Irregular Birth-Death process: stationarity and quasi-stationarity

Transcription

1 Irregular Birth-Death process: stationarity and quasi-stationarity MAO Yong-Hua May 8-12, BNU orks with W-J Gao and C Zhang)

2 CONTENTS 1 Stationarity and quasi-stationarity 2 birth-death process with four boundaries 3 Known results 4 Exit boundary: mininal process 5 Regular boundary: minimal vs maximal 6 Reference

3 Stationarity or quasi-stationarity Markov process X t on state space (E, E ) with life time ζ. We want to find probability µ such that P µ [X t A t < ζ] = µ(a), A E. If ζ = a.e. then µ is called the stationary distribution. If ζ < a.e. then µ is called quasi-stationary distribution, or conditional stationary distribution. Equivalently, where u i p ij (t) = u j e λt, i p ij (t) = P i [X t = j, t < ζ].

4 Initial Consider a branching process whose branching mechanism ξ satisfying m = Eξ 1 lim t 0 p i0 (t) = 1. Yaglom (1947) considered the following limit problem i, j 1 lim P P i [X t = j, t < τ 0 ] i[x t = j τ 0 > t] = lim t t P i [t < τ 0 ] p ij (t) = lim t k 1 p ik(t). This is to consider the ergodicity of the process conditional on {1, 2, }? Or the limit behavior before the particles die out? A.M. Yaglom (1947). Certain limit theorems of the theory of branching processes (in Russian). Doklady Akademii Nauk SSSR 56

5 Branching process Theorem Let Z 0 = 1, Z n = Z n 1 k=1 ξ n,k. Denote f(s) = Es ξ, g(s) = lim n E[sZn Z n > 0]. If m < 1, then g(s) exists and satisfies g(f(s)) = mg(s) + 1 m. In particular, lim n P[Z n = i Z n > 0] = b i > 0, b i = 1. i 1 Furthermore, there exists a family of QSD s: g(s) = 1 (1 g(s)) α, α (0, 1].

6 Finite Markov chain Darroch-Seneta (1965, 1967) considered a jump process on finite state space ( ) 0 0 Q =. a Q Perron-Frobenius theorem: there exist λ > 0 and positive vectors u, v such that uq = λu, Qv = λv. Thus and Q = λvu + O(λ ), λ > λ, P (t) = e tλ vu + O(e tλ ). J.N. Darroch and E. Seneta (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab. 4,

7 QSD for finite Markov chain i, j 1, lim n independent of i. p ij (t) k p ik(t) = lim t e tλ v i u j + o(e tλ ) e tλ k v iu k + O(e tλ ) = Assume k u k = 1 then u is conditional limit distribution. On the other hand, from u j k u k uq = λu. λ invariant we have up (t) = e λt u, P u [t < ζ] = up (t)1 = e λt. Thus P u [X t = i t < ζ] = u.

8 Infinite state space Assume ζ = τ 0 <. Theorem Equivalent: (a) QSD u exists. (b) u is x(> 0)-invariant probability (uq = xu) for Q such that x = i 1 u iq i0. Define decay rate Then x λ. λ = lim t 1 t log p ij(t). D. Vere-Jones (1969). Some limit theorems for evanescent processes. Austral. J. Statist. 11

9 Birth-death process On {0, 1, 2,... }, birth rates b i > 0(i 0), death rates a 0 0, a i > 0(i 1): Define Q = (a 0 + b 0 ) b 0 a 1 (a 1 + b 1 ) b 1 a 2 (a 2 + b 2 ) b µ 0 = 1, µ i = b 0 b i 1 a 1 a i, i 1; ν i = 1 µ i b i, i 0;, R = ν i i=0 j=0 i µ j, S = ν i i=0 j=i+1 µ j.

10 Feller s 4 boundaties Natrual boundary: R = S = ; Entrance boundary: R =, S < ; Exit boundary: R <, S = ; Regular boundary: R <, S <. W. Feller (1959)The birth and death processes as diffusion processes. J. Math. Pures Appl., 38

11 A diagram Done and to do boundary unique λ > 0 σ ess = α > 0 QSD Natural V δ < δ n 0 X a 0 > 0, λ > 0 Entrance V V V V a 0 > 0 Exit-Min X V V V? Regular-Min X V V V? Regular-Max X V V?? Min=minimal process, Max=maximal process, QSD=quasi-stationary distribution σ ess = essential spectrum. Decay rate: 1 λ = lim t t log p ij(t) p ij ( ) and uniform decay rate: 1 α = lim t t log sup i p ij (t) p ij ( ). j

12 Main theorem Theorem?=V.

13 How to find a QSD? Let us take as an example when a 0 > 0, R = and S. Consider Qg(x) = xg(x), where g(x) = (g i (x), i 0, g 0 (x) = 1, g i (0) = 1 and g 1 (x) = 0. Then a i g i 1 (a i + b i )g i + b i g i+1 = xg i, i 0; by letting u i (x) = µ i g i (x), we obtain The difficulty is to prove that i 0 uq = xu. u i (x) = a 0 x.

