Quasi-Stationary Distributions in Linear Birth and Death Processes

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1 Quasi-Stationary Distributions in Linear Birth and Death Processes Wenbo Liu & Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University Hunan , China Abstract The quasi-stationary distributions {a j } for a linear birth and death process is determined by two methods. The first method obtains our desired results by computing directly while the second method bases on the relationship between {a j } and its limiting of probability generating function. In addition, we also obtain the stationary distribution for a linear birth, death and immigration process with the second method. Keywords: Linear birth and death process, Quasi-stationary distributions, Probability generating function, Stationary distribution 1. Introduction We are interested in the long-term behavior of absorbing Marov processes. The processes will be absorbed eventually, however, they often persist for extended periods of time and in fact appear to reach an equilibrium before reaching the absorbing state. The ey to analyze the behavior of the processes before they die out is to simplify condition on their not having been absorbed, which leads us to consider quasi-stationary distributions and limiting conditional distributions, rather than the classical stationary and limiting distributions for irreducible processes. Quasi-stationary distributions and limiting conditional distributions for Marov processes have been studied by Vere-Jones (1969), Flaspohler (1974), Pollett (1988), Darlington and Pollett (000) and Moler et.al (000) in the general setting of absorbing continuous-time denumerable Marov chains. As we all now, birth and death processes are the most important class of Marov processes and their relatively simple structure can mae us study them with a rather extensive analysis. Quasi-stationary distribution for birth and death processes have been studied by Cavender (1978), Pollett (1988), Van Doorn (1991), Schoutens (000) and Clancy and Pollett (003). Again, the linear birth and death processes are the most important class of birth and death processes. In this paper we will study the quasi-stationary distributions in the setting of a linear birth and death process on a semi-infinite lattice of integers, the finite boundary point being an absorbing state which is reached with certainly. In this article, we will calculate the quasi-stationary distributions for a linear birth and death process by two different methods. Meanwhile, we also obtain the stationary distribution for a linear birth, death and immigration process with our methods.. Preliminaries Let E be the set {0,1,,... } of non-negative integers, and let {λ n, n 0} and {μ n, n 0} be sequences of non-negative numbers. A continuous-time Marov Chain {X(t), t 0} having state E and q-matrix given by λ i if j = i + 1, i 0 μ q ij = i if j = i 1, i 1 (1) (λ i + μ i ) if j = i, i 0 0 otherwise, is called a birth and death process on E, with birth coefficients λ n, n 0, and death coefficient μ n, n 0. Suppose λ 0 = μ 0 = 0, λ n > 0, μ n > 0, n 1, then Q will be conservative and 0 is an absorbing state and C={1,,... } is irreducible for the minimal Q-function, F, and hence for any Q-function. As usual we define the potential coefficients π = {π i, i C} and π 1 = 1, π n = λ 1λ λ n 1 μ μ 3 μ n, n () The transition probability {P ij (t), i, j E, t 0} of a birth and death process satisfies the following conditions P ij (t) 1, (3) j P ij (t) 0, (4) Published by Canadian Center of Science and Education 7

