On critical branching processes with immigration in varying environment

Transcription

1 On critical branching processes with immigration in varying environment Márton Ispány Faculty of Informatics, University of Debrecen Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Hungary III Workshop on Branching Processes and their Applications Badajoz, Spain April 7-10, 2015

2 Outline Sequence of branching processes with immigration (BPI) BPI in varying environment (BPIVE) Asymptotic in the mean Criticality and classification Limit theorem: strictly positive offspring variance Deterministic and fluctuation limit theorems: vanishing offspring variance Conclusions and future works

3 Sequence of branching processes with immigration Galton Watson branching processes with immigration (BPI) X (n) k = X (n) k 1 j=1 ξ (n) k,j + ε (n) (n) k, k, n N, X 0 = 0, where, for each n N, both the offsprings {ξ (n) k,j : k, j N} and the immigrations {ε (n) k : k N} are identically distributed, and they are independent, nonnegative, integer valued random variables. Parameters: m n := Eξ (n) 1,1, λ n := Eε (n) 1, σ 2 n := Varξ (n) 1,1, b2 n := Varε (n) 1. Classification: m n < 1 subcritical m n = 1 critical m n > 1 supercritical

4 Asymptotic result I: strictly positive offspring variance Sriram AS (1994) Suppose that m n = 1 + αn 1 + o(n 1 ) with α R, and σ 2 n σ 2 > 0 as n. Let X n t n 1 X n := X (n) nt. Then L X as n, where X := (X t ) t R+ is a (nonnegative) diffusion process with initial value X 0 = 0 and with generator Lf (x) = (λ + αx)f (x) σ2 xf (x), f C c (R + ). This process can also be characterized as the unique solution to the SDE dx t = (λ + αx t ) dt + σ (X t ) + dw t, t R +, X 0 = 0. This is a square-root process or a CBI process, related to the squared Bessel process and Cox-Ingersoll-Ross model in the financial mathematics.

5 Asymptotic result II: vanishing offspring variance I, Pap, van Zuijlen JAP (2005) Suppose that m n = 1 + αn 1 + o(n 1 ) with α R, and σn 2 = βn 1 + o(n 1 ) as Then, we have fluctuation limit theorem n 1/2 (X n EX n, M n ) n. Let M n t := nt Mn k. L (X, M) as n, where M := (M t ) t R+ is a time changed Wiener process, i.e., M t = W T (t), t R +, with T (t) := b 2 t + βλ t s 0 0 e αu duds, (W t ) t R+ is a standard Wiener process; and X t := t 0 e α(t s) dm s is an Ornstein Uhlenbeck process driven by M.

6 BPI in varying environment (BPIVE) Goal: To study the nearly criticality in one model! Galton Watson branching process with immigration X k = X k 1 j=1 ξ k,j + ε k, k N, X 0 = 0, where the offsprings {ξ k,j : k, j N} are identically distributed for each k N, respectively, and they and the immigrations {ε k : k N} are independent, nonnegative, integer valued random variables. The offspring and the immigration distributions may vary from generation to generation. The process (X k ) k Z+ is called branching process with immigration in varying environment (BPIVE) or time varying BPI. Parameters: m k := Eξ k,1, λ k := Eε k, σ 2 k := Varξ k,1, b 2 k := Varε k.

7 Applications General inhomogeneous branching processes: Domain of peer-to-peer file sharing networks, Adar and Huberman (2000), Zhao et al. (2005) Modeling biodiversity or macroevolution, Aldous and Popovic (2005), Haccou and Iwasa (1996) Epidemic type Aftershock Sequence (ETAS) in seismology, Farrington et al. (2003) Heterogeneous INAR models (Bernoulli offsprings): Understanding and predicting consumers buying behaviour, Böckenholt (1999) Modeling the premium in bonus-malus scheme of car insurance, Gourieroux and Jasiak (2004)

8 Asymptotic for the mean We have the following deterministic time varying linear recursion for the mean E(X k ) = m k E(X k 1 ) + λ k, k N, where the sequence (m k ) k N determines the asymptotic behaviour of the process. Define the bottom and top Lyapunov exponents as sup n 1 inf r(n, k) =: γ b γ t := inf n k n n 1 sup k where the partial growing rate function is defined as Classification: r(n, k) := k+n 1 j=k log m j γ t < 0 subcritical γ b 0 γ t??? r(n, k), γ b > 0 supercritical The supercritical case was studied by Goettge (1976), Cohn and Hering (1983), Jagers and Nerman (1985), D Souza and Biggins (1992, 1993), D Souza (1994).

