Limit theorems for continuous time branching flows 1

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1 (Version: 212/4/11) arxiv: v1 [math.pr 12 Apr 212 Limit theorems for continuous time branching flows 1 Hui He and Rugang Ma 2 Beijing ormal University Abstract: We construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching erprocesses over the positive half line studied in Li (212). Mathematics Subject Classification (21): Primary 6J68, 6J8; secondary 6G57 Key words and phrases: Stochastic flow, branching process, continuous-time, discretestate, erprocess, nonlocal branching. 1 Introduction A genealogical tree is naturally associated with a Galton-Watson branching process. A continuous-state branching process (CB-process) can be obtained as the small particle limit of rescaled Galton-Watson processes; see, e.g., Lamperti (1967). The genealogical structures of binary branching CB-processes were investigated by introducing continuum random trees in the pioneer work of Aldous (1991, 1993). Continuum random trees corresponding to general branching mechanisms were constructed in Le Gall and Le Jan (1998a, 1998b) and were studied further in Duquesne and Le Gall (22). By pruning a Galton-Watson tree, Aldous and Pitman (1998) and Abraham at al. (211) constructed a tree-valued Markov process. Tree-valued processes associated with general CB-processes were studied in Abraham and Delmas (21) by pruning arguments. Motivated by the study of genealogy trees for critical branching processes conditioned on non-extinction, Bakhtin (211) studied a flow of binary branching continuous-state branching processes with immigration (CBI-processes) driven by a time-space Gaussian white noise. He also pointed out the connection of the model with a erprocess conditioned on non-extinction. In Li (212), a class of path-valued branching processes were constructed and studied using the techniques of stochastic equations and erprocesses. The work is closely related to those of Bertoin and Le Gall (26) and Dawson and Li (212). In a special case, the path-valued branching processes in Li (212) can be coded by 1 Supported by SFC and 985 Project. 2 Corresponding author. marugang@mail.bnu.edu.cn. 1

2 the tree-valued processes of Abraham and Delmas (21). In He and Ma (212), two flows of discrete time and state Galton-Watson branching processes were introduced. There it was showed that suitable rescaled sequences of those flows converge to special forms of the flows of Dawson and Li (212) and Li (212), respectively. The limit theorems in He and Ma (212) were given in the setting of the corresponding erprocesses. From those limit theorems the convergence of the finite-dimensional distributions of corresponding the path-valued processes was derived. The results give a better understanding of the connection between discrete and continuum tree-valued branching processes. In this paper, we introduce a kind of flows of continuous time and discrete state branching processes. We shall prove the scaling limit theorems for those flows of the type of He and Ma (212). In Section 2 a short review is given to the path-valued branching processes and nonlocal branching erprocesses studied in Li (212). In Section 3 we construct a continuous time and discrete state branching processes as the strong solution of a stochastic integral equation. In Section 4 the construction is extended to branching flows by considering stochastic equation systems. In Section 5 we prove that suitable rescaled sequences of those flows converge to the nonlocal branching erprocess. From the limit theorem we also derive the convergence of the finite-dimensional distributions of corresponding the path-valued processes. Let={,1,2, }and + ={1,2, }. Let M[,1 betheset offiniteborel measures on [,1 endowed with the topology of weak convergence. We identify M[,1 with the set F[,1 of positive right continuous increasing functions on [,1. Let B[,1 be the Banach space of bounded Borel functions on [,1 endowed with the remum norm. Let C[,1 denote its subspace of continuous functions. We use B[,1 + and C[,1 + to denote the subclasses of positive elements and C[,1 ++ to denote the subset of C[,1 + of functions bounded away from zero. For µ M[,1 and f B[,1 write µ,f = fdµ if the integral exists. Let D([, ),M[,1) denote the space of càdlàg paths from [, ) to M[,1 endowed with the Skorokhod topology. Throughout the paper, we only consider continuous time processes, so we shall often omit this phrase in the sequel. 2 Preliminaries In this section, we recall some results established in Li (212) on flows of CB-processes and nonlocal branching erprocesses over the positive half line. By a branching mechanism φ we mean a function φ on [, ) with the representation φ(z) = bz σ2 z 2 + (e zu 1+zu)m(du), (2.1) where σ and b are constants and (u u 2 )m(du) is a finite measure on (, ). Consider a family of branching mechanisms {φ q : q [,1} that is admissible in the sense that each φ q is given by (2.1) with parameters (b,m) = (b q,m q ) depending on q [,1 and for each z the function q φ q (z) is decreasing and continuously differentiable with the 2

