1001.0044.txt

arXiv:1001.0044v3  [math.PR]  31 Mar 2011A law of large numbers approximation for
Markov population processes with countably
many types
A. D. Barbour∗and M. J. Luczak†
Universit¨ at Z¨ urich and London School of Economics
Abstract
When modelling metapopulation dynamics, the influence of a s in-
gle patch on the metapopulation depends on the number of indi vidu-
als in the patch. Since the population size has no natural upp er limit,
this leads to systems in which there are countably infinitely many
possible types of individual. Analogous considerations ap ply in the
transmission of parasitic diseases. In this paper, we prove a law of
large numbers for quite general systems of this kind, togeth er with
a rather sharp bound on the rate of convergence in an appropri ately
chosen weighted ℓ1norm.
Keywords: Epidemic models, metapopulation processes, countably many
types, quantitative law of large numbers, Markov population proce sses
AMS subject classification: 92D30, 60J27, 60B12
Running head: A law of large numbers approximation
1 Introduction
There are many biological systems that consist of entities that diffe r in their
influence according to the number of active elements associated wit h them,
∗Angewandte Mathematik, Universit¨ at Z¨ urich, Winterthurertra sse 190, CH-8057
Z¨URICH; ADB was supported in part by Schweizerischer Nationalfond s Projekt Nr. 20–
107935/1.
†London School of Economics; MJL was supported in part by a STICE RD grant.
1and can be divided into types accordingly. In parasitic diseases (Bar bour &
Kafetzaki 1993, Luchsinger 2001a,b, Kretzschmar 1993), the in fectivity of a
host depends on the number of parasites that it carries; in metapo pulations,
the migration pressure exerted by a patch is related to the number of its
inhabitants (Arrigoni 2003); the behaviour of a cell may depend on the num-
ber of copies of a particular gene that it contains (Kimmel & Axelrod 2 002,
Chapter 7); and so on. In none of these examples is there a natura l upper
limit to the number of associated elements, so that the natural set ting for
a mathematical model is one in which there are countably infinitely man y
possible types of individual. In addition, transition rates typically incr ease
with the number of associated elements in the system — for instance , each
parasite has an individual death rate, so that the overall death ra te of par-
asites grows at least as fast as the number of parasites — and this le ads
to processes with unbounded transition rates. This paper is conce rned with
approximations to density dependent Markov models of this kind, wh en the
typical population size Nbecomes large.
In density dependent Markov population processes with only finitely
many types of individual, a law of large numbers approximation, in the f orm
ofasystemofordinarydifferentialequations, wasestablishedbyK urtz(1970),
together with a diffusion approximation (Kurtz, 1971). In the infinit e di-
mensional case, the law of large numbers was proved for some spec ific mod-
els (Barbour & Kafetzaki 1993, Luchsinger 2001b, Arrigoni 2003 , see also
L´ eonard 1990), using individually tailored methods. A more general result
was then given by Eibeck & Wagner (2003). In Barbour & Luczak (20 08),
the law of large numbers was strengthened by the addition of an err or bound
inℓ1that is close to optimal order in N. Their argument makes use of an
intermediate approximation involving an independent particles proce ss, for
which the law of large numbers is relatively easy to analyse. This proce ss is
then shown to be sufficiently close to the interacting process of act ual inter-
est, by means of a coupling argument. However, the generality of t he results
obtained is limited by the simple structure of the intermediate proces s, and
the model of Arrigoni (2003), for instance, lies outside their scop e.
In this paper, we develop an entirely different approach, which circu m-
vents the need for an intermediate approximation, enabling a much w ider
class of models to be addressed. The setting is that of families of Mar kov
population processes XN:= (XN(t), t≥0),N≥1, taking values in the
countable space X+:={X∈ZZ+
+;/summationtext
m≥0Xm<∞}. Each component repre-
2sents the number of individuals of a particular type, and there are c ountably
many types possible; however, at any given time, there are only finit ely
many individuals in the system. The process evolves as a Markov proc ess
with state-dependent transitions
X→X+Jat rate NαJ(N−1X), X∈ X+, J∈ J,(1.1)
where each jump is of bounded influence, in the sense that
J ⊂ {X∈ZZ+;/summationdisplay
m≥0|Xm| ≤J∗<∞},for some fixed J∗<∞,(1.2)
so that the number of individuals affected is uniformly bounded. Dens ity
dependence is reflected in the fact that the arguments of the fun ctionsαJ
are counts normalised by the ‘typical size’ N. Writing R:=RZ+
+, the func-
tionsαJ:R →R+are assumed to satisfy
/summationdisplay
J∈JαJ(ξ)<∞, ξ∈ R0, (1.3)
whereR0:={ξ∈ R:ξi= 0 for all but finitely many i}; this assumption
implies that the processes XNare pure jump processes, at least for some
non-zero length of time. To prevent the paths leaving X+, we also assume
thatJl≥ −1 for each l, and that αJ(ξ) = 0 ifξl= 0 for any J∈ Jsuch
thatJl=−1. Some remarks on the consequences of allowing transitions J
withJl≤ −2 for some lare made at the end of Section 4.
Thelawoflargenumbersisthenformallyexpressed intermsofthesy stem
ofdeterministic equations
dξ
dt=/summationdisplay
J∈JJαJ(ξ) =:F0(ξ), (1.4)
to be understood componentwise for those ξ∈ Rsuch that
/summationdisplay
J∈J|Jl|αJ(ξ)<∞,for alll≥0,
thus by assumption including R0. Here, the quantity F0represents the in-
finitesimal average drift of the components of the random proces s. However,
in this generality, it is not even immediately clear that equations (1.4) h ave
a solution.
3In order to make progress, it is assumed that the unbounded comp onents
in the transition rates can be assimilated into a linear part, in the sens e
thatF0can be written in the form
F0(ξ) =Aξ+F(ξ), (1.5)
again to be understood componentwise, where Ais a constant Z+×Z+
matrix. These equations are then treated as a perturbed linear sy stem
(Pazy 1983, Chapter 6). Under suitable assumptions on A, there exists a
measure µonZ+, defining a weighted ℓ1norm/⌊ard⌊l · /⌊ard⌊lµonR, and a strongly
/⌊ard⌊l·/⌊ard⌊lµ–continuoussemigroup {R(t), t≥0}oftransitionmatriceshaving point-
wise derivative R′(0) =A. IfFis locally /⌊ard⌊l·/⌊ard⌊lµ–Lipschitz and /⌊ard⌊lx(0)/⌊ard⌊lµ<∞,
this suggests using the solution xof the integral equation
x(t) =R(t)x(0)+/integraldisplayt
0R(t−s)F(x(s))ds (1.6)
as an approximation to xN:=N−1XN, instead of solving the deterministic
equations (1.4) directly. We go on to show that the solution XNof the
stochastic system can be expressed using a formula similar to (1.6), which
has an additional stochastic component in the perturbation:
xN(t) =R(t)xN(0)+/integraldisplayt
0R(t−s)F(xN(s))ds+/tildewidemN(t),(1.7)
where
/tildewidemN(t) :=/integraldisplayt
0R(t−s)dmN(s), (1.8)
andmNis the local martingale given by
mN(t) :=xN(t)−xN(0)−/integraldisplayt
0F0(xN(s))ds. (1.9)
The quantity mNcanbe expected to be small, at least componentwise, under
reasonable conditions.
To obtain tight control over /tildewidemNin all components simultaneously, suf-
ficient to ensure that sup0≤s≤t/⌊ard⌊l/tildewidemN(s)/⌊ard⌊lµis small, we derive Chernoff–like
boundsonthedeviations ofthemost significant components, witht hehelpof
a family of exponential martingales. The remaining components are t reated
usingsomegeneral a prioriboundsonthebehaviourofthestochasticsystem.
4This allows us to take the difference between the stochastic and det erministic
equations (1.7) and (1.6), after which a Gronwall argument can be c arried
through, leading to the desired approximation.
The main result, Theorem 4.7, guarantees an approximation error o f or-
derO(N−1/2√logN) in the weighted ℓ1metric/⌊ard⌊l·/⌊ard⌊lµ, except on an event of
probability of order O(N−1logN). More precisely, for each T >0, there
exist constants K(1)
T,K(2)
T,K(3)
Tsuch that, for Nlarge enough, if
/⌊ard⌊lN−1XN(0)−x(0)/⌊ard⌊lµ≤K(1)
T/radicalbigg
logN
N,
then
P/parenleftBig
sup
0≤t≤T/⌊ard⌊lN−1XN(t)−x(t)/⌊ard⌊lµ> K(2)
T/radicalbigg
logN
N/parenrightBig
≤K(3)
TlogN
N.(1.10)
Theerrorboundissharper, byafactoroflog N, thanthatgiveninBarbour&
Luczak(2008),andthetheoremisapplicabletoamuch widerclassof models.
