cid:18)Z t + δt (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds (cid:19) + 9 ǫ ϕ ( δ ) E "(cid:18)Z t (cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds (cid:19) + 9 ǫ E Z t (cid:16) λ ∗ k X j =1 ( F j ( t + δ − s ) − F j ( t − s )) (cid:17)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) ds . (3.47)The first and second terms are bounded, respectively, by9 ǫ ( λ ∗ ) δ sup s ∈ [0 ,T ] E (cid:2)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) (cid:3) and 9 ǫ ϕ ( δ ) T sup s ∈ [0 ,T ] E (cid:2)(cid:12)(cid:12) ˆΥ N ( s ) (cid:12)(cid:12) (cid:3) , while the third term satisfies (3.45). By Lemma 3.10 and our assumption (2.8), it is clear that(3.43) holds for Ξ N . This completes the proof of the lemma. (cid:3) Recall the representation of ˆΥ N ( t ) in (3.6). Lemma 3.10.
Under Assumptions 2.2 and 2.3, sup N sup t ∈ [0 ,T ] E (cid:2) ˆΥ N ( t ) (cid:3) < ∞ . (3.48) Proof.
We use (3.6) and the two integral representations in (3.8) and (3.9). We first obtain thefollowing estimates. It is clear that sup t ∈ [0 ,T ] E (cid:2) | ˆ M NA ( t ) | (cid:3) ≤ λ ∗ T and there exists a constant C such that for all N , sup t ∈ [0 ,T ] E (cid:2)(cid:0) ˆ I N (0)¯ λ ( t ) (cid:1) (cid:3) ≤ ( λ ∗ ) C . Also by the independence of { λ i ( · ) } andby the decomposition of ˆ A N using (3.1), and by (3.2), we havesup t ∈ [0 ,T ] E (cid:2) | ˆ I N ( t ) | (cid:3) = sup t ∈ [0 ,T ] E (cid:20)Z t [ λ − ¯ λ ] ( t − s ) ¯Υ N ( s ) ds (cid:21) ≤ λ ∗ Z T v ( s ) ds < ∞ , and by the PRM representation in (3.30) and by (3.2), we obtainsup t ∈ [0 ,T ] E (cid:2) | ˆ I N ( t ) | (cid:3) = sup t ∈ [0 ,T ] Z t ¯ λ ( t − s ) ¯Υ N ( s ) ds ≤ C ′ ( λ ∗ ) T , for some constant C ′ > S ( t ) ≤ I N ( t ) ≤ λ ∗ ( ¯ I N (0) + ¯ A N ( t )) ≤ λ ∗ by (2.4), weapply Gronwall’s inequality and obtainsup N sup t ∈ [0 ,T ] E (cid:2) ˆ S N ( t ) (cid:3) < ∞ and sup N sup t ∈ [0 ,T ] E (cid:2) ˆ I N ( t ) (cid:3) < ∞ , and thus the claim in (3.48). (cid:3) To show that the limit process ˆ I has a continuous version in C , given the consistent finitedimensional distributions of ˆ I , it suffices to show that the continuity of the covariance function.Note that E (cid:2)(cid:12)(cid:12) ˆ I ( t + δ ) − ˆ I ( t ) (cid:12)(cid:12) (cid:3) = Z t + δt ¯ λ ( t + δ − s ) ¯Υ( s ) ds + Z t (cid:0) ¯ λ ( t + δ − s ) − ¯ λ ( t − s ) (cid:1) ¯Υ( s ) ds . The continuity property follows immediately under Assumption 2.2 (ii).We are now ready to complete the proof of Theorem 2.1 for the joint convergence of (cid:0) ˆ S N , ˆ I N (cid:1) . Proof of Theorem 2.1.
