Limit theorems on counting measures for a branching random walk with immigration in a random environment
aa r X i v : . [ m a t h . P R ] F e b Limit theorems on counting measures for a branching random walk withimmigration in a random environment
Mengxue Li a , Chunmao Huang b , Xiaoqiang Wang a a School of Mathematics and Statistics, Shandong University, Weihai, Shandong, 264209, China b Department of Mathematics, Harbin Institute of Technology (Weihai), Shandong, 264209, China
Abstract
We consider a branching random walk with immigration in a random environment, where the environment is a sta-tionary and ergodic sequence indexed by time. We focus on the asymptotic properties of the sequence of measures( Z n ) that count the number of particles of generation n located in a Borel set of real line. In the present work, a seriesof limit theorems related to the above counting measures are established, including a central limit theorem, a moderatedeviation principle and a large deviation result as well as a convergence theorem of the free energy. Keywords:
Branching random walk with immigration, random environment, central limit theorem, large deviation,moderate deviation
1. Introduction and main results
As one of the frontier field of stochastic processes, the branching random walk with a random environment in time(BRWRE) has been making a lot of progress in recent years. For instance, Gao and Liu (2018)[7] generalized theasymptotic expansions in the central theorem for BRWRE and obtained related results in the second and third orders.Wang and Huang (2019)[6] gave the su ffi cient and necessary conditions for existence of quenched moments as wellas weighted moments of BRWRE. Huang et al. (2019) [3] established large and moderate deviations of BRWRE inhigh-dimensional real space. As an extension of BRWRE, a branching random walk with immigration in randomenvironment in time (BRWIRE) also has attracted extensive attention. However there are too few relevant resultsconstraining some applications, hence it is interesting to study BRWIRE both in theory and in applications. In thispaper we consider a branching random walk with immigration in a time-dependent random environment. Let ξ = ( ξ n )representing the random environment in time be a stationary and ergodic sequence of random variables of distribution τ . Suppose that each realization of ξ n is related to two distributions η n = η ( ξ n ) and ι n = ι ( ξ n ) on N × R . Given the environment ξ , the process can be described as follows. At time 0, there exists an initial particle ∅ ofgeneration 0, located at S ∅ = ∈ R . At time 1, it is replaced by new born particles with the relative displacements L i = L i ( ∅ ) , i = , , · · · , N ( ∅ ) , where N ( ∅ ) is the number of o ff spring of ∅ . At the same time, V new immigrants comeand join located at S i , i = , , · · · , V . All of the new born particles and new immigrants make up the first generationof particles. In general, each particle u of generation n with locations S u has N ( u ) o ff spring ui , i = , , · · · , N ( u ) , located at S ui = S u + L i ( u ) . At the same time, V n new immigrants 0 n i , i = , , · · · , V n come and join with locations S n i . Then all the o ff spring and new immigrants form the particles of generation n +
1. In addition, we define that therandom vector X ( u ) = ( N ( u ) , L ( u ) , L ( u ) , · · · ) is of distribution η n and that the random vector Y n = ( V n , S n , S n , · · · )is of distribution ι n . And they are all independent of each other conditional on ξ. Let P ξ so-called quenched law bethe conditional probability given the environment ξ . The total probability can be written as P (d x , d ξ ) = P ξ (d x ) τ (d ξ )and it is usually called the annealed law. Also, let P ξ, Y be the conditional probability given the environment ξ and theimmigrant sequence Y = ( Y n ). E ξ, Y , E ξ and E denote the expectations with respect to P ξ, Y , P ξ and P respectively. Email addresses: (Mengxue Li), [email protected] (Chunmao Huang), [email protected] (Xiaoqiang Wang)