3 derivative ψ θ (z) = ( / θ)φ θ (z) of the form ψ θ (z) = h θ z + (1 e zu )n θ (du), (2.2) where h θ and n θ (du) is a σ-finite kernel from [,1 to (, ) satisfying [ h θ + θ 1 un θ (du) <. Let m(dz,dθ) be the measure on (, ) [,1 defined by m([c,d [,q) = m q [c,d, q [,1,d > c >. LetW(ds,du)beawhitenoiseon(, ) 2 basedonthelebesguemeasure, Ñ(ds,dz,dθ,du) be a compensated Poisson random measure on (, ) 2 [,1 (, ) with intensity dsm(dz, dθ)du. By the results in Li (212), the following stochastic equation t Y (q) t Y t (q) = Y (q) b q Y (q)ds+σ W(ds,du) t Y (q) + zñ(ds,dz,dθ,du) (2.3) [,q has a unique solution flow {Y t (q) : t,q [,1}. For each q [,1, the onedimensional process {Y t (q) : t } is a CB-process with branching mechanism φ q. The flow is increasing in q [,1. It was verified in Li (212) that {(Y t (q)) t : q [,1} can be identified as a path-valued branching process. Moreover, the flow induces a càdlàg M[,1-valued erprocess {Y t : t } which is the unique solution of the following martingale problem: For every G C 2 (R) and f C[,1, t G( Y t,f ) = G( Y,f )+ G ( Y s,f )ds Y s (dx) f(x θ)h θ dθ [,1 [,1 t b G ( Y s,f ) Y s,f ds+ 1 t 2 σ2 G ( Y s,f ) Y s,f 2 ds t [ + ds Y s (dx) G( Y s,f +zf(x)) [,1 G( Y s,f ) zf(x)g ( Y s,f ) m (dz) t [ + ds Y s (dx) dθ G( Y s,f +zf(x θ)) [,1 [,1 G( Y s,f ) n θ (dz)+ local mart. (2.4) Let f Ψ(,f) be the operator on C + [,1 defined by Ψ(x,f) = f(x θ)h θ dθ + dθ (1 e zf(x θ) )n θ (dz). (2.5) [,1 [,1 3

4 Then the erprocess {Y t : t } has local branching mechanism φ and nonlocal branching mechanism Ψ. Its transition semigroup (Q t ) t is given by { } e ν,f Q t (µ,dν) = exp µ,v t f, f C + [,1, (2.6) M[,1 where t V t f is the unique locally bounded positive solution of V t f(x) = f(x) t [φ (V s f(x)) Ψ(x,V s f)ds, t,x [,1. (2.7) The reader may refer Li (212) for the derivations of the erprocess {Y t : t }. 3 Stochastic equations for discrete state branching processes In this section, we give a construction of the continuous time and discrete state branching process as the solution of a stochastic integral equation driven by Poisson random measure. Stochastic integral equations of this type were used in Li and Ma (28) to construct catalytic branching processes. We here give all the details for completeness. Let g = g(z) = i= p iz i be a probability generating function with g (1) <. Let (ds,dz,du) be a Poisson random measure on (, ) (, ) with intensity σdsπ(dz)du, where σ > is a constant and π(dz) := i= p iδ i (dz). Suppose that X is a non-negative integer-valued random variable satisfying E[X <. We assume X is independent of (ds, dz, du) and consider the stochastic integral equation X t = X + t X (z 1)(ds,dz,du). (3.1) By a solution of (3.1) we mean a non-negative càdlàg progressive process {X t : t } satisfyingtheequation a.s. foreach t. Wesaypathwise uniqueness ofsolutionholdsfor (3.1) if any two solutions of the equation with the same initial state are indistinguishable. Theorem 3.1 Suppose that {Xt} 1 and {Xt} 2 are two solutions of (3.1) satisfying E[ X 1+ X 2 <. Then we have E[ X 2 t X1 t E[ X2 X1 exp{σt(g (1)+1)}. (3.2) Consequently, the pathwise uniqueness of solution holds for (3.1). Proof. The pathwise uniqueness for (3.1) follows from Theorem 2.1 of Dawson and Li (212). We present a proof of the result here for completeness. Let ξ t = X 2 t X1 t for t. From (3.1) we have t ξ t = X 2 X1 + X 2 X 1 (z 1)1 {X 1 X 2 }(ds,dz,du) 4