9 Asymptotic for nearly critical recursion Notation: λ k λ stands for the Cesaro convergence n 1 n λ k λ as n. For a deterministic time varying linear recursion where x k = m k x k 1 + λ k, k N, x 0 = 0, 1 m k = 1 + αk 1 + δ k for some α R and δ k < ; 2 λ k λ as k for some λ 0, where (λ k ) k Z+ is a non-negative sequence, we have 1 n 1 x n λ(1 α) 1 if α < 1, i.e. EX n = O(n); 2 (n ln n) 1 x n λ if α = 1, i.e. EX n = O(n ln n); 3 n α x n κ if α > 1, i.e. EX n = O(n α ), where κ > 0 is a constant, as n.

10 Proof of the asymptotic Representation for the unique solution, see Elaydi (1.2.4), k k x k = m i λ j j=1 i=j+1 Method of the proof: perturbation argument. Introduce the new sequence y k := k α x k, k N. Then, we have y k = (1 + δ k )y k 1 + k α λ k, where δ k <. This recursion is a small perturbation of the recursion z k = z k 1 + k α λ k. Hence, their asymptotic behaviors are similar. Thus, we have x n n α n k α λ k

11 Proof of the asymptotic In case of α < 1, by Toeplitz theorem, we have n 1 x n n α 1 n k α λ k λ 1 α since n α 1 n k α 1 0 s α ds = (1 α) 1

12 Second order linear difference equations I. An inhomogeneous first order d.e. can be transformed to a homogeneous second order d.e. Suppose that Then where m k = 1 + αk 1 + βk 2, λ k = λ + γk 1. x k + a(k)x k 1 + b(k)x k 2 = 0, k = 2, 3,..., a(k) a 0 + a 1 k 1 + a 2 k 2, b(k) b 0 + b 1 k 1 + b 2 k 2 with a 0 = 2, a 1 = α, a 2 = γ β and b 0 = 1, b 1 = α, b 2 = α + β γ. Asymptotic theory: Wong and Li (1992), goes back to Birkhoff ( 30).

13 Second order linear difference equations II. Characteristic polynomial: ϱ 2 + a 0 ϱ + b 0 = ϱ 2 2ϱ + 1 = 0 = ϱ 1,2 = 1 The common value is a root of the auxiliary equation Indicial polynomial: a 1 ϱ + b 1 = 0 = αϱ + α = 0 κ(κ 1)ϱ 2 + (a 1 κ + a 2 ) + b 2 = κ 2 (α + 1)κ + α = 0 Roots: κ 1 = 1 and κ 2 = α. Two linearly independent asymptotic solutions if α 1 with x (i) n ϱ n n κ i i = 1, 2 If α = 1 then we have an extra log n factor.

14 Criticality and classification A BPIVE is called asymptotically critical if m n 1 as n. More precisely, if the parametrization m k = 1 + αk 1 + δ k, α R, δ k < holds then BPIVE is called nearly critical and it has criticality index α. In this case, γ b = γ t = 0. Classification (regimes) for nearly critical TVBPI: α < 1 proper α = 1 logarithmically α > 1 polinomially nearly critical In the sequel, we investigate proper nearly critical BPIVE. If α = 0 then a BPIVE is called strongly critical.

