Branching processes and and their applications Badajoz, April 11-13, 2012 Serik Sagitov Chalmers University and University of Gothenburg Linear-fractional branching processes with countably many types http://arxiv.org/abs/1111.4689 Supported by the Swedish Research Council grant 621-2010-5623 General linear-fractional generating functions 2 3 LF branching processes with countably many types 4 6 Classification of branching processes with countably many types 7 9 Main result 10 13 Associated branching renewal system 14 17 Example 18 19 R-positively recurrent case 20 22 References 23 24 1
General linear-fractional generating functions Notation for vectors of possibly infinite dimension s = (s 1, s 2,...) h = (h 1, h 2,...) g = (g 1, g 2,...) 0 = (0, 0,...) Definition 1 Random vector Z has a linear-fractional distribution LF(h, g, m) if (h 0, h 1, h 2,...) is a probability distribution on {0, 1, 2,...} (g 1, g 2,...) is a probability distribution on {1, 2,...} m is a positive constant so that in particular E(s Z ) = h 0 + i=1 h is i 1 + m m i=1 g is i, (1) P(Z = 0) = h 0. 2
General linear-fractional generating functions E(s Z ) = h 0 + h 1 s 1 +h 2 s 2 +... 1+m m(g 1 s 1 +g 2 s 2 +...) Such a multivariate linear-fractional distribution can be viewed as a multivariate geometric distribution modified at zero: Z = X + (Y 1 +... + Y N ) 1 {X 0} (2) in terms of mutually independent random entities (X, N, Y 1, Y 2,...). Here vectors X and Y j have multivariate Bernoulli distributions P(X = e i ) = h i, i 1, P(X = 0) = h 0, P(Y j = e i ) = g i, i 1, j 1, and N is a geometric random variable with mean m. 3
Definition 2 LF branching processes with countably many types LF branching process with parameters (H, g, m) sub-stochastic matrix H with non-negative rows h 1, h 2,... probability distribution g on {1, 2,...} positive constant m if particles of type i reproduce according to a LF(h i, g, m) distribution, i = 1, 2,.... Remark Strong limitation on the reproduction law: parameters (g, m) are ignorant on mother s type. This is needed for iterations of the corresponding generating functions to be also linear-fractional. 4
LF branching processes with countably many types The matrix of mean offspring numbers for a LF branching process with parameters (H, g, m) is computed as M = H + mh1 t g, (3) where 1 t is the column version of 1 = (1, 1,...). It follows, H = M m 1 + m M1t g. (4) Proposition 1 Consider a linear-fractional branching process with parameters (H, g, m) starting from a type i particle. The vector of its n-th generation sizes has a LF(h (n) i, g (n), m (n) ) distribution with 5
LF branching processes with countably many types m (n) g (n) = mg(i + M + + M n 1 ), (5) m (n) = m n 1 j=0 gm j 1 t, (6) H (n) = M n m(n) 1 + m (n) Mn 1 t g (n), (7) where H (n) is the matrix with rows h (n) 1, h(n) 2,.... In particular, P(Z (n) 0) = (1 + m (n) ) 1 M n 1 t, (8) where P(Z (n) 0) is a vector with elements P(Z (n) 0 Z (0) = e i ). 6
Classification of BP with countably many types Given that M is irreducible and aperiodic and all powers M n = (m (n) ij ) i,j=1 are element-wise finite, which is always true in the linear-fractional case, the asymptotical behavior of these powers is described by the Perron-Frobenius theory for countable matrices. Due to Theorem 6.1 in [Seneta] all elements of the matrix power series M(s) = s n M n (9) n=0 have a common convergence radius 0 R <. Furthermore, one of the two alternatives holds: R-transient case: R-recurrent case: n=0 m(n) ii n=0 m(n) ii R n < for all i R n = for all i 7
Classification of BP with countably many types In the R-recurrent case there exist unique up to constant multipliers positive vectors u and v such that RMu t = u t, RvM = v. Renormalization Rv j m ji /v i transforms M into a stochastic matrix The R-recurrent case is further divided in two sub-cases: R-null case, when vu t = R-positive case, when vu t < In the R-null case (and clearly also in the R-transient case) R n m (n) ij 0, i, j = 1, 2,... 8
Classification of BP with countably many types In the R-positive case one can scale eigenvectors so that vu t = 1 and R n m (n) ij u i v j, i, j = 1, 2,.... These suggest a double classification of BP with a mean matrix M usual classification based on the value of the growth rate ρ = 1/R: subcritical ρ < 1, critical ρ = 1, or supercritical ρ > 1 an additional classification due to recurrence property of the branching process with respect to the infinite type space. Definition 3 A branching process will be called R-transient, R-recurrent, R-null recurrent, or R-positively recurrent depending on the respective property of its matrix of mean offspring numbers M. 9
Consider a LF branching process with parameters (H, g, m). The monotone function f(s) = Main result s n gh n 1 t (10) n=1 grows from zero to infinity as s goes from zero to infinity. Therefore equation mf(r) = 1 (11) always has a unique positive solution R. Observe that mf(rs) is the generating function for a probability distribution with mean β = m nr n gh n 1 t. (12) n=1 10