Preprint submitted to Elsevier February 23, 2021 et T be the family tree with defining elements { N ( u ) } and T n = { u ∈ T : | u | = n } be the set of particles ofgeneration n with | u | representing the length of u . For n ∈ N , define Z n ( · ) = X u ∈ T n δ S u ( · ) , the counting measure of particles of generation n . In this paper, we concentrate on the asymptotic properties of thesequence of counting measures ( Z n ) via establishing limit theorems. Similarly, let T X be the family tree withoutimmigration and T Xn = { u ∈ T X : | u | = n } be the set of particles of generation n originated from the initial particle ∅ .For convenience, we use the symbol ‘ − ’ for notations without immigration. According to the Laplace transform of Z n ( · ), for n ∈ N and t ∈ R , we define˜ Z n ( t ) = X u ∈ T n e tS u , W n ( t ) = ˜ Z n ( t ) Π n ( t ) , ¯ Z n ( t ) = X u ∈ T Xn e tS u and ¯ W n ( t ) = ¯ Z n ( t ) Π n ( t ) (1.1) m n ( t ) = E ξ N ( u ) X i = e tL i ( u ) ( u ∈ N ∗ n ) , Y n ( t ) = V n X i = e tS ni , Π ( t ) = Π n ( t ) = n − Y i = m i ( t ) ( n ≥ , (1.2)In particular, set m n (0) = m n , Π n = Π n (0), W n = W n (0) and ¯ W n = ¯ W n (0) for short. We can see that ¯ W n ( t ) isthe intrinsic non-negative martingale of BRWRE, hence it converges almost surely (a.s.) to some limit ¯ W ( t ) with E ξ ¯ W ( t ) ≤
1. Also, it is not di ffi cult to verify that W n ( t ) is a non-negative sub-martingale with respect to the probability P ξ, Y . We consider that ( Z n ) is a supercritical branching process i.e. E log m >
0. Based on the necessary momentconditions (for example, E log + V < ∞ and E log + E ξ N p < ∞ for some p > W n converges a.s.to a limit W ∈ (0 , ∞ ). To study the intrinsic properties and for simplicity, we further assume that P ξ ( N = = , P ξ ( N = < E ξ N X i = L i = a . s .. (1.3)Firstly, we establish a central limit theorem concerning ( Z n ). Set σ n = m n E ξ N ( u ) X i = L i ( u ) ( u ∈ N ∗ n ) and b n = n − X i = σ i / . Theorem 1.1 (Central limit theorem) . If E σ ∈ (0 , ∞ ) , E log + Y ( ± δ ) < ∞ for some δ > and E log + E ξ N p < ∞ forsome p > , then a.s., Z n ( −∞ , b n x ] Z n ( R ) → Φ ( x ) ∀ x ∈ R , where Φ ( x ) = √ π R x −∞ e − t / d t. We next establish a moderate deviation principle associated to ( Z n ). Let a n = n α with α ∈ (0 . , Theorem 1.2 (Moderate deviation principle) . If E m P Ni = e δ | L i | < ∞ , E log + Y ( ± δ ) < ∞ and E log + E ξ ( P Ni = e δ | L i | ) p < ∞ for some δ > and p > , then it is a.s. that the sequence of probabilities A Z n ( a n A ) / Z n ( R ) satisfies a principleof moderate deviation with rate function x σ : for each measurable subset A of R , − σ inf x ∈ A ◦ x ≤ lim inf n →∞ na n log Z n ( a n A ) Z n ( R ) ≤ lim sup n →∞ na n log Z n ( a n A ) Z n ( R ) ≤ − σ inf x ∈ ¯ A x , where σ = E σ , A ◦ denotes the interior of A, and ¯ A its closure. Z n . We shall begin with the study of convergence of the free energy log ˜ Z n n . Assume that E | L | < ∞ , E | log m ( t ) | < ∞ , E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ′ ( t ) m ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ and E | log Y ( t ) | < ∞ (1.4)for all t ∈ R . Therefore, the function Λ ( t ) : = E log m ( t ) is well-defined and di ff erentiable everywhere on R . Define t − = inf { t ∈ R : t Λ ′ ( t ) − Λ ( t ) ≤ } and t + = sup { t ∈ R : t Λ ′ ( t ) − Λ ( t ) ≤ } . (1.5)For BRWRE, it is known from [2, Theorem 3.1] that a.s. for all t ∈ R ,lim n →∞ n log ¯ Z n ( t ) = ¯ Λ ( t ) : = Λ ( t ) i f t ∈ ( t − , t + ) , t Λ ′ ( t + ) i f t ≥ t + , t Λ ′ ( t − ) i f t ≤ t − . (1.6) Theorem 1.3 (Convergence of the free energy) . Assume (1.4) . Let ˜ Λ ( t ) : = max { ¯ Λ ( t ) , } . It is a.s. that for all t ∈ R , lim n →∞ n log ˜ Z n ( t ) = ˜ Λ ( t ) . (1.7)Applying Theorem 1.3 and the G¨artner-Ellis theorem, we obtain the following large deviation results. Denote t = inf { t ∈ R : Λ ( t ) < } and t = sup { t ∈ R : Λ ( t ) < } . According to the relationship between t , t , t − and t + andconsidering the properties of the function Λ ( t ), we can distinguish three cases: case I. t = −∞ and t = ∞ ; case II. t < t − < t <