We first prove the joint convergence (cid:0) ˆ I N (0) , ˆ M NA , ˆ I N , ˆ I N , ˆ I N (cid:1) ⇒ (cid:0) ˆ I (0) , ˆ M A , ˆ I , ˆ I , ˆ I (cid:1) in R × D as N → ∞ . (3.49)By the independence of the variables associated with the initially and newly infected individuals,it suffices to show the joint convergences (cid:0) ˆ I N (0) , ˆ I N (cid:1) ⇒ (cid:0) ˆ I (0) , ˆ I (cid:1) in R × D as N → ∞ , and (cid:0) ˆ M NA , ˆ I N , ˆ I N (cid:1) ⇒ (cid:0) ˆ M A , ˆ I , ˆ I (cid:1) in D as N → ∞ . (3.50)The convergence of (cid:0) ˆ I N (0) , ˆ I N (cid:1) is straightforward. We focus on the convergence of (cid:0) ˆ M NA , ˆ I N , ˆ I N (cid:1) .Recall the compensated PRM e Q ( ds, du, dλ ) on R + × R + × D . Define an auxiliary process f M NA ( t ) := 1 √ N Z t Z ∞ Z D u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) . (3.51)Recall the process e I N ( t ) defined in (3.20), where ˘ Q can be replaced by e Q . Also, recall the process e I N ( t ) in (3.32) using the compensated PRM Q , which can be equivalently (in distribution) writtenas follows using e Q : e I N ( t ) = 1 √ N Z t Z ∞ Z D ¯ λ ( t − s ) u ≤ N ¯Υ( s ) e Q ( ds, du, dλ ) . (3.52)Then it is easy to show that joint convergence (cid:0) f M NA , e I N , e I N (cid:1) ⇒ (cid:0) ˆ M A , ˆ I , ˆ I (cid:1) in D as N → ∞ . (3.53)Indeed, the joint finite dimensional distributions can be calculated similarly as in (3.21) and (3.33),and tightness of the joint processes follow directly from tightness of the individual processes (asshown in Lemmas 3.5 and 3.6 for the processes e I N and e I N , respectively). Similarly to the proofs in Lemma 3.8 and Lemma 3.9, we can prove that f M NA − ˆ M NA → D , as N → ∞ .Hence ( f M NA − ˆ M NA , e I N − ˆ I , e I N − ˆ I ) → D , as N → ∞ . Combined with (3.53), this establishes (3.50).Observe that the equations (3.8) and (3.9) coupled with (3.6) define uniquely the processes (cid:0) ˆ S N , ˆ I N (cid:1) as the solution of a two-dimensional integral equation driven by (cid:0) ˆ I N (0) , ˆ M NA , ˆ I N , ˆ I N , ˆ I N (cid:1) and the fixed functions ¯ λ ( t ), ¯ λ ( t ), ¯ S ( t ) and ¯ I ( t ). The mapping which to those data associates thesolution is continuous in the Skorohod J topology, see Lemma 8.1 in [18]. Thus, by the jointconvergence in (3.49), we apply the continuous mapping theorem to conclude (2.17). (cid:3) Convergence of ( ˆ E N , ˆ I N , ˆ R N ) . We have the following representations for the processes( ˆ E N , ˆ I N , ˆ R N ):ˆ E N ( t ) = ˆ E N (0) G c ( t ) + ˆ E N ( t ) + ˆ E N ( t ) + Z t G c ( t − s ) ˆΥ N ( s ) ds, (3.54)ˆ I N ( t ) = ˆ I N (0) F c ,I ( t ) + ˆ E N (0)Ψ ( t ) + ˆ I N , ( t ) + ˆ I N , ( t ) + ˆ I N ( t ) + Z t Ψ( t − s ) ˆΥ N ( s ) ds, (3.55)ˆ R N ( t ) = ˆ I N (0) F ,I ( t ) + ˆ E N (0)Φ ( t ) + ˆ R N , ( t ) + ˆ R N , ( t ) + ˆ R N ( t ) + Z t Φ( t − s ) ˆΥ N ( s ) ds, (3.56)where ˆ E N ( t ) := 1 √ N E N (0) X j =1 ( ζ j >t − G c ( t )) , ˆ I N , ( t ) := 1 √ N I N (0) X k =1 ( η ,Ik >t − F c ,I ( t )) , ˆ I N , ( t ) := 1 √ N E N (0) X j =1 ( ζ j + η j >t − Ψ ( t )) , ˆ R N , ( t ) := 1 √ N I N (0) X k =1 ( η ,Ik ≤ t − F ,I ( t )) , ˆ R N , ( t ) := 1 √ N E N (0) X j =1 ( ζ j + η j ≤ t − Φ ( t )) , and ˆ E N ( t ) := 1 √ N A N ( t ) X i =1 (cid:0) τ Ni + ζ i >t − G c ( t − τ Ni ) (cid:1) , ˆ I N ( t ) := 1 √ N A N ( t ) X i =1 (cid:0) τ Ni + ζ i ≤ t τ Ni + ζ i + η i >t − Ψ( t − τ Ni ) (cid:1) , ˆ R N ( t ) := 1 √ N A N ( t ) X i =1 (cid:0) τ Ni + ζ i + η i ≤ t − Φ( t − τ Ni ) (cid:1) . Lemma 3.11.