15 Limit theorem: Assumptions I (2015) Suppose that (i) m n = 1 + αn 1 + δ n with α < 1 and (ii) λ n λ 0 as n ; (iii) σ 2 n σ 2 0 as n ; (iv) n 1 b 2 n 0 as n ; δ n < ; n=1 moreover the following Lindeberg conditions hold 1 n ( ) (L1) E ξ k,1 m k 2 1 n { ξk,1 m k >θn} 0 for all θ > 0, (L2) 1 n n 2 as n. ( ) E ε k λ k 2 1 { εk λ k >θn} 0 for all θ > 0

16 Limit theorem: Result I (2015) Let Xt n := X nt. Then, weakly in the Skorokhod space D(R +, R), where (X t ) t R+ n 1 X n L X as n, satisfies the SDE dx t = (λ + αt 1 X t ) dt + σ X t dw t, t > 0, where (W t ) t R+ is a standard Wiener process, with initial condition X 0 = 0. Formal SDE: the drift β(t, x) := (λ + αt 1 x) is not Lipschitz and it is not defined at t = 0. Solution: take the process Y t := t α X t and apply the Ito s formula (formally) dy t = λt α dt + σ t α Y t dw t, t > 0,

17 Associated martingale differences Let F k denote the σ-algebra generated by X 0, X 1,..., X k. We have the conditional expectation Clearly, E(X k F k 1 ) = m k X k 1 + λ k. k N. M k := X k E(X k F k 1 ) = X k m k X k 1 λ k defines a martingale difference sequence with respect to the filtration (F k ) k Z+. On the other hand, X k 1 M k := (ξ k,j m k ) + ε k λ k j=1 Thus, we have the heteroscedastic property: E(M 2 k F k 1) = σ 2 k X k 1 + b 2 k

18 Heuristic for the limit theorem I. For all k N, we have X k = m k X k 1 + λ k + M k = X k 1 + λ k + α X k 1 k Let 0 < s < t. Then, by iteration, X nt n where = X ns n + 1 n nt 1 k= ns ( λ k + α X ) k 1 k nt 1 1 (δ k X k 1 + ε k λ k ) n k= ns W n,k := X 1 k 1 nxk 1 j=1 + nt 1 k= ns ξ k,j m k σ k N (1, n 1 ) + δ k X k 1 + M k Xk 1 σ k n W n,k

19 Heuristic for the limit theorem II. The blue part can be approximated by stochastic integral equation t ( X t = X s + λ + α X ) t u du + σ X u dw u s u s On the other hand, the red part is vanishing in probability since 1 n n δ k E(X k 1 ) C n n δ k k 0 by Kronecker lemma and, by assumption (iv), ( ) 1 n Var (ε k λ) = 1 n n n 2 bk 2 0

20 Weak convergence to a diffusion process I. For each n N, let (U n k ) k N be a sequence of R d valued adapted random variables w.r.t. a filtration (F n k ) k Z +. Introduce the random step functions: nt Ut n := Uk n, t R +, n N. Let (U t ) t R+ be a d dimensional diffusion process d U t = β(t, U t ) dt + γ(t, U t ) dw t, t R +, where β : R + R d R d and γ : R + R d R d r are continuous functions and (W t ) t R+ is an r-dimensional standard Wiener process. Assume that the SDE has a unique weak solution with U 0 = x 0 for all x 0 R d. Let (U t ) t R+ be a solution with U 0 = 0.

21 Weak convergence to a diffusion process II. I & Pap, (2010) Suppose that, for each T > 0, Uniform convergence on compacts in probability (ucp) nt sup E ( U n ) t k F n k 1 β(s, U n s ) ds P 0, t [0,T ] 0 nt sup E ( U n k t [0,T ] (Un k ) Fk 1 n ) t γ(s, Us n )γ(s, Us n ) ds P 0, 0 and the conditional Lindeberg condition Then nt E ( Uk n 2 1 { U n k >θ} Fk 1) n P 0 for all θ > 0. U n L U as n.

22 Sketch for the proof of the limit theorem Introduce the new process Y k := k α X k, k N, Y 0 := 0 and define U n k := nα 1 (Y k Y k 1 ), k, n N. Then nt Ut n := Uk n = nα 1 Y nt We prove, by general limit theorem, that weakly in the Skorokhod space D(R +, R) n α 1 Y nt = U n t L U t := Y t as n

23 Sketch for the proof of the limit theorem This implies, for 0 t 1 < t 2 <... < t m, Hence n α 1 ( ) L Y nt1,..., Y ntm (Y t1,..., Y tm ) n 1 ( ) L X nt1,..., X ntm (X t1,..., X tm ) shows the convergence of finite dimensional distributions. Then, we prove tigthness by checking the conditional Lindeberg condition.