However, the method of proof involves moment arguments, which r equire
somewhat stronger assumptions on the initial state of the system , and, in
models such as that of Barbour & Kafetzaki (1993), onthe choice of infection
distributions allowed. The conditions under which the theorem holds c an be
divided into three categories: growth conditions on the transition r ates, so
that the a prioribounds, which have the character of moment bounds, can
be established; conditions on the matrix A, sufficient to limit the growth of
the semigroup R, and (together with the properties of F) to determine the
weights defining the metric in which the approximation is to be carried o ut;
and conditions on the initial state of the system. The a priori bounds are
derived in Section 2, the semigroup analysis is conducted in Section 3, and
the approximation proper is carried out in Section 4. The paper conc ludes
in Section 5 with some examples.
The form (1.8) of the stochastic component /tildewidemN(t) in (1.7) is very simi-
lar to that of a key element in the analysis of stochastic partial differ ential
equations; see, for example, Chow (2007, Section 6.6). The SPDE a rguments
used for its control are however typically conducted in a Hilbert spa ce con-
text. Our setting is quite different in nature, and it does not seem cle ar how
to translate the SPDE methods into our context.
52 A priori bounds
We begin by imposing further conditions on the transition rates of th e pro-
cessXN, sufficient to constrain its paths to bounded subsets of X+dur-
ing finite time intervals, and in particular to ensure that only finitely ma ny
jumps can occur in finite time. The conditions that follow have the flav our
of moment conditions on the jump distributions. Since the index j∈Z+is
symbolic in nature, we start by fixing an ν∈ R, such that ν(j) reflects in
some sense the ‘size’ of j, with most indices being ‘large’:
ν(j)≥1 for allj≥0 and lim
j→∞ν(j) =∞. (2.1)
We then define the analogues of higher empirical moments using the q uanti-
tiesνr∈ R, defined by νr(j) :=ν(j)r,r≥0, setting
Sr(x) :=/summationdisplay
j≥0νr(j)xj=xTνr, x∈ R0, (2.2)
where, for x∈ R0andy∈ R,xTy:=/summationtext
l≥0xlyl. In particular, for X∈ X+,
S0(X) =/⌊ard⌊lX/⌊ard⌊l1. Note that, because of (2.1), for any r≥1,
#{X∈ X+:Sr(X)≤K}<∞for allK >0. (2.3)
To formulate the conditions that limit the growth of the empirical mom ents
ofXN(t) witht, we also define
Ur(x) :=/summationdisplay
J∈JαJ(x)JTνr;Vr(x) :=/summationdisplay
J∈JαJ(x)(JTνr)2, x∈ R.(2.4)
The assumptions that we shall need are then as follows.
Assumption 2.1 There exists a νsatisfying (2.1)andr(1)
max,r(2)
max≥1such
that, for all X∈ X+,
/summationdisplay
J∈JαJ(N−1X)|JTνr|<∞,0≤r≤r(1)
max,(2.5)
the case r= 0following from (1.2)and(1.3); furthermore, for some non-
negative constants krl, the inequalities
U0(x)≤k01S0(x)+k04,
U1(x)≤k11S1(x)+k14, (2.6)
Ur(x)≤ {kr1+kr2S0(x)}Sr(x)+kr4,2≤r≤r(1)
max;
6and
V0(x)≤k03S1(x)+k05,
Vr(x)≤kr3Sp(r)(x)+kr5,1≤r≤r(2)
max, (2.7)
are satisfied, where 1≤p(r)≤r(1)
maxfor1≤r≤r(2)
max.
The quantities r(1)
maxandr(2)
maxusually need to be reasonably large, if Assump-
tion 4.2 below is to be satisfied.
Now, for XNas in the introduction, we let tXNndenote the time of its
n-th jump, with tXN
0= 0, and set tXN∞:= lim n→∞tXNn, possibly infinite. For
0≤t < tXN∞, we define
S(N)
r(t) :=Sr(XN(t));U(N)
r(t) :=Ur(xN(t));V(N)
r(t) :=Vr(xN(t)),
(2.8)
once again with xN(t) :=N−1XN(t), and also
τ(N)
r(C) := inf {t < tXN
∞:S(N)
r(t)≥NC}, r≥0,(2.9)
where the infimum of the empty set is taken to be ∞. Our first result shows
thattXN∞=∞a.s., and limits the expectations of S(N)
0(t) andS(N)
1(t) for any
fixedt.
In what follows, we shall write F(N)
s=σ(XN(u),0≤u≤s), so that
(F(N)
s:s≥0) is the natural filtration of the process XN.
Lemma 2.2 Under Assumptions 2.1, tXN∞=∞a.s. Furthermore, for any
t≥0,
E{S(N)
0(t)} ≤(S(N)
0(0)+Nk04t)ek01t;
E{S(N)
1(t)} ≤(S(N)
1(0)+Nk14t)ek11t.
Proof. Introducing the formal generator ANassociated with (1.1),
ANf(X) :=N/summationdisplay
J∈JαJ(N−1X){f(X+J)−f(X)}, X∈ X+,(2.10)
we note that NUl(x) =ANSl(Nx). Hence, if we define M(N)
lby
M(N)
l(t) :=S(N)
l(t)−S(N)
l(0)−N/integraldisplayt
0U(N)
l(u)du, t ≥0,(2.11)
7for 0≤l≤r(1)
max, it is immediate from (2.3), (2.5) and (2.6) that the process
(M(N)
l(t∧τ(N)
1(C)), t≥0) is a zero mean F(N)–martingale for each C >0.
In particular, considering M(N)
1(t∧τ(N)
1(C)), it follows in view of (2.6) that
E{S(N)
1(t∧τ(N)
1(C))} ≤S(N)
1(0)+E/braceleftBigg/integraldisplayt∧τ(N)
1(C)
0{k11S(N)
1(u)+Nk14}du/bracerightBigg
≤S(N)
1(0)+/integraldisplayt
0(k11E{S(N)
1(u∧τ(N)
1(C))}+Nk14)du.
Using Gronwall’s inequality, we deduce that
E{S(N)
1(t∧τ(N)
1(C))} ≤(S(N)
1(0)+Nk14t)ek11t,(2.12)
uniformly in C >0, and hence that
P/bracketleftBig
sup
0≤s≤tS1(XN(s))≥NC/bracketrightBig
≤C−1(S1(xN(0))+k14t)ek11t(2.13)
also. Hence sup0≤s≤tS1(XN(s))<∞a.s. for any t, limC→∞τ(N)
1(C) =∞
a.s., and, from (2.3) and (1.3), it thus follows that tXN∞=∞a.s. The bound
onE{S(N)
1(t)}is now immediate, and that on E{S(N)
0(t)}follows by applying
the same Gronwall argument to M(N)
0(t∧τ(N)
1(C)).
The next lemma shows that, if any T >0 is fixed and Cis chosen large
enough, then, with high probability, N−1S(N)
0(t)≤Cholds for all 0 ≤t≤T.
Lemma 2.3 Assume that Assumptions 2.1 are satisfied, and that S(N)
0(0)≤
NC0andS(N)
1(0)≤NC1. Then, for any C≥2(C0+k04T)ek01T, we have
P[{τ(N)
0(C)≤T}]≤(C1∨1)K00/(NC2),
whereK00depends on Tand the parameters of the model.
Proof. It is immediate from (2.11) and (2.6) that
S(N)
0(t) =S(N)
0(0)+N/integraldisplayt
0U(N)
0(u)du+M(N)
0(t)
≤S(N)
0(0)+/integraldisplayt
0(k01S(N)
0(u)+Nk04)du+ sup
0≤u≤tM(N)
0(u).(2.14)
8Hence, from Gronwall’s inequality, if S(N)
0(0)≤NC0, then
S(N)
0(t)≤/braceleftbigg
N(C0+k04T)+ sup
0≤u≤tM(N)
0(u)/bracerightbigg
ek01t.(2.15)
Now, considering the quadratic variation of M(N)
0, we have
E/braceleftBigg
{M(N)
0(t∧τ(N)
1(C′))}2−N/integraldisplayt∧τ(N)
1(C′)
0V(N)
0(u)du/bracerightBigg
= 0 (2.16)
for anyC′>0, from which it follows, much as above, that
E/parenleftBig
{M(N)
0(t∧τ(N)
1(C′))}2/parenrightBig
≤E/braceleftbigg
N/integraldisplayt
0V(N)
0(u∧τ(N)
1(C′))du/bracerightbigg
≤/integraldisplayt
0{k03ES(N)
1(u∧τ(N)
1(C′))+Nk05}du.