Main result Theorem 1 Consider a LF branching process with parameters (H, g, m) assuming that its mean matrix M is irreducible and aperiodic. The convergence parameter R of M is defined by (11) and M is always R-recurrent If β =, then M is R-null recurrent. If β <, then M is R-positively recurrent and R n M n u t v, n. (13) 11
Main result Here positive vectors u t = (1 + m)β 1 v = m 1 + m k=1 R k H k 1 t, (14) R k gh k (15) k=0 are such that vu t = v1 t = 1 and gu t = 1+m mβ. Remark Formulae (14) and (15) are new even in the case of finitely many types. 12
The proof is based on M(s) = H(s) + Main result m 1 mf(s) H(s)1t g(h(s) + I), (16) where M(s) = n=0 sn M n and H(s) = n=0 sn H n. Apply the renewal theorem taken from Chapter XIII.4 in [Feller-2]: Proposition 2 Let A(s) = n=0 a ns n be a probability generating function and B(s) = n=0 b ns n is a generating function for a non-negative sequence so that A(1) = 1 while B(1) (0, ). Then the non-negative sequence defined by t n s n = B(s) 1 A(s) is such that t n B(1) A (1) n=0 as n. 13
Associated branching renewal system If the initial particle of the LF branching process with parameters (H, g, m) has distribution g P(Z (0) = e i ) = g i, i = 1, 2,..., (17) then we get a branching renewal property implying an alternative description of this reproduction system in terms of individuals defined as sequences of first-born particles. 14
Associated branching renewal system Individual s life length L has a discrete phase-type distribution governed by the pair (H, g). Its tail probabilities are computed as P(L > n) = gh n 1 t, n 0, (18) allowing to express the generating function (10) in the form f(s) = s n P(L > n). (19) n=1 Clearly, and R is found from f(s) = 1 E(sL ) 1 s 1 E(R L ) = R 1 + m m 1 m. 15
Associated branching renewal system The life length has always a positive mean (finite or infinite) λ := E(L) = 1 + f(1). (20) Every year during its life (except the last year) the individual produces a geometric number of offspring with mean m. Thus the mean total offspring number per individual is µ = m(λ 1) = mf(1). (21) Depending on µ < 1, µ = 1, or µ > 1, the branching process should be called sub-, critical, or supercritical resulting exactly in the same classification defined in terms of the convergence parameter R. 16
Associated branching renewal system Total population sizes Z (n) = Z (n) 1 t form a discrete time CMJ process. There are many triplets (H, g, m) resulting in the same CMJ process with the same values of R and β. Mean age at childbearing β = k=1 kmrk P(L > k) The difference between such related branching processes is in the labeling rules for particles and will be seen in their Perron-Frobenius eigenvectors. Remark Making births very rare events we can approximate the linear-fractional CMJ processes by the continuous time CMJ processes with Poisson reproduction, recently studied by Lambert. 17
There is a particular labeling system allowing for more explicit formulae for the Perron-Frobenius eigenvectors. Example We relabel individuals by looking into the future and assigning label i to individuals whose remaining life length equals i for i = 1, 2,.... For example, an individual with label 1 must die in the next year. The relabeled process is again a linear-fractional branching process with a modified triplet of parameters ( H, g, m), where H = 0 0 0... 0 0... 1 0 0... 0 0... 0 1 0... 0 0.............. 0 0 0... 0 1.............., g n = P(L = n). 18
Observe that the corresponding mean matrix M is not fully irreducible, since the type 1 is final type. Example However, the Perron-Frobenius theorem can be extended to this case as well and the corresponding Perron-Frobenius eigenvectors could be computed using (14) and (15) as It follows ů i = β 1 (1 +... + R i 1 ), i = 1, 2,..., v j = m(1 + m) 1 E(R L j ; L j), j = 1, 2,.... m (n) ij ρ n+j (ρ 1 +... + ρ i )m(1 + m) 1 β 1 E(ρ L ; L j), thus in the critical case m (n) ij iλ 1 β 1 P(L j). 19
R-positively recurrent case Proposition 3 In the subcritical case when ρ < 1, or equivalently µ < 1 we have P(Z (n) 0) ρ n (1 + m) 1 (1 µ)u, and a convergence in distribution [ Z (n) Z (n) 0, Z (0) = e i ] d Y. Limit Y has a LF( h, g, m) distribution with parameters h = (1 + m)(1 µ) 1 v (µ + m)(1 µ) 1 g, (22) g = λ 1 (1 µ)g(i M) 1 (23) being independent of the initial type. In particular, for Y = Y1 t P(Y = k) = m k 1 (1 + m) k, k = 1, 2,.... (24) 20
R-positively recurrent case Proposition 4 In the critical case when ρ = 1 we have P(Z (n) 0) n 1 (1 + m) 1 βu, and [ n 1 Z (n) Z (n) 0, Z (0) ] d = e i Xv, where X is exponentially distributed with mean (1 + m)β 1. If furthermore a vector w is such that vw t = 0, then [ n 1/2 Z (n) w t Z (n) 0, Z (0) = e i ] d Y (1 + m)β 1 w 2 v t, (25) where w 2 = (w 2 1, w 2 2,...) and Y has a Laplace distribution with parameter 1. 21
R-positively recurrent case Proposition 5 In the supercritical case when ρ > 1 we have P(Z (n) 0) (ρ 1)(1 + m) 1 βu, and [ ρ n Z (n) Z (n) 0, Z (0) ] d = e i Xv, where X is exponentially distributed with mean (1 + m)(ρ 1) 1 β 1. If furthermore a vector w is such that vw t = 0, then [ ρ n/2 Z (n) w t Z (n) 0, Z (0) = e i ] d Y (1 + m)(ρ 1) 1 β 1 w 2 v t, where w 2 = (w 2 1, w 2 2,...) and Y has a Laplace distribution with parameter 1. 22
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