5 t X 1 X 2 (z 1)1 {X 1 >X 2 }(ds,dz,du). Let τ m = inf{t : X 1 t m or X 2 t m}. Then we have t τm X 2 E[ ξ t τm E[ ξ +E (z +1)1 {X 1 X 1 X 2 }(ds,dz,du) t τm X 1 +E (z +1)1 {X 1 X 2 >X 2 }(ds,dz,du) t τm = E[ ξ +E ds ξ 1 {ξ }(z +1)σπ(dz) t τm +E ds ( ξ )1 {ξ <}(z +1)σπ(dz) E[ ξ + By Gronwall s inequality we get t E[ ξ s τm σ(g (1)+1)ds. E[ ξ t τm E[ ξ exp{σt(g (1)+1)}. Then (3.2) follows by Fatou s lemma. By Theorem 2.5 in Dawson and Li (212), there is a unique strong solution to (3.1). Here we give a simple direct proof of the existence of the solution. We first take an n + and consider the following stochastic equation X t = X + t X n (z 1)(ds,dz,du). (3.3) Proposition 3.2 Let {Xt n } be a solution of (3.3). Then we have [ E E[X exp{σg (1)t}, t. (3.4) s t X n s Proof. From (3.3) we have [ E s t X n s [ t X n n E[X +E z(ds,dz,du) [ t = E[X +E ds (X. n n)zσπ(dz) Thus t E[ s t Xs n is a locally bounded function. Moreover, [ E s t X n s E[X + t 5 [ ds E r s X n r zσπ(dz)

6 = E[X +σg (1) t [ E r s X n r ds. By Gronwall s lemma we get the result. By a modification of the proof of Theorem 3.1 we get the following Proposition. Proposition 3.3 Suppose that {X n,1 t } and {X n,2 t } are two solutions of (3.3). Then we have E[ X n,2 t X n,1 t E[ X n,2 X n,1 exp{σt(g (1)+1)}. (3.5) Consequently, the pathwise uniqueness of solution holds for (3.3). Proposition 3.4 For each n 1, there is a solution {X n t : t } of (3.3). Proof. Let {S k : k = 1,2, } be the set of jump times of the Poisson process t n t (ds,dz,du). We have clearly S k as k. For t < S 1, set Xt n = X. Suppose that Xt n has been defined for t < S k and let X n Xt n = XS n Sk n k + (z 1)(ds,dz,du), S k t < S k+1. {S k } From the construction of XS n k we see XS n k XS n k 1 1. And since XS n k 1 = implies XS n k =, XS n k. By induction that defines a non-negative process {Xt n : t } which is clearly a solution to (3.3). Proposition 3.5 Let {X n t : t } be the solution of (3.3) with n = 1,2,. Then the sequence {X n t : t } is tight in D([, ),). Proof. By Proposition 3.2, it is easy to see that [ t C t := E n 1 s t is locally bounded. Then for every fixed t, the sequence of random variables X n t is tight. Moreover, in view of (3.3), if {τ n } is a sequence of stopping times bounded above by T, we have [ E[ Xt+τ n n Xτ n t+τ n n = E τ n [ t E X n s X n n (z 1)(ds,dz,du) ds (Xs+τ n n n)(z +1)σπ(dz) 6

7 σ(g (1)+1) t E[X n s+τ n ds E[X exp{σg (1)(t+T)}σ(g (1)+1)t, where the last inequality follows by Proposition 3.2. Consequently, as t, E[ Xt+τ n n Xτ n n. n 1 Then {Xt n : t } is tight in D([, ),) by the criterion of Aldous (1978); see also Ethier and Kurtz (1986, pp ). Theorem 3.6 There is a solution {X t : t } of (3.1). Proof. For each n 1, let {Xt n : t } be the solution of (3.3). Define τ n = inf{t : Xt n n}. From Proposition 3.2 it follows that [ E[Xt τ n n E E[X exp{σg (1)t}, t. Then we have s t X n s E[X n t τ n 1 {τn t} E[X exp{σg (1)t}. By the right continuity of {X n t } we have Xn τ n n, so np[{τ n t} E[X exp{σg (1)t}, t. That implies τ n almost surely as n. On the other hand, {X n t } satisfies the equation (3.1) for t < τ n. By the pathwise uniqueness of the solution of (3.1) we get, for any i,j, X i t = X j t, t < τ i τ j. Let {X t } be the process such that X t = Xt n for all t < τ n and n 1. It is easily seen that {X t } is a solution of (3.1). Theorems 3.1 and 3.6 imlpy that (3.1) has unique strong solution and the solution {X t : t } is a strong Markov process; see, e.g., Ikeda and Watanabe (1989, pp and p.215). Let B() denote the set of bounded measurable functions on. By Itô s formula it is easy to see that {X t : t } has generator A defined by Af(x) = σx [f(x+i 1) f(x)p i, i= x, f B(). Then {X t : t } is a continuous time Galton-Watson branching process. 7