0; case III. 0 < t < t + < t . Theorem 1.4 (Large deviations) . Assume (1.4) . Let ˜ Λ ∗ ( x ) = sup t ∈ R { xt − ˜ Λ ( t ) } be the Legendre transform of ˜ Λ ( t ) .(a) For case I, it is a.s. that the sequence of measures A Z n ( nA ) satisfies a principle of large deviation with ratefunction ˜ Λ ∗ ( x ) : for each measurable subset A of R , − inf x ∈ A ◦ ˜ Λ ∗ ( x ) ≤ lim inf n →∞ n log Z n ( nA ) ≤ lim sup n →∞ n log Z n ( nA ) ≤ − inf x ∈ ¯ A ˜ Λ ∗ ( x ) , where A ◦ denotes the interior of A, and ¯ A its closure.(b) For cases II and III, it is a.s. that for each closed subset F of R , lim sup n →∞ n log Z n ( nF ) ≤ − inf x ∈ F ˜ Λ ∗ ( x ) , and for each open subset G of R , lim inf n →∞ n log Z n ( nG ) ≥ − inf x ∈ G ∩E ˜ Λ ∗ ( x ) , where E = ( Λ ′ ( t ) , Λ ′ ( t + )) for case I and E = ( Λ ′ ( t − ) , Λ ′ ( t )) for case III, is the set of exposed points of the ratefunction ˜ Λ ∗ .
2. Sketch of proof
The proofs will be based on the convergence of the sub-martingale W n ( t ). The following proposition describes thea.s. and L p -convergence of W n ( t ) under the probability P ξ, Y as well as its L p -convergence rate. For t ∈ R fixed, set f t ( x ) = x Λ ( xt ) − Λ ( t ). 3 roposition 2.1. Let p > . If Λ ( t ) > , E log + Y ( t ) m ( t ) < ∞ , f t ( p ) < and E log + E ξ ¯ Z ( t ) p < ∞ , then sup n E ξ, Y W n ( t ) p < ∞ a.s., so that W n ( t ) converges a.s. and in L p under P ξ, Y to some limit W ( t ) which satisfies the decompositionW ( t ) = ¯ W ( t ) + ∞ X k = Π k ( t ) − V k − X i = ¯ W (0 k − i , t ) e tS k − i , (2.1) where the notation ¯ W ( u , t ) denotes the limit of the intrinsic non-negative martingale ¯ W n ( u , t ) of the BRWRE originatedfrom the particle u ∈ U , and the L p -convergence rate is a.s., lim sup n →∞ n log (cid:16) E ξ, Y | W n + ( t ) − W n ( t ) | p (cid:17) / p ≤ ( max {− Λ ( t ) , f t ( p ) } if p ∈ (1 , , max {− Λ ( t ) , f t ( p ) , f t (2) } if p > . (2.2) Proof.
We first prove (2.2), which implies sup n E ξ, Y W n ( t ) < ∞ a.s. Here we just provide the proof for p ∈ (1 , p >
2, (2.2) can be obtained by a technique of induction.ee According to the structure of the family tree T , wedecompose W n ( t ) = ¯ W n ( t ) + n X k = Π k ( t ) − V k − X i = ¯ W n − k (0 k − i , t )e tS k − i . (2.3)Set Y − = (1 , , , · · · ) and ¯ W n (0 − i , t ) = ¯ W n ( t ). Using Minkowski’s inequality and Burholder’s inequality, we get (cid:16) E ξ, Y | W n + ( t ) − W n ( t ) | p (cid:17) / p ≤ C (cid:16) E T n ξ (cid:12)(cid:12)(cid:12) ¯ W ( t ) − (cid:12)(cid:12)(cid:12) p (cid:17) / p n X k = Π k ( t ) − Y k − ( t ) γ k , n ( p , t ) + Π n + ( t ) − Y n ( t ) , where γ k , n ( p , t ) = Q n − i = k [ m i ( t ) − m i ( pt ) / p ], C > T represents the shift operator: T ξ = ( ξ , ξ , · · · ) if ξ = ( ξ , ξ , · · · ). Applying [5, Lemma 2.1] leads tolim sup n →∞ n log (cid:16) E T n ξ (cid:12)(cid:12)(cid:12) ¯ W ( t ) − (cid:12)(cid:12)(cid:12) p (cid:17) / p n X k = Π k ( t ) − Y k − ( t ) γ k , n ( p , t ) ≤ max {− Λ ( t ) , f t ( p ) } a . s . Besides, it is not di ffi cult to see that lim sup n →∞ n log[ Π n + ( t ) − Y n ( t )] ≤ − Λ ( t ) a.s. So (2.2) is proved. In order toobtain the decomposition (2.1), we can prove that ∞ X k = Π k ( t ) − V k − X i = e tS k − i sup n ¯ W n (0 k − i , t ) < ∞ a . s . (2.4)Then let n tend to infinity in (2.3) and use the dominated convergence theorem. Proof of Theorem 1.1.