2 P ij (0) = δ ij, (5) P ij (s + t) = P i (s)p j (t) (6) P ij (t) = q i P j (t), (7) P ij (t) = P i (t)q j, (8) for i, j E and t, s 0, δ ij is the Kronecer delta. The equations (6), (7) and (8) are called the Chapman- Kolmogorov equations, the bacward equations(be) and the forward equations(fe), respectively. We now, each Q- function, P ij (t), satisfies (7) since Q is conservative but might not satisfy (8). We define T = inf{t 0:X(t) = 0} (9) the absorption (hitting) time at 0. We shall only be interested in processes for which E i T < for all i 1. We will assume the eventual absorption at 0 is certain, which is equivalent to assuming 1 =. (10) λ n π n n=0 Imposing (10) implies that the condition 1 n π i = (11) λ n=0 n π n i=0 is satisfied, and as a consequence (see for example Reuter (1957), Kemperman (196)), the transition probabilities P ij (t) constitute, under some obvious side conditions, the unique solution of the equation (7) with initial conditions (5). Also, imposing (10) (and hence (11)) implies that the process is non-explosive and therefore honest (see for example Reuter (1957), Kemperman (196)), that is P ij (t) = 1, i E, t 0. (1) Given a transition function P ij (t), a set {u i, i E} of non-negative numbers such that u i P ij (t) = u j for all j E and t 0, i E j=0 is called an invariant measure for P ij (t). If, furthermore, i E u i = 1, then u = {u i, i E} is called a stationary distribution. Suppose that P ij (t) is an irreducible transition function, then the limits u j = lim t P ij (t) exist and are independent of i for all j E. The set {u i, i E} of numbers is an invariant measure and either (a) (b) u j = 0 for all j E, or u j > 0 for all j E and u j = 1. (13) j E We call A = {a j } the limiting conditional distribution (LCD) if for each j 1 a j = lim P i (X(t) = j T > t) and a j = 1 (14) t provided the limits exists for some (and hence for all) i. We say a measure A = {a j } on C is a quasi-stationary distribution (QSD) if a i = 1 and for each j 1 and t > 0 a j = P A (X(t) = j T > t). (15) For the above concepts on LCD and QSD, we can refer to Paes (1995) or Ferrari et al. (1995). From Van Doorn (1991) and Vere-Jones (1969) we now that any QSD is a (proper) LCD and any (proper) LCD is a QSD. In this paper, we focus on the process which starts from an single state i and derive the corresponding results. 8 ISSN E-ISSN

3 Our study will use three particular families of distributions: geometric, Pascal and Poisson. Their probabilities generating functions are respectively: (1 p)s g(s) =, for the geometric distribution with parameter p (0 < p < 1) (16) 1 sp g(s) = e ρ(s 1), for the Poisson distribution with parameter ρ (17) ( ) r 1 p g(s) =, for the Pascal distribution with parameters r and p (r > 0, 0 < p < 1) (18) 1 sp 3. Quasi-stationary distributions of birth and death processes In this section we mainly study the quasi-stationary distributions for a linear birth and death process and its generator determined by λ i = iλ, μ i = iμ, i 0, (19) 0 <λ<μ. The constants λ and μ are called the birth and death rates. It is clear that (10) and (11) are all satisfied. The properties of this process have been established in detail in Karlin and McGregor (1958). We now that Van Doorn (1991) obtained the quasi-stationary distributions by Karlin and McGregor s spectral representation of the transition probabilities of the process. However, the smallest point in the support of ψ is a slight difficult to be