8 Let (1) (ds,dz,du) and (2) (ds,dz,du) be two mutually independent Poisson random measures on(, ) (, ) with the same intensity σdsπ(dz)du. Consider the following two stochastic equations and X (1) t = X (1) + X (2) t = X (2) + t t X (1) X (2) (z 1) (1) (ds,dz,du) (z 1) (2) (ds,dz,du). Clearly, X (1) t and X (2) t are mutually independent. Set X t = X (1) t. Since the random measure (ds,dz) := (1) (ds,dz,du) + (2) (ds,dz,du) {<u X (1) } {<u X (2) } t +X (2) has predictable compensator σx dsπ(dz), by representation theorems for semimartingales, on an extension of the original probability space, there is a Poisson random measure on (, ) (, ) with intensity σdsπ(dz)du such that X t = X + t X (z 1)(ds,dz,du); see, e.g., Ikeda and Watanabe (1989, p.93). Then the solution of (3.1) is a branching process (continuous time and discrete state). This gives another derivation of the branching property of {X t : t }. 4 The flow of discrete state branching processes In this section, we give a formulation of the discrete state branching flow as the solution flow of a set of stochastic integral equations. Let {g θ : θ } be a family of probability generating functions, that is, for each θ, g θ (z) = p i (θ)z i, z 1, i= where p i (θ) and k= p i(θ) = 1. Moreover, we assume θ g θ (1) is continuous and p i (θ 2 ) p i (θ 1 ) holds for any θ 2 θ 1 and i +. Define a family of probability measures {π θ : θ } on by π θ (dz) = p i (θ)δ i (dz) i= 8

9 Then we have π θ2 + π θ1 + for any θ 2 θ 1. Let π(dz,dθ) be the measure on + [, ) defined by π(a [,θ) = π θ (A), A +,θ. otice that the positive function θ b(θ) := π θ ({}) is decreasing. Letq X (q)beadeterministicpositiverightcontinuousincreasingfunctionon[, ) and take values in. Let (ds,dz,dθ,du) be a Poisson random measure on (, ) [, ) (, ) withintensity σds π(dz,dθ)du and (ds,dθ,du) apoisson randommeasure on (, ) 3 with intensity σdsdθdu. Suppose that (ds,dz,dθ,du) and (ds,dθ,du) are independent of each other. Consider stochastic integral equation X t (q) = X (q)+ ote that for each q, t + [,q t b(q) X (q) {<θ b(q)} X (q) (z 1)(ds,dz,dθ,du) (ds,dθ,du). (4.1) (ds,dθ,du) is a Poisson random measure with intensity σb(q)dsdu = σ π ( [,q)dsdu, where π (dz,dθ) is a measure on [, ) defined by π (A [,q) = π q ({})δ (A), A, θ. By representation theorems for semi-martingales, there is a Poisson random measure 1 (ds,dz,dθ,du) on (, ) [, ) (, ) with intensity σds π (dz,dθ)du such that for every E B(, ), t b(q) E (ds,dθ,du) = t [,q E 1 (ds,dz,dθ,du); see, e.g., Ikeda and Watanabe (1989, p.93). Define 2 (ds,dz,du) by 2 (ds,dz,du) = (ds,dz,dθ,du) + 1 (ds,dz,dθ,du). { θ q} { θ q} Then 2 is a Poisson random measure on (, ) (, ) with intensity σdsπ q (dz)du and the equation (4.1) can be rewrited as X t (q) = X (q)+ t X (q) (z 1) 2 (ds,dθ,du). By Theorem 3.6 we see that for each q, the equation (4.1) has a unique strong solution {X t (q) : t }. 9