By Proposition 2.1, W = lim n →∞ W n exists and satisfies the decomposition W = ¯ W (0) + P ∞ k = Π − k V k − ¯ W (0 k − i , Ψ n ( t ) = Π − n ˜ Z n ( i t ) be the characteristic function of the measure Π − n Z n . It su ffi cesto show that Ψ n ( t / b n ) → g ( t ) W a.s., where g ( t ) is the characteristic function of the standard normal distribution.Using (2.3) with t replaced by i t / b n , we have Ψ n ( t / b n ) = ¯ Ψ n ( t / b n ) + n X k = Π − k V k − X i = ¯ Ψ n − k (0 k − i , t / b n ) e i tbn S k − i , (2.5)where ¯ Ψ n − k (0 k − i , t ) = Q n − l = k [ m − l m l ( i t )] ¯ W n − k (0 k − i , i t ). Since ¯ Ψ n ( t / b n ) → g ( t ) ¯ W (0) a.s. by the central limit theoremfor BRWRE (see [2, Theorem 10.1]), we can prove that ¯ Ψ n − k (0 k − i , t / b n ) → g ( t ) ¯ W (0 k − i ,
0) a.s. Noticing (2.4), we let n tend to infinity in (2.5) and use the dominated convergence theorem. Proof of Theorem 1.2.
Let Γ n ( t ) = log R e tx Z n ( a n dx ) Z n ( R ) , then na n Γ n (cid:18) a n n t (cid:19) = na n log Π n ( a n n t ) Π n + na n (cid:20) log W n (cid:18) a n n t (cid:19) − log W n (cid:21) . (2.6)4y [3, Lemma 6.1], lim n →∞ na n log[ Π − n Π n ( a n n t )] = σ t a.s. Now we shall prove that W n ( t ) converges uniformly a.s.to the limit W ( t ) on the interval D ε = [ − ε, ε ] for some ε >
0. Notice thatsup t ∈ D ε | W n + ( t ) − W n ( t ) | ≤ | W n + − W n | + Z D ε (cid:12)(cid:12)(cid:12) W ′ n + ( t ) − W ′ n ( t ) (cid:12)(cid:12)(cid:12) dt . By Proposition 2.1, it can be seen that P n | W n + − W n | < ∞ a.s. It remains to show that X n sup t ∈ D ε (cid:16) E ξ, Y (cid:12)(cid:12)(cid:12) W ′ n + ( t ) − W ′ n ( t ) (cid:12)(cid:12)(cid:12) p (cid:17) / p < ∞ a . s . (2.7)for suitable p ∈ (1 , t ∈ D ε (cid:16) E ξ, Y (cid:12)(cid:12)(cid:12) W ′ n + ( t ) − W ′ n ( t ) (cid:12)(cid:12)(cid:12) p (cid:17) / p ≤ n X k = sup t ∈ D ε (cid:26) Π k ( t ) − Y k − ( t ) (cid:16) E T k ξ (cid:12)(cid:12)(cid:12) ¯ W ′ n + − k ( t ) − ¯ W ′ n − k ( t ) (cid:12)(cid:12)(cid:12) p (cid:17) / p (cid:27) + n X k = sup t ∈ D ε (cid:26) Π k ( t ) − ˜ Y k − ( t ) (cid:16) E T k ξ (cid:12)(cid:12)(cid:12) ¯ W n + − k ( t ) − ¯ W n − k ( t ) (cid:12)(cid:12)(cid:12) p (cid:17) / p (cid:27) + sup t ∈ D ε n Π n + ( t ) − ˜ Y n ( t ) o = : I , n + I , n + I , n , where ˜ Y n ( t ) = P V n − i = e tS n − i | S n − i − P n − l = m l ( t ) − m ′ l ( t ) | . By tedious calculations, it can be seen that P n I i , n < ∞ a.s., i = , ,
3, then (2.7) is proved.