Using (2.12), we thus find that
E/parenleftBig
{M(N)
0(t∧τ(N)
1(C′))}2/parenrightBig
≤k03
k11N(C1+k14T)(ek11t−1)+Nk05t,(2.17)
uniformlyforall C′. Doob’smaximal inequality appliedto M(N)
0(t∧τ(N)
1(C′))
now allows us to deduce that, for any C′,a >0,
P/bracketleftBig
sup
0≤u≤TM(N)
0(u∧τ(N)
1(C′))> aN/bracketrightBig
≤1
Na2/braceleftbiggk03
k11(C1+k14T){ek11T−1}+k05T/bracerightbigg
=:C1K01+K02
Na2,
say, so that, letting C′→ ∞,
P/bracketleftBig
sup
0≤u≤TM(N)
0(u)> aN/bracketrightBig
≤C1K01+K02
Na2
also. Taking a=1
2Ce−k01Tand putting the result into (2.15), the lemma
follows.
In the next theorem, we control the ‘higher ν-moments’ S(N)
r(t) ofXN(t).
9Theorem 2.4 Assume thatAssumptions 2.1are satisfied, andthat S(N)
1(0)≤
NC1andS(N)
p(1)(0)≤NC′
1. Then, for 2≤r≤r(1)
maxand for any C >0, we
have
E{S(N)
r(t∧τ(N)
0(C))} ≤(S(N)
r(0)+Nkr4t)e(kr1+Ckr2)t,0≤t≤T.(2.18)
Furthermore, if for 1≤r≤r(2)
max,S(N)
r(0)≤NCrandS(N)
p(r)(0)≤NC′
r,
then, for any γ≥1,
P[ sup
0≤t≤TS(N)
r(t∧τ(N)
0(C))≥NγC′′
rT]≤Kr0γ−2N−1, (2.19)
where
C′′
rT:= (Cr+kr4T+/radicalbig
(C′
r∨1))e(kr1+Ckr2)T
andKr0depends on C,Tand the parameters of the model.
Proof. Recalling (2.11), use the argument leading to (2.12) with the martin-
galesM(N)
r(t∧τ(N)
1(C′)∧τ(N)
0(C)), for any C′>0, to deduce that
ES(N)
r(t∧τ(N)
1(C′)∧τ(N)
0(C))
≤S(N)
r(0)+/integraldisplayt
0/parenleftBig
{kr1+Ckr2}E/braceleftBig
S(N)
r(u∧τ(N)
1(C′)∧τ(N)
0(C))/bracerightBig
+Nkr4/parenrightBig
du,
for 1≤r≤r(1)
max, sinceN−1S(N)
0(u)≤Cwhenu≤τ(N)
0(C): define k12= 0.
Gronwall’s inequality now implies that
ES(N)
r(t∧τ(N)
1(C′)∧τ(N)
0(C))≤(S(N)
r(0)+Nkr4t)e(kr1+Ckr2)t,(2.20)
for 1≤r≤r(1)
max, and (2.18) follows by Fatou’s lemma, on letting C′→ ∞.
Now, also from (2.11) and (2.6), we have, for t≥0 and each r≤r(1)
max,
S(N)
r(t∧τ(N)
0(C))
=S(N)
r(0)+N/integraldisplayt∧τ(N)
0(C)
0U(N)
r(u)du+M(N)
r(t∧τ(N)
0(C))
≤S(N)
r(0)+/integraldisplayt
0/parenleftBig
{kr1+Ckr2}S(N)
r(u∧τ(N)
0(C))+Nkr4/parenrightBig
du
+ sup
0≤u≤tM(N)
r(u∧τ(N)
0(C)).
10Hence, from Gronwall’s inequality, for all t≥0 andr≤r(1)
max,
S(N)
r(t∧τ(N)
0(C))≤/braceleftBig
N(Cr+kr4t)+ sup
0≤u≤tM(N)
r(u∧τ(N)
0(C))/bracerightBig
e(kr1+Ckr2)t.
(2.21)
Now, as in (2.16), we have
E/braceleftBigg
{M(N)
r(t∧τ(N)
1(C′)∧τ(N)
0(C))}2−N/integraldisplayt∧τ(N)
1(C′)∧τ(N)
0(C)
0V(N)
r(u)du/bracerightBigg
= 0,
(2.22)
from which it follows, using (2.7), that, for 1 ≤r≤r(2)
max,
E/parenleftBig
{M(N)
r(t∧τ(N)
1(C′)∧τ(N)
0(C))}2/parenrightBig
≤E/braceleftBigg
N/integraldisplayt∧τ(N)
1(C′)∧τ(N)
0(C))
0V(N)
r(u)du/bracerightBigg
≤/integraldisplayt
0{kr3ES(N)
p(r)(u∧τ(N)
1(C′)∧τ(N)
0(C))+Nkr5}du
≤N(C′
r+kp(r),4T)kr3
kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2t)−1)+Nkr5T,
this last by (2.20), since p(r)≤r(1)
maxfor 1≤r≤r(2)
max. Using Doob’s
inequality, it follows that, for any a >0,
P/bracketleftBig
sup
0≤u≤TM(N)
r(u∧τ(N)
0(C))> aN/bracketrightBig
≤1
Na2/braceleftbiggkr3(C′
r+kp(r),4T)
kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2T)−1)+kr5T/bracerightbigg
=:C′
rKr1+Kr2
Na2.
Takinga=γ/radicalbig
(C′
r∨1) and putting the result into (2.21) gives (2.19), with
Kr0= (C′
rKr1+Kr2)/(C′
r∨1).
Note also that sup0≤t≤TS(N)
r(t)<∞a.s. for all 0 ≤r≤r(2)
max, in view of
Lemma 2.3 and Theorem 2.4.
In what follows, we shall particularly need to control quantities of t he
form/summationtext
J∈JαJ(xN(s))d(J,ζ), where xN:=N−1XNand
d(J,ζ) :=/summationdisplay
j≥0|Jj|ζ(j), (2.23)
11forζ∈ Rchosen such that ζ(j)≥1 grows fast enough with j: see (4.12).
Defining
τ(N)(a,ζ) := inf/braceleftBigg
s:/summationdisplay
J∈JαJ(xN(s))d(J,ζ)≥a/bracerightBigg
,(2.24)
infinite if there is no such s, we show in the following corollary that, under
suitable assumptions, τ(N)(a,ζ) is rarely less than T.
Corollary 2.5 Assume that Assumptions 2.1 hold, and that ζis such that
/summationdisplay
J∈JαJ(N−1X)d(J,ζ)≤ {k1N−1Sr(X)+k2}b(2.25)
for some 1≤r:=r(ζ)≤r(2)
maxand some b=b(ζ)≥1. For this value
ofr, assume that S(N)
r(0)≤NCrandS(N)
p(r)(0)≤NC′
rfor some constants
CrandC′
r. Assume further that S(N)
0(0)≤NC0,S(N)
1(0)≤NC1for some
constants C0,C1, and define C:= 2(C0+k04T)ek01T. Then
P[τ(N)(a,ζ)≤T]≤N−1{Kr0γ−2
a+K00(C1∨1)C−2},
for anya≥ {k2+k1C′′
rT}b, whereγa:= (a1/b−k2)/{k1C′′
rT},Kr0andC′′
rT
are as in Theorem 2.4, and K00is as in Lemma 2.3.
Proof. In view of (2.25), it is enough to bound the probability
P[ sup
0≤t≤TS(N)
r(t)≥N(a1/b−k2)/k1].
However, Lemma 2.3 and Theorem 2.4 together bound this probability by
N−1/braceleftbig
Kr0γ−2
a+K00(C1∨1)C−2/bracerightbig
,
whereγais as defined above, as long as a1/b−k2≥k1C′′
rT.
If (2.25) is satisfied,/summationtext
J∈JαJ(xN(s))d(J,ζ)isa.s. bounded on0 ≤s≤T,
becauseS(N)
r(s) is. The corollary shows that the sum is then bounded by
{k2+k1C′′
r,T}b, except on an event of probability of order O(N−1). Usually,
one can choose b= 1.