10 Theorem 4.1 Suppose that q p. Let {X t (q)} be the solution of (4.1) and {X t (p)} be the solution of the equation with q replaced by p. Then we have P{X t (q) X t (p) for all t } = 1. Proof. Let ζ t = X t (p) X t (q) for t. From (4.1) we have t X (p) ζ t = ζ + (z 1)(ds,dz,dθ,du) + [,p X (q) t X (q) t (z 1)(ds,dz,dθ,du) + (p,q t b(p) X (p) b(q) b(q) X (p) X (q) (ds,dθ,du) (ds,dθ,du). (4.2) Let τ m = inf{t : X t (q) m or X t (p) m}. It is easy to construct a sequence of functions {f n } on R such that f n (z) 1 for z and f n(z) = f n (z) = for z. Moreover, f n (z) z + := z increasingly as n. By (4.2) and Itô s formula, t τm X (p) f n (ζ t τm ) = [f n (ζ +z 1) f n (ζ )1 {ζ >}(ds,dz,dθ,du) + [,p X (q) t τm X (q) + [f n (ζ z +1) f n (ζ )(ds,dz,dθ,du) + + σ + (p,q t τm b(q) X (q) X (p) t τm b(p) X (p) t τm [f n (ζ 1) f n (ζ )1 {ζ >} (ds,dθ,du) [f n (ζ 1) f n (ζ ) (ds,dθ,du) b(q) ζ 1 {ζ >}ds (z 1)π p (dz)+martingale. + Taking the expectation in both sides and letting n gives t E[ζ t τ + m σ(g p (1) 1+b(p)) E[ζ s τ + m ds. Then E[ζ t τ + m = for all t. Since τ m as m, that proves the desired comparison result. Proposition 4.2 There is a locally bounded positive function (t,u) C(t,u) on [, ) 2 so that, for any t and p q u <, { } { } E [X s (q) X s (p) C(t,u) X (q) X (p)+g q(1) g p(1). (4.3) s t Proof. Let ξ t = X t (q) X t (p). From (4.1) we get t X (q) ξ s ξ + (z 1)(ds,dz,dθ,du) s t + [,q X (p) 1

11 Then [ E + + s t t X (p) + (p,q t b(p) X (p) b(q) (z 1)(ds,dz,dθ,du) (ds,dθ,du). t ξ s ξ +σ[g p (1) 1+b(q) E[ξ s ds t +σ[g q (1) g p (1) E[X s (p)ds. Since t E[X t (p) is locally bounded, by Gronwall s inequality we get the desired estimate. From the discussion above, given a constant σ > and a family of probability generating functions {g θ : θ }, we obtain a continuous time and discrete state branching process flow {X t (q) : t,q } as the solution of equation (4.1). For any t define the random function X t F[,1 by X t (1) = X t (1) and X t (q) = inf{x t (u) : rational u (q,1}, q < 1. (4.4) By Proposition 4.2, for each q [,1 we have P{ X t (q) = X t (q) for all t } = 1. Then { X t (q) : t } is also càdlàg and solves (4.1) for every q [,1. 5 Scaling limits of the discrete branching flows In this section, we prove some limit theorems for the discrete state branching flows, which will lead to the continuous state branching flows of Li (212). We shall present the limit theorems in the settings of measure-valued processes and path-valued processes. Suppose that for each k 1, there is a positive constant σ k and a family of generating functions {g (k) θ : θ } satisfying the assumptions specified at the beginning of the last section. Then we can define π (k) θ (dz) and π (k) (dz,dθ) in the same way as there. Moreover, assume θ b k (θ) := g (k) θ () is differentiable. Let {X (k) t (q) : t } be the corresponding (k) solution of (4.1) and { X t (q) : t,q [,k} be defined in the same way as in (4.4). Define From (4.1) we have Y (k) t (q) = 1 k Y (k) t (q) = Y (k) (q)+ 1 k t X (k) t (kq), q [,1. (5.1) + [,kq 11 (k) ky (q) (z 1)(ds,dz,dθ,du)

12 1 k t bk (kq) (k) ky (q) (ds,dθ,du). (5.2) One can use a standard stopping time argument to show that for any q [,1, the function t E[Y (k) t (q) is locally bounded. Then by an argument similar to the proof of Proposition 3.2 we have Proposition 5.1 For any t and q [,1, we have [ E Y s (k) (q) Y (k) (q) exp s t The random function Y (k) t that Y (k) t ([,q) = Y (k) {Y (k) t {tσ k ( (g (k) kq ) (1) 1+b k (kq) )}. (5.3) F[,1 induces a random measure Y (k) t M[,1 so t (q) for q [,1. We are interested in the asymptotic behavior of : t } as k. For any f C 1 [,1 one can use Fubini s theorem to see Y (k) t,f = f(1)y (k) t (1) 1 f (q)y (k) t (q)dq. (5.4) Fix an integer n 1 and let q i = i/2 n for i =,1,,2 n. By (5.2) we have 2 n i=1 where and f (q i )Y (k) t (q i ) = = F (k) n F (k) n 2 n i=1 + 1 k 1 k f (q i )Y (k) (q i ) 2 n i=1 2 n i=1 2 n i=1 t + 1 k 1 k f(q i ) f (q i ) t (k) ky (q i) + [,kq i t bk (kq i ) (k) ky (q i) f (q i )Y (k) (q i ) + [,k t bk () (k) ky (1) 2n (s,θ,u) = i=1 2n (s,θ,u) = i=1 (k) ky (1) (z 1)(ds,dz,dθ,du) (ds,dθ,du) F n (k) (s,θ,u)(z 1)(ds,dz,dθ,du) F (k) n (s,θ,u) (ds,dθ,du), (5.5) f (q i )1 {θ kqi }1 {u ky (k) (q i)} f (q i )1 {θ bk (kq i )}1 {u ky (k) (q i)}. 12