calculated. Here, we will calculate the quasi-stationary distributions with two other ways. The two methods base on the specificity that we have now the transition function of the process. The explicit form for P ij (t) follows from Anderson (1991) or Karlin and McGregor (1958): i j = min(i, j) and P ij (t) = γ (1 σ j γ )i+ j i j ( ) i 1 (1 σ) i+ j ( 1) σ γ 1 σ 1 σ γ 1 σ γ σ = γe (μ λ)t,γ= λ μ, (a) = { a(a + 1) (a + ) 1 1 = 0, (i) j ( j )!, (0) and its probability generating function P ij (t)z j = j=0 ( ) i rμ 1, i 1, z 1, (1) rλ 1 ( ) 1 z r = e (μ λ)t. μ λz 3.1 The first method In this section we obtain QSD (15) by computing LCD (14). Obviously, when the process starts from an single state i, (15) and (14) are equivalent. Aslo (14) can be written as P ij (t) a j = lim t P i (T > t). () First, we introduce the first method, that is, the quasi-stationary distributions of the process is obtained directly by taing the limit for (). Let β = e (μ λ)t, then the equation (0) becomes P ij (t) = γ j (1 β) i+ j (1 γβ) i+ j Again, since T is absorption time and hence i j ( i )( 1) ( γ γ β β + γβ γ(1 β) ) (i) j ( j )!. (3) Then P i (T > t) = 1 P i (T t) = 1 P i0 (t) = 1 P ij (t) j 1 P i0 (t) = 1 j (1 β)i+ (1 γβ) i (1 β) i γ (1 γβ) j i j ( i (1 β)i (1 γβ) i. (4) ( γ γ )( 1) β β + γβ ) (i) j γ(1 β) ( j )!. (5) Published by Canadian Center of Science and Education 9

4 Obviously, the expressions is 0 0 type when tae the limit, we use L Hôspital s Rule to tacle it. After a series of operations, we have 1 i j ( ) i (5) i(1 γ) γ j (i + j jγ) ( 1) (i) j i j ( ) i ( j )! + γ j 1 (γ 1) ( 1) (i) j ( j )! = Let i j, then, setting γ j 1 i(γ 1) A j = i j ( ) i γ j 1 i(γ 1) ( 1) (i) j [ ] (i + j jγ)γ + (γ 1) as t. ( j )! j ( ) i ( 1) (i) j [ ] (i + j jγ)γ + (γ 1). (6) ( j )! By (6), it is not hard to find A 1 = 1 γ, A = (1 γ)γ, A 3 = (1 γ)γ, recursively, we conjecture A j = (1 γ)γ j 1, j 1. Below, we will use the method of mathematical induction to testify. Suppose A j = (1 γ)γ j 1, then, A j+1 = (1 γ)γ j. Actually, from (6) and assumption, we see easily j ( ) i ( 1) (i) j [ ] (i + j jγ)γ + (γ 1) = i(1 γ) (7) ( j )! Obviously, the left of (7) is independent of j, and so A j+1 = γ j i(γ 1) j+1 ( ) i ( 1) (i) j+1 [ ] (i + j + 1 ( j + 1)γ)γ + (γ 1) = ( j + 1 )! γ j i(γ 1) ( i)(1 γ) = (1 γ)γ j (8) Therefore, we obtain a j = (1 γ)γ j 1 = (1 λ μ )(λ μ ) j 1, j 1. (9) The result is in eeping with Van Doorn (1991). Remar: The above proof course only considers the situation i j, actually, because of the feature of (i) j ( j )!, we now the result is invariable when i j. 3. The second method In this section we will introduce the second method and the idea partly refers to Karlin and Iavaré (198). From (1), we see that [ ] i 1 β + (β γ)z, i 1, z 1, (30) 1 βγ zγ(1 β) β = e (μ λ)t,γ = λ μ. To yield our result more conveniently,we induct the following lemma. Lemma 3.1. lim t β 1 (G i (z, t) 1) = ia(z), i 1, (1 γ)(z 1) A(z) =, 0 z 1. 1 zγ Proof: From (30), we have [ ] i (1 γ)(z 1)β 1 + = 1 βγ zγ + βzγ i ( )[ ] i (1 γ)(z 1)β 1 βγ zγ + βzγ and so, and thus, G i (z, t) 1 = i =1 ( )[ ] i (1 γ)(z 1)β, 1 βγ zγ + βzγ i(1 γ)(z 1) i(1 γ)(z 1) lim t β 1 (G i (z, t) 1) = lim = = ia(z). t 1 βγ zγ + βzγ 1 zγ 30 ISSN E-ISSN