123 Semigroup properties
We make the following initial assumptions about the matrix A: first, that
Aij≥0 for alli/ne}ationslash=j≥0;/summationdisplay
j/negationslash=iAji<∞for alli≥0,(3.1)
and then that, for some µ∈RZ+
+such that µ(m)≥1 for each m≥0, and
for some w≥0,
ATµ≤wµ. (3.2)
We then use µto define the µ-norm
/⌊ard⌊lξ/⌊ard⌊lµ:=/summationdisplay
m≥0µ(m)|ξm|onRµ:={ξ∈ R:/⌊ard⌊lξ/⌊ard⌊lµ<∞}.(3.3)
Note that there may be many possible choices for µ. In what follows, it is
important that Fbe a Lipschitz operator with respect to the µ-norm, and
this has to be borne in mind when choosing µ.
Setting
Qij:=AT
ijµ(j)/µ(i)−wδij, (3.4)
whereδis the Kronecker delta, we note that Qij≥0 fori/ne}ationslash=j, and that
0≤/summationdisplay
j/negationslash=iQij=/summationdisplay
j/negationslash=iAT
ijµ(j)/µ(i)≤w−Aii=−Qii,
using (3.2) for the inequality, so that Qii≤0. Hence Qcan be augmented to
a conservative Q–matrix, in the sense of Markov jump processes, by adding a
coffin state ∂, and setting Qi∂:=−/summationtext
j≥0Qij≥0. LetP(·) denote the semi-
group of Markov transition matrices corresponding to the minimal p rocess
associated with Q; then, in particular,
Q=P′(0) and P′(t) =QP(t) for all t≥0 (3.5)
(Reuter 1957, Theorem 3). Set
RT
ij(t) :=ewtµ(i)Pij(t)/µ(j). (3.6)
13Theorem 3.1 LetAsatisfy Assumptions (3.1)and(3.2). Then, with the
above definitions, Ris a strongly continuous semigroup on Rµ, and
/summationdisplay
i≥0µ(i)Rij(t)≤µ(j)ewtfor alljandt. (3.7)
Furthermore, the sums/summationtext
j≥0Rij(t)Ajk= (R(t)A)ikare well defined for all
i,k, and
A=R′(0)andR′(t) =R(t)Afor allt≥0.(3.8)
Proof. We note first that, for x∈ Rµ,
/⌊ard⌊lR(t)x/⌊ard⌊lµ≤/summationdisplay
i≥0µ(i)/summationdisplay
j≥0Rij(t)|xj|=ewt/summationdisplay
i≥0/summationdisplay
j≥0µ(j)Pji(t)|xj|
≤ewt/summationdisplay
j≥0µ(j)|xj|=ewt/⌊ard⌊lx/⌊ard⌊lµ, (3.9)
sinceP(t) is substochastic on Z+; henceR:Rµ→ Rµ. To show strong
continuity, we take x∈ Rµ, and consider
/⌊ard⌊lR(t)x−x/⌊ard⌊lµ=/summationdisplay
i≥0µ(i)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
j≥0Rij(t)xj−xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/summationdisplay
i≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleewt/summationdisplay
j≥0µ(j)Pji(t)xj−µ(i)xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤(ewt−1)/summationdisplay
i≥0/summationdisplay
j≥0µ(j)Pji(t)xj+/summationdisplay
i≥0/summationdisplay
j/negationslash=iµ(j)Pji(t)xj+/summationdisplay
i≥0µ(i)xi(1−Pii(t))
≤(ewt−1)/summationdisplay
j≥0µ(j)xj+2/summationdisplay
i≥0µ(i)xi(1−Pii(t)),
from which it follows that lim t→0/⌊ard⌊lR(t)x−x/⌊ard⌊lµ= 0, by dominated conver-
gence, since lim t→0Pii(t) = 1 for each i≥0.
The inequality (3.7) follows from the definition of Rand the fact that P
is substochastic on Z+. Then
(ATRT(t))ij=/summationdisplay
k/negationslash=iQikµ(i)
µ(k)ewtµ(k)
µ(j)Pkj(t)+(Qii+w)ewtµ(i)
µ(j)Pij(t)
=µ(i)
µ(j)[(QP(t))ij+wPij(t)]ewt,
14with (QP(t))ij=/summationtext
k≥0QikPkj(t) well defined because P(t) is sub-stochastic
andQis conservative. Using (3.5), this gives
(ATRT(t))ij=µ(i)
µ(j)d
dt[Pij(t)ewt] =d
dtRT
ij(t),
and this establishes (3.8).
4 Main approximation
LetXN,N≥1, beasequence ofpure jumpMarkov processes asinSection 1,
withAandFdefined as in (1.4) and (1.5), and suppose that F:Rµ→ Rµ,
withRµas defined in (3.3), for some µsuch that Assumption (3.2) holds.
Suppose also that Fis locally Lipschitz in the µ-norm: for any z >0,
sup
x/negationslash=y:/bardblx/bardblµ,/bardbly/bardblµ≤z/⌊ard⌊lF(x)−F(y)/⌊ard⌊lµ//⌊ard⌊lx−y/⌊ard⌊lµ≤K(µ,F;z)<∞.(4.1)
Then, for x(0)∈ RµandRas in (3.6), the integral equation
x(t) =R(t)x(0)+/integraldisplayt
0R(t−s)F(x(s))ds. (4.2)
has a unique continuous solution xinRµon some non-empty time interval
[0,tmax), such that, if tmax<∞, then/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞ast→tmax(Pazy 1983,
Theorem 1.4, Chapter 6). Thus, if Awere the generator of R, the function x
would be a mild solution of the deterministic equations (1.4). We now wish
to show that the process xN:=N−1XNis close to x. To do so, we need a
corresponding representation for XN.
To find such a representation, let W(t),t≥0, be a pure jump path on X+
that has only finitely many jumps up to time T. Then we can write
W(t) =W(0)+/summationdisplay
j:σj≤t∆W(σj),0≤t≤T, (4.3)
where ∆W(s) :=W(s)−W(s−)andσj,j≥1, denote thetimes when Whas
its jumps. Now let Asatisfy (3.1) and (3.2), and let R(·) be the associated
semigroup, as defined in (3.6). Define the path W∗(t), 0≤t≤T, from the
equation
W∗(t) :=R(t)W(0)+/summationtext
j:σj≤tR(t−σj)∆j−/integraltextt
0R(t−s)AW(s)ds,
(4.4)
15where ∆ j:= ∆W(σj). Note that the latter integral makes sense, because
each of the sums/summationtext
j≥0Rij(t)Ajkis well defined, from Theorem 3.1, and
because only finitely many of the coordinates of Ware non-zero.
Lemma 4.1 W∗(t) =W(t)for all0≤t≤T.
Proof. Fix any t, and suppose that W∗(s) =W(s) for alls≤t. This is
clearly the case for t= 0. Let σ(t)> tdenote the time of the first jump
ofWaftert. Then, for any 0 < h < σ(t)−t, using the semigroup property
forRand (4.4),
W∗(t+h)−W∗(t)
= (R(h)−I)R(t)W(0)+/summationdisplay
j:σj≤t(R(h)−I)R(t−σj)∆j (4.5)
−/integraldisplayt
0(R(h)−I)R(t−s)AW(s)ds−/integraldisplayt+h
tR(t+h−s)AW(t)ds,
where, in the last integral, we use the fact that there are no jumps ofW
between tandt+h. Thus we have
W∗(t+h)−W∗(t)
= (R(h)−I)

R(t)W(0)+/summationdisplay
j:σj≤tR(t−σj)∆j−/integraldisplayt
0R(t−s)AW(s)ds


−/integraldisplayt+h
tR(t+h−s)AW(t)ds
= (R(h)−I)W(t)−/integraldisplayt+h
tR(t+h−s)AW(t)ds. (4.6)
But now, for x∈ X+,
/integraldisplayt+h
tR(t+h−s)Axds= (R(h)−I)x,
from (3.8), so that W∗(t+h) =W∗(t) for all t+h < σ(t), implying that
W∗(s) =W(s) for all s < σ(t). On the other hand, from (4.4), we have
W∗(σ(t))−W∗(σ(t)−) = ∆W(σ(t)), so that W∗(s) =W(s) for alls≤σ(t).
Thus we can prove equality over the interval [0 ,σ1], and then successively
over the intervals [ σj,σj+1], until [0 ,T] is covered.