Under Under Assumptions 2.1, 2.2 and 2.3, (cid:0) ˆ E N , ˆ I N , , ˆ I N , , ˆ R N , , ˆ R N , (cid:1) ⇒ (cid:0) ˆ E , ˆ I , , ˆ I , , ˆ R , , ˆ R , (cid:1) in D as N → ∞ , jointly with the convergence (cid:0) ˆ I N , , ˆ I N , (cid:1) ⇒ (cid:0) ˆ I , , ˆ I N , (cid:1) in (3.12) , where (cid:0) ˆ E , ˆ I , , ˆ I , , ˆ R , , ˆ R , (cid:1) isas given in Theorem 2.2.Proof. By the independence of the sequences { λ j } j ≥ and { λ ,Ik } k ≥ , it suffices to prove the jointconvergence of (cid:0) ˆ I N , , ˆ I N , , ˆ R N , (cid:1) and (cid:0) ˆ I N , , ˆ E N , ˆ I N , , ˆ R N , (cid:1) separately. Recall the processes ˜ I N , and ˜ I N , defined in (3.13) and (3.14), respectively. Similarly for define (cid:0) ˜ E N , ˜ I N , , ˜ I N , , ˜ R N , , ˜ R N , (cid:1) by replacing E N (0) and I N (0) by N ¯ E (0) and N ¯ I (0), respectively. Bythe FCLT for random elements in D (see Theorem 2 in [12], applied to the processes ˜ I N , and ˜ I N , under Assumption 2.2(i) (a) and (b)) and the FCLT for empirical processes (see Theorem 14.3 in[4], applied to the processes (cid:0) ˜ E N , ˜ I N , , ˜ I N , , ˜ R N , , ˜ R N , (cid:1) ), and by the definitions in (2.2) and (2.3), weobtain the joint convergences (cid:0) ˜ I N , , ˜ I N , , ˜ R N , (cid:1) ⇒ (cid:0) ˆ I , , ˆ I , , ˆ R , (cid:1) in D as N → ∞ , and (cid:0) ˜ I N , , ˜ E N , ˜ I N , , ˜ R N , (cid:1) ⇒ (cid:0) ˆ I , , ˆ E , ˆ I , , ˆ R , (cid:1) in D as N → ∞ . It then suffices to show that (cid:0) ˜ I N , − ˆ I N , , ˜ I N , − ˆ I N , , ˜ I N , − ˆ I N , , ˜ R N , − ˆ R N , , ˜ E N − ˆ E N , ˜ I N , − ˆ I N , , ˜ R N , − ˆ R N , , (cid:1) ⇒ D as N → ∞ . The convergence for (cid:0) ˜ I N , − ˆ I N , , ˜ I N , − ˆ I N , (cid:1) ⇒ D is shown in (3.16). For theother process, the convergence follows from the same argument as in the proofs of Lemmas 5.1 and7.1 in [18]. This completes the proof. (cid:3) Lemma 3.12.