5 Moreover, from (1), we have P i (T > t) = P i (X(t) = j) = G i (1, t) G i (0, t). To establish (14), we use lemma 3.1 to see that for 0 z 1, j=1 P i (X(t) = j T > t)z j = j=1 G i (z, t) G i (0, t) G i (1, t) G i (0, t) = (G i(z, t) 1)β 1 (G i (0, t) 1)β 1 (G i (1, t) 1)β 1 (G i (0, t) 1)β 1 A(z) A(0) A(1) A(0) = z zγ 1 zγ as t, and hence, from (16) we see that a j = lim t P i (X(t) = j T > t) is geometric distribution with parameter γ = λ μ,sowe obtain a j = (1 λ μ )(λ μ ) j 1, j 1. (31) Obviously, (31) is in accord with (9) and Van Doorn (1991). Certainly, some authors led to the same conclusion by other techniques, we may refer to Cavender (1978), Pollett (1988) and Van Doorn (1991). 4. Stationary distribution of linear birth,death and immigration processes In this section, we will adopt the method of section (3.) to compute stationary distribution for a linear birth, death and immigration process defined by λ i = a + iλ, μ i = iμ, i 0, (3) a > 0,λ 0,μ 0. The constants a,λ,μare called the immigration, birth, and death rates. Obviously, the (10) and (11) are all satisfied. And therefore P i (T > t) = P i (X(t) = j) = G i (1, t) = 1. j=0 We discuss the following several situations to calculate the stationary distribution of the processes. Case 1. 0 <λ<μ. The probability generating function [ ] i [ ] δ 1 β z(γ β) 1 γ, i 0, (33) 1 βγ γz(1 β) 1 βγ γz(1 β) β = e (μ λ)t,γ= λ μ,δ= a λ. From (33), we have ( ) δ 1 γ G i (z, t) as t, 1 δz and from (18), we now u j = lim t P ij (t) is Pascal distribution with parameters γ and δ, hence u j = C δ 1 j+δ 1 γ j (1 γ) δ = (1 γ) δ γ j (δ) j, j 0. (34) j! Case. 0 = λ<μ. The probability generating function is (1 β + zβ) i e δ(1 β)(z 1), (35) β = e μt,δ= a μ. So, G i (1, t) e δ(z 1) as t. (36) Published by Canadian Center of Science and Education 31

6 From (36), we now that u j is Poisson distribution with parameter δ, and so u j = 1 j! δ j e δ, j 0. Case 3. 0 <λ<μ, 0 = μ<λ, 0 <λ= μ. For these three conditions, we all have G i (z, t) 0, (37) and so, u j = lim P ij (t) = 0. (38) t From (38) and (13a), we now u j is invariant measure. References Anderson, W. J. (1991). Continuous-Time Marov Chains: An Applications-Oriented Approach, Springer, New Yor. Cavender, J. A. (1978). Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob, 10, Clancy, D. and Pollett, P. K. (003). A note on quasi-stationary distributions of birth-death processes and the sis logistic epidemic. J. Appl. Prob, 40, Darlington S. J. and Pollett, P. K. (000). Quasistationarity in continuous-time Marov chains absorption is not certain. J. Appl. Prob, 37, Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob, 3, Flaspohler, D. C. (1974). Quai-stationary distributions in absorbing continuous-time denumerable Marov chains. Ann. Inst. Statist. Math, 6, Karlin, S. and McGregor, J. L. (1958). Linear growth, birth and death processes. J. Math. Mech. (now Indiana Univ. Math. J), 7, Karlin, S. and Tavare, S. (198). Linear birth and death processes with illing. J. Appl. Prob, 19, Kemperman, J. H. B. (196). An analytical approach to the differential equations of the birth-and-death process. Michigan Math. J, 9, Moler J. A., Plo F and San Miguel, M. (000). Minimal quasi-stationary distributions under null R-recurrence. Test, 9, Paes, A. G. (1995). Quasi-stationary laws for Marov processes: examples of an always proximate absorbing state. Adv. Appl. Prob, 7, Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob, 0, Reuter, G. E. H. (1957). Denumerable Marov processes and the associated contraction semi-groups on l. Acta. Math, 91, Schoutens W. (000). Birth and death processes, orthogonal polynomials and limiting conditional distributions. Math. Sci, 5, Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob, 3, Vere-Jones, D. (1969). Some limit theorems for evanescent processes, Austral. J. Statist, 11, ISSN E-ISSN