14 van Doorn s theorem (1991) Assume a 0 > 0 and R =. Let 1 λ = lim t t log p ij(t). Theorem (1) If S = and λ > 0, then for every 0 < x λ, there is a QSD. (2) If S <, then there is exactly one QSD at x = λ. The case when S = and x = λ > 0 is due to Good (1968). E.A. van Doorn (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab. 23 P. Good (1968). The limiting behavior of transient birth and death processes conditioned on survival. J. Austral. Math. Soc. 8

15 Our aim Extend the definition of QSD; Work out the QSD for the minimal process for exit or regular boundaries!

16 How Duality Spectral theory Boundary theory Fastest strong stationary time

17 Exit boundary First, we deal with the minimal process with exit boundary, that is R <, S =.

18 Duality Assume that a 0 = 0 and R <, S =. Let ã i = b i, bi = a i+1, i 0. There is a kind of dual relationship between Q and Q. Let ν 0 ν 1 ν 2 ν 1 ν 2 M = ν Then M Q = QM. Thus Q is such that ã 0 > 0 and R =, S <. More Q and Q has the same spectrum.

19 Boundedness of eigenfunction for Q From the above relation, we have µ i g i (λ) == [ g i (λ) g i 1 (λ)] = lim g i (λ). i i 0 i 0 If we can prove that then this gives QSD. g (λ) = lim i g i (λ) <,

20 Spectral representation for g i (λ) Lemma Assume ã 0 > 0 and R =, S <. Then Let X = ( X t, t 0) be the Q-process and τ 0 := inf{t > 0 : Xt = 0} be the hitting time of state 0. Then g i (λ) = E i e λ τ 0. Lemma Let λ i,k, k 1 be the eigenvalues of the process X t absorbed at i from right. The by Gong-M.-Zhang (2012), we have g (λ) := lim i g i (λ) = λ 0,k k=1 λ <. 0,k λ

21 Spectrum But λ = λ = λ 1,1 < λ 0,1, by Lemma (Hart-Martínez-San Martín (2003)) If à = and λ i,1 = λ j,1 for some i > j 1, then λ i,1 = λ i+1,1. But we can prove that λ i,1 as i.

22 Regular boundary Next, we deal with the minimal process with regular boundary, that is R <, S <. NO duality: S <, R <. Use approximation!

23 Regular: minimal process Let δ ν, ν 1 be the eigenvalues for minimal process, and δ ν (n), 1 ν n be the eigenvalues for Q (n) = b 0 b a 1 (a 1 + b 1 ) b a 2 (a 2 + b 2 ) a n 1 (a n 1 + b n 1 ). Then δ (n) ν δ ν, ν 1. More, by letting τ n be the hitting time to n, τ n ζ (explosion time).

24 Approximation For Q (n) -process (p (n) ij (t)), we have where As we have u (n) i = n 1 i=0 u (n) i µ i g (n) i n 1 i=0 µ ig (n) i u (n) i = g (n) i p (n) ij (t) = e δ(n) 1 t u (n) i, and g (n) i = E 0e δ (n) 1 τ n 1, E 0 e δ(n) 1 τ i ( µ i E 0 e δ(n) n 1 i=0 µ i 1 τ i ) 1 ) 1. (E 0 e δ(n) 1 τ i = E i e δ(n) 1 τ n 1.

25 FSST Let T (n 1) be the FSST for the process reflecting at n 1, then (E 0 e st (n 1)) 1 n 1 n 1 ( µ i = µ i E0 e sτ ) i 1. i=0 i=0 This equality comes from the boundary theory: ψ (n) ij (λ) = φ(n) ij (λ)i {j<n 1} + X(n) i (λ)x (n) j (λ)µ j λ n 1 k=0 µ kx (n) (λ), where X (n) i (λ) = 1 λe i e λτ n 1. Thus u (n) i = µ ie 0 e δ (n) 1 T (n 1) E 0 e δ(n) 1 τ i n 1 i=0 µ i. k

26 Spectral representation Let 0 = λ (n) 0, λ(n) ν, 1 ν n 1 be the eigenvalues for the process reflecting at n 1. Then E 0 e δ(n) 1 T (n 1) = n ν=1 λ (n) ν λ (n) ν δ (n) 1 If δ 1 = lim n δ (n) 1 <λ 1 = lim n λ (n) 1, then by letting n, we can get u i = lim n u(n) i = π i ν=1 λ ν λ ν δ 1 / i ν=1. λ (i) ν λ (i) ν δ 1, and u i = 1. i=0

27 Separation property for eigenvalues Assume a 0 = 0 and R <, S <. Let For ã i = b i, bi = a i+1, i 0. Qg(x) = xg(x) and Q g(x) = x g(x) with usual convention, g i (x) g (x) and g i (x) g (x). Then the zeros of g (x) and g (x) have the separation property We can prove that 0 < ξ 1 < ξ 1 < ξ 2 < ξ 2 <.... δ ν = ξ ν and λ ν = ξ ν, ν 1 T.S. Chihara (1978). An Introduction to Orthogonal Polynomials. New York: Gordon and Breach.

28 References Fill, J.A. (2009) The passage time distribution for a birth and death chain: strong stationary duality gives a first stochastic proof. J. Theor. Probab. Gong, Y., Mao, Y.-H., Zhang, C. (2012) Hitting time Distributions for denumerable birth and death processes. J. Theor. Probab. A. Hart, S. Martínez, J. San Martín (2003). The λ-classification of continuous-time birth-and-death process Adv. Appl. Probab. Yang, X.-Q. (1990) The construction theory of denumerable Markov processes.

29 QED THANKS!