16Now suppose that Warises as a realization of XN. ThenXNhas transi-
tion rates such that
MN(t) :=/summationdisplay
j:σj≤t∆XN(σj)−/integraldisplayt
0AXN(s)ds−/integraldisplayt
0NF(xN(s))ds(4.7)
is a zero mean local martingale. In view of Lemma 4.1, we can use (4.4) t o
write
XN(t) =R(t)XN(0)+/tildewiderMN(t)+N/integraldisplayt
0R(t−s)F(xN(s))ds,(4.8)
where
/tildewiderMN(t) :=/summationdisplay
j:σj≤tR(t−σj)∆XN(σj)
−/integraldisplayt
0R(t−s)AXN(s)ds−/integraldisplayt
0R(t−s)NF(xN(s))ds.(4.9)
Thus, comparing (4.8) and (4.2), we expect xNandxto be close, for
0≤t≤T < tmax, provided that we can show that supt≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis small,
where/tildewidemN(t) :=N−1/tildewiderMN(t). Indeed, if xN(0) andx(0) are close, then
/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ
≤ /⌊ard⌊lR(t)(xN(0)−x(0))/⌊ard⌊lµ
+/integraldisplayt
0/⌊ard⌊lR(t−s)[F(xN(s))−F(x(s))]/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ
≤ewt/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ
+/integraldisplayt
0ew(t−s)K(µ,F;2ΞT)/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ,(4.10)
by (3.9), with the stage apparently set for Gronwall’s inequality, ass uming
that/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµand sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµare small enough that then
/⌊ard⌊lxN(t)/⌊ard⌊lµ≤2ΞTfor 0≤t≤T, where Ξ T:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ.
Bounding sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis, however, not so easy. Since /tildewiderMNis not
itselfamartingale, wecannotdirectlyapplymartingaleinequalitiestoc ontrol
its fluctuations. However, since
/tildewiderMN(t) =/integraldisplayt
0R(t−s)dMN(s), (4.11)
17we can hope to use control over the local martingale MNinstead. For this
and the subsequent argument, we introduce some further assum ptions.
Assumption 4.2
1. There exists r=rµ≤r(2)
maxsuch that supj≥0{µ(j)/νr(j)}<∞.
2. There exists ζ∈ Rwithζ(j)≥1for alljsuch that (2.25)is satisfied
for some b=b(ζ)≥1andr=r(ζ)such that 1≤r(ζ)≤r(2)
max, and that
Z:=/summationdisplay
k≥0µ(k)(|Akk|+1)/radicalbig
ζ(k)<∞. (4.12)
The requirement that ζsatisfies (4.12) as well as satisfying (2.25) for some
r≤r(2)
maximplies in practice that it must be possible to take r(1)
maxandr(2)
max
to be quite large in Assumption 2.1; see the examples in Section 5.
Note that part 1 of Assumption 4.2 implies that lim j→∞{µ(j)/νr(j)}= 0
for some r= ˜rµ≤rµ+1. We define
ρ(ζ,µ) := max {r(ζ),p(r(ζ)),˜rµ}, (4.13)
wherep(·) is as in Assumptions 2.1. We can now prove the following lemma,
which enables us to control the paths of /tildewiderMNby using fluctuation bounds for
the martingale MN.
Lemma 4.3 Under Assumption 4.2,
/tildewiderMN(t) =MN(t)+/integraldisplayt
0R(t−s)AMN(s)ds.
Proof. From (3.8), we have
R(t−s) =I+/integraldisplayt−s
0R(v)Adv.
Substituting this into (4.11), we obtain
/tildewiderMN(t) =/integraldisplayt
0R(t−s)dMN(s)
18=MN(t)+/integraldisplayt
0/braceleftbigg/integraldisplayt
0R(v)A1[0,t−s](v)dv/bracerightbigg
dMN(s)
=MN(t)+/integraldisplayt
0/braceleftbigg/integraldisplayt
0R(v)A1[0,t−s](v)dv/bracerightbigg
dXN(s)
−/integraldisplayt
0/braceleftbigg/integraldisplayt
0R(v)A1[0,t−s](v)dv/bracerightbigg
F0(xN(s))ds.
It remains to change the order of integration in the double integrals , for
which we use Fubini’s theorem.
In the first, the outer integral is almost surely a finite sum, and at e ach
jump time tXN
lwe havedXN(tXN
l)∈ J. Hence it is enough that, for each i,
mandt,/summationtext
j≥0Rij(t)Ajmis absolutely summable, which follows from Theo-
rem 3.1. Thus we have
/integraldisplayt
0/braceleftbigg/integraldisplayt
0R(v)A1[0,t−s](v)dv/bracerightbigg
dXN(s) =/integraldisplayt
0R(v)A{XN(t−v)−XN(0)}dv.
(4.14)
For the second, the k-th component of R(v)AF0(xN(s)) is just
/summationdisplay
j≥0Rkj(v)/summationdisplay
l≥0Ajl/summationdisplay
J∈JJlαJ(xN(s)). (4.15)
Now, from (3.7), we have 0 ≤Rkj(v)≤µ(j)ewv/µ(k), and
/summationdisplay
j≥0µ(j)|Ajl| ≤µ(l)(2|All|+w), (4.16)
becauseATµ≤wµ. Hence, puttingabsolutevaluesinthesummandsin(4.15)
yields at most
ewv
µ(k)/summationdisplay
J∈JαJ(xN(s))/summationdisplay
l≥0|Jl|µ(l)(2|All|+w).
Now, in view of (4.12) and since ζ(j)≥1 for allj, there is a constant K <∞
such that µ(l)(2|All|+w)≤Kζ(l). Furthermore, ζsatisfies (2.25), so that,
by Corollary 2.5,/summationtext
J∈JαJ(xN(s))/summationtext
l≥0|Jl|ζ(l) is a.s. uniformly bounded in
0≤s≤T. Hence we can apply Fubini’s theorem, obtaining
/integraldisplayt
0/braceleftbigg/integraldisplayt
0R(v)A1[0,t−s](v)dv/bracerightbigg
F0(xN(s))ds=/integraldisplayt
0R(v)A/braceleftbigg/integraldisplayt−v
0F0(xN(s))ds/bracerightbigg
dv,
19and combining this with (4.14) proves the lemma.
We now introduce the exponential martingales that we use to bound the
fluctuations of MN. Forθ∈RZ+bounded and x∈ Rµ,
ZN,θ(t) :=eθTxN(t)exp/braceleftBig
−/integraltextt
0gNθ(xN(s−))ds/bracerightBig
, t≥0,
is a non-negative finite variation local martingale, where
gNθ(ξ) :=/summationdisplay
J∈JNαJ(ξ)/parenleftBig
eN−1θTJ−1/parenrightBig
.
Fort≥0, we have
logZN,θ(t) =θTxN(t)−/integraldisplayt
0gNθ(xN(s−))ds
=θTmN(t)−/integraldisplayt
0ϕN,θ(xN(s−),s)ds, (4.17)
where
ϕN,θ(ξ) :=/summationdisplay
J∈JNαJ(ξ)/parenleftBig
eN−1θTJ−1−N−1θTJ/parenrightBig
,(4.18)
andmN(t) :=N−1MN(t). Note also that we can write
ϕN,θ(ξ) =N/integraldisplay1
0(1−r)D2vN(ξ,rθ)[θ,θ]dr, (4.19)
where
vN(ξ,θ′) :=/summationdisplay
J∈JαJ(ξ)eN−1(θ′)TJ,
andD2vNdenotes thematrixofsecond derivatives withrespect totheseco nd
argument:
D2vN(ξ,θ′)[ζ1,ζ2] :=N−2/summationdisplay
J∈JαJ(ξ)eN−1(θ′)TJζT
1JJTζ2(4.20)
for anyζ1,ζ2∈ Rµ.
Now choose any B:= (Bk, k≥0)∈ R, and define ˜ τ(N)
k(B) by
˜τ(N)
k(B) := inf/braceleftBigg
t≥0:/summationdisplay
J:Jk/negationslash=0αJ(xN(t−))> Bk/bracerightBigg
.
Our exponential bound is as follows.
20Lemma 4.4 For anyk≥0,
P
sup
0≤t≤T∧˜τ(N)
k(B)|mk
N(t)| ≥δ
≤2exp(−δ2N/2BkK∗T).
for all0< δ≤BkK∗T, whereK∗:=J2
∗eJ∗, andJ∗is as in(1.2).
Proof. Takeθ=e(k)β, forβto be chosen later. We shall argue by stopping
the local martingale ZN,θat timeσ(N)(k,δ), where
σ(N)(k,δ) :=T∧˜τ(N)
k(B)∧inf{t:mk
N(t)≥δ}.
Note that eN−1θTJ≤eJ∗, so long as |β| ≤N, so that
D2vN(ξ,rθ)[θ,θ]≤N−2/parenleftBigg/summationdisplay
J:Jk/negationslash=0αJ(ξ)/parenrightBigg
β2K∗.