24 Deterministic limit theorem I (2015) Suppose that 1 m n = 1 + αn 1 + δ n with α < 1 and 2 λ n λ 0 as n ; 3 σ 2 n 0, as n ; 4 n 1 b 2 n 0 as n. Then n 1 X n δ n < ; n=1 L µ X as n, where µ X : R + R + is the unique solution of the ordinary differential equation (ODE) dµ X (t) = (λ + αt 1 µ X (t))dt, t > 0, with initial condition µ X (0) = 0. In fact, µ X (t) = λt/(1 α).

25 Fluctuation limit theorem: Assumptions I (2015) Suppose that (i) m n = 1 + αn 1 + δ n with α < 1 and (ii) λ n λ 0 as n ; (iii) nσ 2 n σ 2 0, as n ; (iv) b 2 n b 2 0 as n ; δ n < ; n=1 moreover the following Lindeberg conditions hold n ( ) (L1) E ξ k,1 m k 2 1 { ξk,1 m k >θn 1/2 } 0 for all θ > 0, 1 n ( ) (L2) E ε k λ k 2 1 n { εk λ k >θn 1/2 } 0 for all θ > 0 as n.

26 Fluctuation limit theorem: Statements Then, weakly in the Skorokhod space D(R +, R), where (M t ) t R+ n 1/2 M n L M as n, is a Wiener process with variance σm 2 := σ 2 λ 1 α + b2. Moreover, suppose that α < 1/2. Then, weakly in the Skorokhod space D(R +, R 2 ), n 1/2 (X n EX n, M n ) L (X, M) as n, where (X t ) t R+ satisfies the SDE with initial condition X 0 = 0. dx t = αt 1 X t dt + dm t, t > 0,

27 Ornstein-Uhlenbeck fluctuation limit The SDE can be written in the form The solution is given as dx t = αt 1 X t dt + σ M dw t, t > 0. t X t = σ M t α s α dw s, t > 0. 0 If α < 1/2 the integral is well-defined in L 2 and Itô s sense as well since t s 2α ds = t1 2α 1 2α <. 0 (X t ) t R+ is an Ornstein Uhlenbeck type process t X t = σ M e α(ln t ln s) dw s, t > 0, with logarithmic exponent function. 0

28 Why α < 1/2? Define the sequence V k := Var(X k ), k N. Then we have the recursion V k = m 2 k V k 1 + EM 2 k, k N. Asymptotic for the variance: 1 n 1 V n λ(1 2α) 1 ((1 α) 1 λσ 2 + b 2 ) if α < 1/2, 2 (n ln n) 1 V n (1 α) 1 λσ 2 + b 2 if α = 1/2, 3 n 2α V n c 0 if α > 1/2, where c R is a constant, since with δ k <. m 2 k = 1 + 2αk 1 + δ k, k N,

29 Conclusions and future works A criticality index was introduced for branching processes with immigration in varying environment. Limit theorems were proved for proper nearly critical branching processes with immigration in varying environment. Estimating the criticality index and the average immigration intensity. Developing tests for hypothesis H 0 : α = α 0, e.g., testing the strong criticality (α 0 = 0). Investigating the logarithmically and polinomially nearly critical BPIVE.

30 References Ispány, M., Pap, G., van Zuijlen, M.C.A. (2005). Fluctuation limit of branching processes with immigration and estimation of the means Adv. Appl. Prob. 37: Ispány, M., Pap, G. (2010). A note on weak convergence of random step processes. Acta Mathematica Hungarica 126(4): Sriram, T.N. (1994). Invalidity of bootstrap for critical branching processes with immigration. Ann. Statist. 22: R. Wong, H. Li (1992) Asymptotic expansions for second-order linear difference equations J. Comp. Appl. Math. 41:

31 Thank you for your attention!