Under Assumptions 2.1, 2.2 and 2.3, (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) ⇒ (cid:0) ˆ E , ˆ I , ˆ R (cid:1) in D as N → ∞ , jointly with the convergence of ˆ I N ⇒ ˆ I in (3.19) and ˆ I N ⇒ ˆ I in (3.31) , where (cid:0) ˆ E , ˆ I , ˆ R (cid:1) is asgiven in Theorem 2.2.Proof. The convergence of (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) follows from the same argument as in the proofs of Lem-mas 7.3 and 7.4 in [18]. We have shown the joint convergence of (cid:0) ˆ I N , ˆ I N (cid:1) in (3.50) using theprocesses (cid:0)e I N , e I N (cid:1) which are defined via the PRM ˘ Q j ( ds, du, dξ ). We define (cid:0) e E N , e I N , e R N (cid:1) byreplacing A N ( t ) by ˘ A N ( t ) which is also defined using the PRM ˘ Q j ( ds, du, dξ ) (see (3.51)). It isnot too hard to establish the joint convergence (cid:0)e I N , e I N , e E N , e I N , e R N (cid:1) ⇒ (cid:0) ˆ I , ˆ I , ˆ E , ˆ I , ˆ R (cid:1) in D as N → ∞ . Then similar to Lemma 3.9, and Lemmas 7.3 and 7.4 in [18], we obtain the jointconvergence (cid:0) ˆ I N , ˆ I N , ˆ E N , ˆ I N , ˆ R N (cid:1) ⇒ (cid:0) ˆ I , ˆ I , ˆ E , ˆ I , ˆ R (cid:1) in D as N → ∞ . (cid:3) The representations of E N ( t ), I N ( t ) and R N ( t ) in (3.54), (3.55) and (3.56), give a natural in-tegral mapping from ( ˆ E N (0) , ˆ R N (0)), (cid:0) ˆ E N , ˆ I N , , ˆ I N , , ˆ R N , , ˆ R N , (cid:1) , (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) , (cid:0) ˆ S N , ˆ J N (cid:1) and thefixed functions G ( t ) , F ,I ( t ) , Ψ ( t ), ¯ λ ( t ), ¯ λ ,I ( t ), ¯ λ ( t ), ¯ S ( t ) and ¯ I ( t ) to the processes (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) .The mapping is continuous in the Skorohod J topology (by a slight modification of Lemma8.1 in [18]). We can then apply the continuous mapping theorem to conclude the convergence (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) ⇒ (cid:0) ˆ E, ˆ I, ˆ R (cid:1) in D . Given their joint convergence with (cid:0) ˆ I N , , ˆ I N , (cid:1) in Lemma 3.11 and(ˆ I N , ˆ I N ) in Lemma 3.12, we can also conclude the joint convergence of the processes (cid:0) ˆ E N , ˆ I N , ˆ R N (cid:1) with ( ˆ S N , ˆ I N ). This completes the proof of Theorem 2.2.4. Appendix
The aim of this Appendix is to establish the following slightly reinforced version of Theorem 13.5from [4].
Theorem 4.1.
Let { X N , N ≥ } be a sequence of elements of D . Assume that there exists acontinuous and non–decreasing function G and α > / such that for any r < s < t , N ≥ , E (cid:2) | X Ns − X Nr | × | X Nt − X Ns | (cid:3) ≤ (cid:0) ψ ( N ) + ( G ( t ) − G ( r )) α (cid:1) , where ψ ( N ) → , as N → ∞ . Then the sequence { X N , N ≥ } is tight. Theorem 4.1 is used in the proof of Lemma 3.2 with ψ ( N ) = sup ≤ t ′ ≤ t ′′ ≤ t ′ +1 n N E (cid:2) | ¯ E N (0) − ¯ E (0) | (cid:3) ( ψ ( t ′′ ) − ψ ( t ′ )) β + 3 E (cid:2) | ¯ E N (0) − ¯ E (0) | (cid:3) ( φ ( t ′′ ) − φ ( t ′ )) α o and G ≡
0, and in the proof of Lemma 3.5 with ψ ( N ) = C ′ /N , G ( t ) = ( C ′ ) δ t , and 2 α = 1 + δ . Proof.
We follow the proof on page 143 of [4], with β = 1 / F ( t ) = 2 / α G ( t ). We first notethat we deduce from our assumption that whenever ψ ( N ) ≤ ( G ( t ) − G ( r )) α , E (cid:2) | X Ns − X Nr | × | X Nt − X Ns | (cid:3) ≤ G ( t ) − G ( r )) α = [ F ( t ) − F ( r )] α . We define w ′′ ( X N , δ ) as in [4] (see in particular the formula (12.27)), where however the interval[0 ,
1] is replaced by [0 , T ], with
T > C ′ > P (cid:0) w ′′ ( X N , δ ) ≥ ε (cid:1) ≤ C ′ ε w F (2 δ ) α − , provided ψ ( N ) ≤ w G (2 δ ) α , where w F (resp. w G ) is the modulus of continuity of F (resp. G ).Given η > δ ε,η be such that C ′ ε w F (2 δ ε,η ) α − ≤ η and N suchthat ψ ( N ) ≤ w G (2 δ ε,η ) α , for all N ≥ N . We have proved that for any ε , η >
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The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, Collegeof Engineering, Pennsylvania State University, University Park, PA 16802 USA
E-mail address : [email protected] Aix–Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille, France
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