Thus, from (4.19), we have
ϕN,θ(xN(u−))≤1
2N−1Bkβ2K∗, u≤˜τ(N)
k(B),
and hence, on the event that σ(N)(k,δ) = inf{t:mk
N(t)≥δ} ≤(T∧˜τ(N)
k(B)),
we have
ZN,θ(σ(k,δ))≥exp{βδ−1
2N−1Bkβ2K∗T}.
But since ZN,θ(0) = 1, it now follows from the optional stopping theorem
and Fatou’s lemma that
1≥E{ZN,θ(σ(N)(k,δ))}
≥P/bracketleftBig
sup
0≤t≤T∧˜τ(N)
k(B)mk
N(t)≥δ/bracketrightBig
exp{βδ−1
2N−1Bkβ2K∗T}.
We can choose β=δN/B kK∗T, as long as δ/BkK∗T≤1, obtaining
P
sup
0≤t≤T∧˜τ(N)
k(B)mk
N(t)≥δ
≤exp(−δ2N/2BkK∗T).
Repeating with
˜σ(N)(k,δ) :=T∧˜τ(N)
k(B)∧inf{t:−mk
N(t)≥δ},
21and choosing β=δN/B kK∗T, gives the lemma.
Theprecedinglemmagivesaboundforeachindividualcomponentof MN.
We need first to translate this into a statement for all components simulta-
neously. For ζas in Assumption 4.2, we start by writing
Z(1)
∗:= max
k≥1k−1#{m:ζ(m)≤k};Z(2)
∗:= sup
k≥0µ(k)(|Akk|+1)/radicalbig
ζ(k).(4.21)
Z(2)
∗is clearly finite, because of Assumption 4.2, and the same is true for Z(1)
∗
also, since Zof Assumption 4.2 is at least # {m:ζ(m)≤k}/√
k, for each k.
Then, using the definition (2.24) of τ(N)(a,ζ), note that, for every k,
/summationdisplay
J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
|Jk|ζ(k)≤ah(k)
ζ(k),(4.22)
for anyt < τ(N)(a,ζ) and any h∈ R, and that, for any K ⊆Z+,
/summationdisplay
k∈K/summationdisplay
J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
k∈K/summationdisplay
J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
|Jk|ζ(k)
≤a
mink∈K(ζ(k)/h(k)). (4.23)
From (4.22) with h(k) = 1 for all k, if we choose B:= (a/ζ(k), k≥0), then
τ(N)(a,ζ)≤˜τ(N)
k(B) for allk. For this choice of B, we can take
δ2
k:=δ2
k(a) :=4aK∗TlogN
Nζ(k)=4BkK∗TlogN
N(4.24)
in Lemma 4.4 for k∈κN(a), where
κN(a) :=/braceleftbig
k:ζ(k)≤1
4aK∗TN/logN/bracerightbig
={k:Bk≥4logN/K∗TN},
(4.25)
since then δk(a)≤BkK∗T. Note that then, from (4.12),
/summationdisplay
k∈κN(a)µ(k)δk(a)≤2Z/radicalbig
aK∗TN−1logN, (4.26)
withZas defined in Assumption 4.2, and that
|κN(a)| ≤1
4aZ(1)
∗K∗TN/logN. (4.27)
22Lemma 4.5 If Assumptions 4.2 are satisfied, taking δk(a)andκN(a)as
defined in (4.24)and(4.25), and for any η∈ R, we have
1.P
/uniondisplay
k∈κN(a)/braceleftBig
sup
0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≥δk(a)/bracerightBig
≤aZ(1)
∗K∗T
2NlogN;
2.P
/summationdisplay
k/∈κN(a)Xk
N(t) = 0for all0≤t≤T∧τ(N)(a,ζ)
≥1−4logN
K∗N;
3. sup
0≤t≤T∧τ(N)(a,ζ)

/summationdisplay
k/∈κN(a)η(k)|Fk(xN(t))|

≤aJ∗
mink/∈κN(a)(ζ(k)/η(k)).
Proof. For part 1, use Lemma 4.4 together with (4.24) and (4.27) to give
the bound. For part 2, the total rate of jumps into coordinates w ith indices
k /∈κN(a) is
/summationdisplay
k/∈κN(a)/summationdisplay
J:Jk/negationslash=0αJ(xN(t))≤a
mink/∈κN(a)ζ(k),
ift≤τ(N)(a,ζ),using(4.23)with K= (κN(a))c,which, combinedwith(4.25),
proves the claim. For the final part, if t≤τ(N)(a,ζ),
/summationdisplay
k/∈κN(a)η(k)|Fk(xN(t))| ≤/summationdisplay
k/∈κN(a)η(k)/summationdisplay
J:Jk/negationslash=0αJ(xN(t))J∗,
and the inequality follows once more from (4.23).
LetB(1)
N(a) andB(2)
N(a) denote the events
B(1)
N(a) :=

/summationdisplay
k/∈κN(a)Xk
N(t) = 0 for all 0 ≤t≤T∧τ(N)(a,ζ)

;
B(2)
N(a) :=
/intersectiondisplay
k∈κN(a)/braceleftBig
sup
0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≤δk(a)/bracerightBig
,(4.28)
and setBN(a) :=B(1)
N(a)∩B(2)
N(a). Then, by Lemma 4.5, we deduce that
P[BN(a)c]≤aZ(1)
∗K∗T
2NlogN+4logN
K∗N, (4.29)
23of order O(N−1logN) for each fixed a. Thus we have all the components
ofMNsimultaneously controlled, except on a set of small probability. We
now translate this into the desired assertion about the fluctuation s of/tildewidemN.
Lemma 4.6 If Assumptions 4.2 are satisfied, then, on the event BN(a),
sup
0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ≤√aK4.6/radicalbigg
logN
N,
where the constant K4.6depends on Tand the parameters of the process.
Proof. From Lemma 4.3, it follows that
sup
0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ (4.30)
≤sup
0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊lmN(t)/⌊ard⌊lµ+ sup
0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds.
For the first term, on BN(a) and for 0 ≤t≤T∧τ(N)(a,ζ), we have
/⌊ard⌊lmN(t)/⌊ard⌊lµ≤/summationdisplay
k∈κN(a)µ(k)δk(a)+/integraldisplayt
0/summationdisplay
k/∈κN(a)µ(k)|Fk(xN(u))|du.
The first sum is bounded using (4.26) by 2 Z√aK∗T N−1/2√logN, the sec-
ond, from Lemma 4.5 and (4.25), by
TaJ∗
mink/∈κN(a)(ζ(k)/µ(k))≤Z(2)
∗2J∗/radicalbigg
Ta
K∗/radicalbigg
logN
N.
For the second term in (4.30), from (3.7) and (4.16), we note that
/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµ≤/summationdisplay
k≥0µ(k)/summationdisplay
l≥0Rkl(t−s)/summationdisplay
r≥0|Alr||mr
N(s)|
≤ew(t−s)/summationdisplay
l≥0µ(l)/summationdisplay
r≥0|Alr||mr
N(s)|
≤ew(t−s)/summationdisplay
r≥0µ(r){2|Arr|+w}|mr
N(s)|.
24OnBN(a) and for 0 ≤s≤T∧τ(N)(a,ζ), from (4.12), the sum for r∈κN(a)
is bounded using
/summationdisplay
r∈κN(a)µ(r){2|Arr|+w}|mr
N(s)|
≤/summationdisplay
r∈κN(a)µ(r){2|Arr|+w}δr(a)
≤/summationdisplay
r∈κN(a)µ(r){2|Arr|+w}/radicalBigg
4aK∗TlogN
Nζ(r)
≤(2∨w)Z/radicalbig
4aK∗T/radicalbigg
logN
N.
The remaining sum is then bounded by Lemma 4.5, on the set BN(a) and
for 0≤s≤T∧τ(N)(a,ζ), giving at most
/summationdisplay
r/∈κN(a)µ(r){2|Arr|+w}|mr
N(s)|
≤/summationdisplay
r/∈κN(a)µ(r){2|Arr|+w}/integraldisplays
0|Fr(xN(t))|dt
≤(2∨w)saJ∗
mink/∈κN(a)(ζ(k)/µ(k){|Akk|+1})
≤(2∨w)Z(2)
∗2J∗/radicalbigg
Ta
K∗/radicalbigg
logN
N.
Integrating, it follows that
sup
0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds
≤(2T∨1)ewT/braceleftBigg/radicalbig
4aK∗TZ+Z(2)
∗J2J∗/radicalbigg
Ta
K∗/bracerightBigg/radicalbigg
logN
N,
and the lemma follows.
This has now established the control on sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµthat we need,
in order to translate (4.10) into a proof of the main theorem.