13 By the right continuity of q Y (k) t (q) it is easy to see that, as n, and 2 n F (k) n (s,θ,u) F(k) (s,θ,u) := n 2 F(k) n (s,θ,u) F (k) (s,θ,u) := Then by (5.5) we have, almost surely, 1 f (q)y (k) t (q)dq = k 1 k f (q)y (k) (q)dq t [,k t bk () (k) ky (1) (k) ky (1) From (5.2), (5.4) and (5.6) it follows that, almost surely, Y (k) t,f = Y (k),f + 1 t k 1 k + 1 k (k) ky (1) + [,k t bk (k) (k) ky (1) t bk () b k (k) (k) ky (1) f (q)1 {θ kq} 1 {u ky (k) (q)}dq f (q)1 {θ bk (kq)}1 {u ky (k) (q)}dq. F (k) (s,θ,u)(z 1)(ds,dz,dθ,du) F (k) (s,θ,u) (ds,dθ,du). (5.6) [f(1) F (k) (s,θ,u)(z 1)(ds,dz,dθ,du) [f(1) F (k) (s,θ,u) (ds,dθ,du) F (k) (s,θ,u) (ds,dθ,du). (5.7) Proposition 5.2 Suppose that Y (k) (1) converges to some Y (1) as k and [ σ k (g (k) k ) (1) 1+b k () <. k 1 Then {Y (k) t : t }, k = 1,2, is a tight sequence in D([, ),M[,1). Proof. For any t and f C[,1, by Proposition 5.1 it is easy to see that [ t C t := E Y s (k),f k 1 s t is locally bounded. Then for every fixed t, the sequence Y (k) t,f is tight. Let τ k be a bounded stopping time for {Y (k) t : t } and assume the sequence {τ k : k = 1,2, } is bounded above by T. Let f C 1 [,1. By (5.7) we see [ Y (k) (k) E,f Y τ k,f τ k +t 13

14 σ [ t k k E + σ k k E + σ k k E ds [ t ds [ t ds + [,k bk (k) bk () b k (k) (k) ky s+τ (1) k dθ dθ (k) ky s+τ (1) k (k) ky s+τ (1) k (z 1) f(1) F (k) (s+τ k,θ,u) π (k) (dz,dθ)du f(1) F (k) (s+τ k,θ,u) du F (k) (s+τ k,θ,u) du. (5.8) For s,θ,u > let Y 1 (k) s,k (u) = inf{q : Y s (q) > u} and b 1 k (u) = inf{q : b k(q) > u}. It is easy to see that {q : u ky s (k) (q)} = [Y 1 s,k (u/k), ) and {q : θ b k(kq)} = [,b 1 k (θ)/k except for at most countably many u > and θ >, respectively. Then in the above we can replace f(1) F (k) (s,θ,u) by 1 ( f(1) f (q)1 {Y 1 (u/k) q}dq = f Y 1 s,k (u s,k θ/k k ) θ ) k and F (k) (s,θ,u) can be replaced by 1 f (q)1 {q b 1 [ = (θ)/k}1 k {Y 1 (u/k) q}dq s,k f(1 (b 1 k (θ)/k)) f(y 1 s,k (u/k)) 1 {Y 1 s,k (u/k) b 1 k (θ)/k}. Then from (5.8) we have [ Y (k) (k) E τ k +t,f Y τ k,f [ t 1 σ k E ds Y (k) s+τ k (dx) (z 1) f(x θ) π (k) (dz,dθ) + [,k [ t bk (k) 1 +σ k E ds dθ f(x) Y (k) s+τ k (dx) [ t bk () 1 +σ k E ds dθ f(b 1 (k) k (θ)/k) f(x) Y s+τ k (dx) b k (k) t [ f σ k E Y (k) s+τ k (1) ds (z 1)π (k) k (dz) + t [ + f σ k b k (k)e Y (k) s+τ k (1) ds t [ +2 f σ k [b k () b k (k) E Y (k) s+τ k (1) ds ( ) f σ k (g (k) t [ k ) (1) 1+2b k () E Y (k) s+τ k (1) ds { } 2 f Y (k) (1)tσ k A k exp σ k A k (t+t), (5.9) where A k = (g (k) k ) (1) 1+b k () and the last inequality follows by Proposition 5.1. For 14