Proof of Theorem 1.3.
Since ˜ Z n ( t ) ≥ max { ¯ Z n ( t ) , Y n ( t ) } , by (1.6),lim inf n →∞ n log ˜ Z n ( t ) ≥ max { lim inf n →∞ n log ¯ Z n ( t ) , lim inf n →∞ n log Y n ( t ) } = max { ¯ Λ ( t ) , } = ˜ Λ ( t ) a . s . It remains to show the superior limit. For s > max { Λ ( t ) , } , we have E ξ, Y ( P n e − ns ˜ Z n ( t )) < ∞ a.s., which implies thatlim sup n →∞ n log ˜ Z n ( t ) ≤ max { Λ ( t ) , } a.s. For t ∈ ( t − , t + ), Λ ( t ) = ¯ Λ ( t ). So it remains to consider the case t ≥ t + and t ≤ t − . In the following, we just provide the proof for the case t ≥ t + ; the proof for t ≤ t − is similar. Let t ≥ t + . Noticethat lim sup n →∞ n log E ξ, Y ˜ Z n ( t ) ≤ max { Λ ( t ) , } . Denote Λ ∗ + ( x ) = sup t ∈ R { xt − max { Λ ( t ) , }} the Legendre transform ofmax { Λ ( t ) , } . By [1, Theorem 4.5.3], for a > max { Λ ′ ( t + ) , } , lim sup n →∞ n log E ξ, Y ˜ Z n ( t ) ≤ − Λ ∗ + ( a ) < P n P ξ, Y ( Z n [ na , ∞ ) ≥ < ∞ a.s. Therefore, by Borel-Cantelli Lemma, Z n [ na , ∞ ) = R n < na a.s., where R n : = max { S u : u ∈ T n } is the rightmost position of generation n . For t ∈ ( t + , t ], noticing the fact that˜ Z n ( t ) ≤ ˜ Z n ( t ) exp { ( t − t ) R n n } , we deduce that lim sup n →∞ n log ˜ Z n ( t ) ≤ max { Λ ( t ) , } + ( t − t ) a a.s. The proof is finishedby letting a ↓ max { Λ ′ ( t + ) , } and t ↑ t + . References [1] A. Dembo, O. Zeitouni, Large deviations Techniques and Applications. Springer, New York, 1998.[2] C. Huang, Q. Liu, Branching random walk with a random environment in time. arXiv:1407.7623[3] C. Huang, X. Wang, X. Wang, Large and moderate deviations for a R d -valued branching random walk with a random environment in time.Stochastics An International Journal of Probability & Stochastic Processes, 2019(63): 1-25.[4] X. Wang, C. Huang, Convergence of martingale and moderate deviations for a branching random walk with a random environment in Time.Journal of Theoretical Probability, 2017, 30(3): 961-995.[5] X. Wang, C. Huang, Convergence of complex martingale for a branching random walk in a time random environment. Electronic Communi-cations in Probability, 2019, 24(41): 1-14.[6] Y. Wang, Z. Liu, Q. Liu, Y. Li, Asymptotic properties of a branching random walk with a random environment in time. Acta MathematicaScientia, 2019, 39B(5): 1345-1362.[7] Z. Gao, Q. Liu, Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a randomenvironment in time. Bernoulli, 2018, 24(1): 772-800.-valued branching random walk with a random environment in time.Stochastics An International Journal of Probability & Stochastic Processes, 2019(63): 1-25.[4] X. Wang, C. Huang, Convergence of martingale and moderate deviations for a branching random walk with a random environment in Time.Journal of Theoretical Probability, 2017, 30(3): 961-995.[5] X. Wang, C. Huang, Convergence of complex martingale for a branching random walk in a time random environment. Electronic Communi-cations in Probability, 2019, 24(41): 1-14.[6] Y. Wang, Z. Liu, Q. Liu, Y. Li, Asymptotic properties of a branching random walk with a random environment in time. Acta MathematicaScientia, 2019, 39B(5): 1345-1362.[7] Z. Gao, Q. Liu, Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a randomenvironment in time. Bernoulli, 2018, 24(1): 772-800.