25Theorem 4.7 Suppose that (1.2),(1.3),(3.1),(3.2)and(4.1)are all satis-
fied, and that Assumptions 2.1 and 4.2 hold. Recalling the defi nition(4.13)
ofρ(ζ,µ), forζas given in Assumption 4.2, suppose that S(N)
ρ(ζ,µ)(0)≤NC∗
for some C∗<∞.
Letxdenote the solution to (4.2)with initial condition x(0)satisfying
Sρ(ζ,µ)(x(0))<∞. Thentmax=∞.
Fix any T, and define ΞT:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ. If/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤
1
2ΞTe−(w+k∗)T, wherek∗:=ewTK(µ,F;2ΞT), then there exist constants c1,c2
depending on C∗,Tand the parameters of the process, such that for all N
large enough
P/parenleftBigg
sup
0≤t≤T/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ>/parenleftBigg
ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+c1/radicalbigg
logN
N/parenrightBigg
ek∗T/parenrightBigg
≤c2logN
N. (4.31)
Proof. AsS(N)
ρ(ζ,µ)(0)≤NC∗, it follows also that S(N)
r(0)≤NC∗for all
0≤r≤ρ(ζ,µ). Fix any T < tmax, takeC:= 2(C∗+k04T)ek01T, and observe
that, for r≤ρ(ζ,µ)∧r(2)
max, and such that p(r)≤ρ(ζ,µ), we can take
C′′
rT≤/tildewideCrT:={2(C∗∨1)+kr4T}e(kr1+Ckr2)T, (4.32)
in Theorem 2.4, since we can take C∗to bound CrandC′
r. In particular,
r=r(ζ) as defined in Assumption 4.2 satisfies both the conditions on r
for (4.32) to hold. Then, taking a:={k2+k1/tildewideCr(ζ)T}b(ζ)in Corollary 2.5, it
follows that for some constant c3>0, on the event BN(a),
P[τ(N)(a,ζ)≤T]≤c3N−1.
Then, from (4.29), for some constant c4,P[BN(a)c]≤c4N−1logN. Here,
the constants c3,c4depend on C∗,Tand the parameters of the process.
We now use Lemma 4.6 to bound the martingale term in (4.10). It fol-
lows that, on the event BN(a)∩ {τ(N)(a,ζ)> T}and on the event that
/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµ≤ΞTfor all 0≤s≤t,
/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
logN
N/parenrightBigg
+k∗/integraldisplayt
0/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds,
26wherek∗:=ewTK(µ,F;2ΞT). Then from Gronwall’s inequality, on the
eventBN(a)∩{τ(N)(a,ζ)> T},
/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
logN
N/parenrightBigg
ek∗t,
(4.33)
for all 0≤t≤T, provided that
/parenleftBigg
ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
logN
N/parenrightBigg
≤ΞTe−k∗T.
This is true for all Nsufficiently large, if /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤1
2ΞTe−(w+k∗)T,
which we have assumed. We have thus proved (4.31), since, as show n above,
P(BN(a)c∪{τ(N)(a,ζ)> T}c) =O(N−1logN).
We now use this to show that in fact tmax=∞. Forx(0) as above, we
can take xj
N(0) :=N−1⌊Nxj(0)⌋ ≤xj(0), so that S(N)
ρ(ζ,µ)(0)≤NC∗forC∗:=
Sρ(ζ,µ)(x(0))<∞. Then, by (4.13), lim j→∞{µ(j)/νρ(ζ,µ)(j)}= 0, so it fol-
lowseasilyusing boundedconvergence that /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ→0asN→ ∞.
Hence, for any T < t max, it follows from (4.31) that /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ→D0
asN→ ∞, fort≤T, with uniform bounds over the interval, where ‘ →D’
denotes convergence in distribution. Also, by Assumption 4.2, ther e is a con-
stantc5such that /⌊ard⌊lxN(t)/⌊ard⌊lµ≤c5N−1S(N)
rµ(t) for each t, whererµ≤r(2)
maxand
rµ≤ρ(ζ,µ). Hence, using Lemma 2.3 and Theorem 2.4, sup0≤t≤2T/⌊ard⌊lxN(t)/⌊ard⌊lµ
remains bounded in probability as N→ ∞. Hence it is impossible that
/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞asT→tmax<∞,implyingthatinfact tmax=∞forsuchx(0).
Remark . The dependence on the initial conditions is considerably compli-
cated by the way the constant Cappears in the exponent, for instance in the
expression for /tildewideCrTin the proof of Theorem 4.7. However, if kr2in Assump-
tions 2.1 can be chosen to be zero, as for instance in the examples be low, the
dependence simplifies correspondingly.
Therearebiologicallyplausiblemodelsinwhichtherestrictionto Jl≥ −1
is irksome. In populations in which members of a given type lcan fight one
another, a natural possibility is to have a transition J=−2e(l)at a rate
proportional to Xl(Xl−1), which translates to αJ=α(N)
J=γxl(xl−N−1),
a function depending on N. Replacing this with αJ=γ(xl)2removes the
27N-dependence, but yields a process that can jump to negative value s ofXl.
For this reason, it is useful to be able to allow the transition rates αJto
depend on N.
Since the arguments inthis paper are not limiting arguments for N→ ∞,
it does not require many changes to derive the corresponding resu lts. Quan-
tities such as A,F,Ur(x) andVr(x) now depend on N; however, Theorem 4.7
continues toholdwithconstants c1andc2thatdo notdepend on N, provided
thatµ,w,ν, theklmfrom Assumption 2.1 and ζfrom Assumption 4.2 can
be chosen to be independent of N, and that the quantities Z(l)
∗from (4.21)
can be bounded uniformly in N. On the other hand, the solution x=x(N)
of (4.2) that acts as approximation to xNin Theorem 4.7 now itself depends
onN, through R=R(N)andF=F(N). IfA(and hence R) can be taken
to be independent of N, and lim N→∞/⌊ard⌊lF(N)−F/⌊ard⌊lµ= 0 for some fixed µ–
Lipschitz function F, a Gronwall argument can be used to derive a bound
for the difference between x(N)and the (fixed) solution xto equation (4.2)
withN-independent RandF. IfAhas to depend on N, the situation is
more delicate.
5 Examples
We begin with some general remarks, to show that the assumptions are sat-
isfied in many practical contexts. We then discuss two particular ex amples,
those of Kretzschmar (1993) and of Arrigoni (2003), that fitte d poorly or
not at all into the general setting of Barbour & Luczak (2008), th ough the
other systems referred to in the introduction could also be treate d similarly.
In both of our chosen examples, the index jrepresents a number of individ-
uals — parasites in a host in the first, animals in a patch in the second —
and we shall for now use the former terminology for the preliminary, general
discussion.
Transitions that can typically be envisaged are: births of a few para sites,
which may occur either in the same host, or in another, if infection is b eing
represented; births and immigration of hosts, with or without para sites; mi-
gration of parasites between hosts; deaths of parasites; death s of hosts; and
treatment of hosts, leading to the deaths of many of the host’s pa rasites. For
births of parasites, there is a transition X→X+J, whereJtakes the form
Jl= 1;Jm=−1;Jj= 0, j/ne}ationslash=l,m, (5.1)
28indicating that one m-host has become an l-host. For births of parasites
within a host, a transition rate of the form bl−mmXmcould be envisaged,
withl > m, the interpretation being that there are Xmhosts with parasite
burdenm, each of which gives birth to soffspring at rate bs, for some small
values of s. For infection of an m-host, a possible transition rate would be
of the form
Xm/summationdisplay
j≥0N−1Xjλpj,l−m,
since an m-host comes into contact with j-hosts at a rate proportional to
their density in the host population, and pjrrepresents the probability of a
j-host transferring rparasites to the infected host during the contact. The
probability distributions pj·can be expected to be stochastically increasing
inj. Deaths of parasites also give rise to transitions of the form (5.1),
but now with l < m, the simplest form of rate being just dmXmforl=
m−1, though d=dmcould also be chosen to increase with parasite burden.
Treatment of a host would lead to values of lmuch smaller than m, and
a rate of the form κXmfor the transition with l= 0 would represent fully
successful treatment of randomly chosen individuals. Births and d eaths of
hosts and immigration all lead to transitions of the form
Jl=±1;Jj= 0, j/ne}ationslash=l. (5.2)
Fordeaths, Jl=−1, anda typical ratewould be d′Xl. Forbirths, Jl= 1, and
a possible rate would be/summationtext
j≥0Xjb′
jl(withl= 0 only, if new-born individuals
are free of parasites). For immigration, constant rates λlcould be supposed.