15 f C[,1 the above inequality follows by an approximation argument. Then we have [ Y (k) (k) lim E,f Y τ k,f =. t k 1 τ k +t By a criterion of Aldous (1978), the sequence { Y (k) t,f : t } is tight in D([, ),R); see also Ethier and Kurtz(1986, pp ). Then the tightness criterion of Roelly (1986) implies {Y (k) t : t } is tight in D([, ),M[,1). For any z define [ φ (k) θ (z) = kσ k g (k) kθ (e z/k ) e z/k. (5.1) Let us consider the following condition: Condition (5.A) For each l the sequence {φ (k) θ (z)} is Lipschitz with respect to z uniformly on [,l 2 and there is an admissible family of branching mechanisms {φ (k) θ (z) : θ } with ( / θ)φ θ (z) = ψ θ (z) such that φ (k) θ (z) φ θ(z) uniformly on [,l 2 as k. Let {Y t : t } be the càdlàg erprocess with transition semigroup defined by (2.6) and (2.7). Theorem 5.3 Suppose that Condition (5.A) holds and k 1 σ k b k () <. If Y (k) converges weakly to Y M[,1, then {Y (k) t : t } converges to the erprocess {Y t : t } in distribution on D([, ),M[,1). Proof. Under the assumption, we have [ σ k (g (k) k ) (1) 1+b k () k 1 <. By Proposition 5.2 and Skorokhod s representation theorem, to simplify the notation we pass to a subsequence and simply assume {Y (k) t : t } converges a.s. to a process {Z t : t } in the topology of D([, ),M[,1). Since the solution of the martingale problem (2.4) is unique, it suffices to prove the weak limit point {Z t : t } of the sequence {Y (k) t : t } is the solution of the martingale problem. Let Y 1 s,k (u) and b 1 k (u) be defined as in Proposition 5.2. For every G C 2 (R) and f C 1 [,1 we use (5.7) and Itô s formula to get t (k) ky G( Y (k) t,f ) = G( Y (k) (1) { (,f )+σ k ds G Y (k),f + [,k) +k 1 (z 1)[f(1) F (k) (s,θ,u) G( Y (k) } π,f ) (k) (dz,dθ)du t bk (k) (k) ky (1) { ( +σ k ds dθ G Y (k),f k 1 [f(1) F ) (k) (s,θ,u) 15

16 bk () (k) ky dθ { ( G Y (k),f } t G( Y (k),f ) (1) du+σ k ds ) b k (k) +k 1 F(k) (s,θ,u) G( Y (k) }du+local,f ) mart. t (k) ky = G( Y (k) (1) { (,f )+σ k ds G Y (k),f + [,k ) +k 1 (z 1)f(Y 1 s,k (u/k) (θ/k)) t +σ k ds bk (k) dθ (k) ky (1) } t G( Y (k),f ) du+σ k [,1 ds G( Y (k),f ) } π (k) (dz,dθ)du { ( G Y (k),f k 1 f(y 1 bk () b k (k) dθ (k) ky (1) s,k (u/k)) ) { ( G Y (k),f +k 1 [f(b 1 1 k (θ)/k) f(ys,k (u/k))1 {Y 1 s,k (u/k) b 1 G( Y (k) }du+,f ) local mart. t { ( = G( Y (k),f )+kσ k ds Y (k) (dx) G Y (k),f ) [,1 + [,1 +k 1 (z 1)f(x θ) G( Y (k) } π,f ) (k) (dz,kdθ) t { ( ) } +kσ k b k (k) ds G Y (k),f k 1 f(x) G( Y (k),f ) Y (k) (dx) [,1 t x { ( ) +kσ k ds Y (k) (dx) G Y (k),f + k 1 [f(θ) f(x) [,1 1 k (θ)/k} ) G( Y (k) }b,f ) k (kdθ)+ local mart. t { ( = G( Y (k),f )+kσ k ds Y (k) (dx) G Y (k),f ) [,1 [,1 +k 1 (z 1)f(x θ) G( Y (k) } π,f ) (k) (dz,kdθ) t 1 +kσ k ds Y (k) (dx) ǫ k (s,x,θ) π (k) (dz,kdθ) {} x +local mart., (5.11) where ǫ k (s,θ,x) = { ( ) ( )} G Y (k),f k 1 f(x) G Y (k),f k 1 f(θ) { ( ) G Y (k),f +k 1 [f(θ) f(x) G( Y (k) }.,f ) It is elementary to see that kσ k {} 1 x ǫ k (s,x,θ) π (k) (dz,kdθ) tends to zero uniformly as k. Let G(x) = e x, by letting k in (5.11) we get (2.4) for f C 1 [,1. A simple approximation shows the martingale problem (2.4) 16