Finally, for migration of individual parasites between hosts, transit ions are
of the form
Jl=Jm=−1;Jl+1= 1;Jm−1= 1;Jj= 0, j/ne}ationslash=l,m,l+1,m−1,
(5.3)
a possible rate being γmXmN−1Xl.
For all the above transitions, we can take J∗= 2 in (1.2), and (1.3) is
satisfied in biologically sensible models. (3.1) and (3.2) depend on the wa y in
which the matrix Acan be defined, which is more model specific; in practice,
(3.1) is very simple to check. The choice of µin (3.2) is influenced by the
need to have (4.1) satisfied. For Assumptions 2.1, a possible choice o fνis to
takeν(j) = (j+1) for each j≥0, withS1(X) then representing the num-
ber of hosts plus the number of parasites. Satisfying (2.5) is then e asy for
29transitions only involving the movement of a single parasite, but in gen eral
requires assumptions as to the existence of the r-th moments of the distri-
butions of the numbers of parasites introduced at birth, immigratio n and
infection events. For (2.6), in which transitions involving a net reduc tion
in the total number of parasites and hosts can be disregarded, th e parasite
birth events are those in which the rates typically have a factor mXmfor
transitions with Jm=−1, withmin principle unbounded. However, at such
events, an m-individual changes to an m+sindividual, with the number s
of offspring of the parasite being typically small, so that the value of JTνr
associated with this rate has magnitude mr−1; the product mXmmr−1, when
summed over m, then yields a contribution of magnitude Sr(X), which is al-
lowable in(2.6). Similar considerations showthat theterms N−1S0(X)Sr(X)
accommodate the migration rates suggested above. Finally, in orde r to have
Assumptions 4.2 satisfied, it is in practice necessary that Assumptio ns 2.1
are satisfied for large values of r, thereby imposing restrictions on the dis-
tributions of the numbers of parasites introduced at birth, immigra tion and
infection events, as above.
5.1 Kretzschmar’s model
Kretzschmar (1993) introduced a model of a parasitic infection, in which the
transitions from state Xare as follows:
J=e(i−1)−e(i)at rate Niµxi, i ≥1;
J=−e(i)at rate N(κ+iα)xi, i≥0;
J=e(0)at rate Nβ/summationtext
i≥0xiθi;
J=e(i+1)−e(i)at rate Nλxiϕ(x), i ≥0,
wherex:=N−1X,ϕ(x) :=/⌊ard⌊lx/⌊ard⌊l11{c+/⌊ard⌊lx/⌊ard⌊l1}−1withc >0, and/⌊ard⌊lx/⌊ard⌊l11:=/summationtext
j≥1j|x|j; here, 0≤θ≤1, andθidenotes its i-th power (our θcorresponds
to the constant ξin [7]). Both (1.2) and (1.3) are obviously satisfied. For
Assumptions (3.1), (3.2) and (4.1), we note that equation corresp onding
to (1.5) has
Aii=−{κ+i(α+µ)};AT
i,i−1=iµandAT
i0=βθi, i≥2;
A11=−{κ+α+µ};AT
10=µ+βθ;
A00=−κ+β, i≥1,
30with all other elements of the matrix equal to zero, and
Fi(x) =λ(xi−1−xi)ϕ(x), i≥1;F0(x) =−λx0ϕ(x).
Hence Assumption (3.1) isimmediate, andAssumption (3.2)holds for µ(j) =
(j+1)s, for any s≥0, withw= (β−κ)+. For the choice µ(j) =j+1,F
maps elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
K(µ,F;Ξ) =c−2λΞ(2c+Ξ).
For Assumptions 2.1, choose ν=µ; then (2.5) is a finite sum for each
r≥0. Turning to (2.6), it is immediate that U0(x)≤βS0(x). Then, for
r≥1,
/summationdisplay
i≥0λϕ(N−1X)Xi{(i+2)r−(i+1)r} ≤λS1(X)
S0(X)/summationdisplay
i≥0rXi(i+2)r−1
≤r2r−1λSr(X),
since, by Jensen’s inequality, S1(X)Sr−1(X)≤S0(X)Sr(X). Hence we can
takekr2=kr4= 0 and kr1=β+r2r−1λin (2.6), for any r≥1, so that
r(1)
max=∞. Finally, for (2.7),
V0(x)≤(κ+β)S0(x)+αS1(x),
so thatk03=κ+β+αandk05= 0, and
Vr(x)≤r2(κS2r(x)+αS2r+1(x)+µS2r−1(x)+22(r−1)λS2r−1(x))+βS0(x),
so that we can take p(r) = 2r+1,kr3=β+r2{κ+α+µ+22(r−1)λ}, and
kr5= 0 for any r≥1, and so r(2)
max=∞. In Assumptions 4.2, we can clearly
takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 17.
5.2 Arrigoni’s model
Inthemetapopulation model ofArrigoni (2003), thetransitions f romstate X
are as follows:
J=e(i−1)−e(i)at rateNixi(di+γ(1−ρ)), i ≥2;
J=e(0)−e(1)at rateNx1(d1+γ(1−ρ)+κ);
J=e(i+1)−e(i)at rateNibixi, i ≥1;
J=e(0)−e(i)at rateNxiκ, i ≥2;
J=e(k+1)−e(k)+e(i−1)−e(i)at rateNixixkργ, k ≥0, i≥1;
31as before, x:=N−1X. Here, the total number N=/summationtext
j≥0Xj=S0(X) of
patches remains constant throughout, and the number of animals in any one
patch changes by at most one at each transition; in the final (migra tion)
transition, however, the numbers in two patches change simultane ously. In
the above transitions, γ,ρ,κare non-negative, and ( di),(bi) are sequences of
non-negative numbers.
Once again, both (1.2) and (1.3) are obviously satisfied. The equatio n
corresponding to (1.4) can now be expressed by taking
Aii=−{κ+i(bi+di+γ)};AT
i,i−1=i(di+γ);AT
i,i+1=ibi, i≥1;
A00=−κ,
with all other elements of Aequal to zero, and
Fi(x) =ργ/⌊ard⌊lx/⌊ard⌊l11(xi−1−xi), i≥1;F0(x) =−ργx0/⌊ard⌊lx/⌊ard⌊l11+κ,
where we have used the fact that N−1/summationtext
j≥0Xj= 1. Hence Assumption (3.1)
is again immediate, and Assumption (3.2) holds for µ(j) = 1 with w= 0,
forµ(j) =j+ 1 with w= max i(bi−di−γ−κ)+(assuming ( bi) and (di)
to be such that this is finite), or indeed for µ(j) = (j+1)swith any s≥2,
with appropriate choice of w. With the choice µ(j) =j+1,Fagain maps
elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
K(µ,F;Ξ) = 3ργΞ.
To check Assumptions 2.1, take ν=µ; once again, (2.5) is a finite sum
for each r. Then, for (2.6), it is immediate that U0(x) = 0. For any r≥1,
using arguments from the previous example,
Ur(x)≤r2r−1/braceleftBigg/summationdisplay
i≥1ibixi(i+1)r−1+/summationdisplay
i≥1/summationdisplay
k≥0iργxixk(k+1)r−1/bracerightBigg
≤r2r−1{max
ibiSr(x)+ργS1(x)Sr−1(x)}
≤r2r−1{max
ibiSr(x)+ργS0(x)Sr(x)},
so that, since S0(x) = 1, we can take kr1=r2r−1(maxibi+ργ) andkr2=
kr4= 0 in (2.6), and r(1)
max=∞. Finally, for (2.7), V0(x) = 0 and, for r≥1,
Vr(x)
≤r2/braceleftBig
22(r−1)max
ibiS2r−1(x)+max
i(i−1di)S2r(x)+γ(1−ρ)S2r−1(x)
+ργ(22(r−1)S1(x)S2r−2(x)+S0(x)S2r−1(x))/bracerightBig
+κS2r(x),
32so that we can take p(r) = 2r, and (assuming i−1dito be finite)
kr3=κ+r2{22(r−1)(max
ibi+ργ)+max
i(i−1di)+γ},
andkr5= 0 for any r≥1, andr(2)
max=∞. In Assumptions 4.2, we can again
takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 16.
Acknowledgement
We wish to thank a referee for recommendations that have substa ntially
streamlined our arguments. ADB wishes to thank both the Institut e for
MathematicalSciencesoftheNationalUniversityofSingaporeand theMittag–
Leffler Institute for providing a welcoming environment while part of t his
work was accomplished. MJL thanks the University of Z¨ urich for th eir hos-
pitality on a number of visits.
References
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metapopulation model. Adv. Appl. Prob. 35691–720.
[2]Barbour, A. D. andKafetzaki, M. (1993). A host–parasite model
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