17 actually holds for any f C[,1. By the proof of Theorem 7.13 in Li (211) we get the result. Let { a 1 < a 2 < < a n = 1} be an ordered set of constants. Denote by {Y t,ai : t } and {Y (k) t,a i : t } the restriction of {Y t : t } and {Y (k) t : t } to [,a i, respectively. Let Y t (a i ) := Y t [,a i and Y (k) t (a i ) := Y (k) t [,a i for every t, i = 1,2,,n. By arguments similar to those in He and Ma (212) we get following results. Theorem 5.4 Suppose that Condition (5.A) is satisfied and k 1 σ k b k () <. If Y (k) converges weakly to Y M[,1, then {(Y (k) t,a 1,,Y (k) t,a n ) : t } converges to {(Y t,a1,,y t,an ) : t } in distribution on D([, ),M[,a 1 M[,a n ). Corollary 5.5 Suppose that Condition (5.A) is satisfied and k 1 σ k b k () <. If (Y (k) (a 1 ),,Y (k) (a n )) converges to (Y (a 1 ),,Y (a n )), then {(Y (k) t (a 1 ),,Y (k) t (a n )) : t } converges to {(Y t (a 1 ),,Y t (a n )) : t } in distribution on D([, ),R n + ). Acknowledgement. We would like to give our sincere thanks to Professor Zenghu Li for his encouragement and helpful discussions. References Abraham, R. and Delmas, J.-F. (21): A continuum tree-valued Markov process. Ann. Probab. To appear. Abraham, R., Delmas, J.-F. and He, H. (211): Pruning Galton-Watson Trees and Tree-valued Markov Processes. To appear in Annales de l Institut Henri Poincare: Probabilites et Statistiques. Aldous, D. (1978): Stopping times and tightness. Ann. Probab. 6, Aldous, D. (1991): The continuum random tree I. Ann. Probab. 19, Aldous, D. (1993): The continuum random tree III. Ann. Probab. 21, Aldous, D. and Pitman, J. (1998): Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. H. Poincaré Probab. Statist. 34, Bakhtin, Y. (211): SPDE approximation for random trees. Markov Process. Related Fields 17, Bertoin, J. and Le Gall, J.-F. (26): Stochastic flows associated to coalescent processes III: Limit theorems. Illinois J. Math. 5, Dawson, D.A. and Li, Z. (26): Skew convolution semigroups and affine Markov processes. Ann. Probab. 34,

18 Dawson, D.A. and Li, Z. (212): Stochastic equations, flows and measure-valued processes. Ann. Probab. 4, Duquesne, T. and Le Gall, J.-F. (22): Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281. Ethier, S.. and Kurtz, T.G. (1986): Markov Processes: Characterization and Convergence. Wiley, ew York. Fu, Z. and Li, Z. (21): Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 12, He, H. and Ma, R. (212): Some limit theorems for flows of branching processes. Ikeda,. and Watanabe, S. (1989): Stochastic Differential Equations and Diffusion Processes. Second Edition. orth-holland/kodasha, Amsterdam/Tokyo. Kawazu, K. and Watanabe, S. (1971): Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16, Lamperti, J. (1967): The limit of a sequence of branching processes. Z. Wahrsch. verw. Geb. 7, Le Gall, J.-F. and Le Jan, Y. (1998a): Branching processes in Lévy processes: The exploration process. Ann. Probab. 26, Le Gall, J.-F. and Le Jan, Y. (1998b): Branching processes in Lévy processes: Laplace functionals of snakes and erprocesses. Ann. Probab. 26, Li, Z. (26): A limit theorem for discrete Galton-Watson branching processes with immigration. J. Appl. Probab. 43, Li, Z. (211): Measure-Valued Branching Markov Processes. Springer, Berlin. Li, Z. (212): Path-valued branching processes and nonlocal branching erprocesses. Ann. Probab. To appear. Li, Z. and Ma, C. (28): Catalytic discrete state branching models and related limit theorems. J. Theoret. Probab. 21, Roelly, S. (1986): A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17, Hui He and Rugang Ma: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing ormal University, Beijing 1875, People s Republic of China. hehui@bnu.edu.cn, marugang@mail.bnu.edu.cn 18