Linear-fractional branching processes with countably many types

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1 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 Supported by the Swedish Research Council grant 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 Associated branching renewal system Example R-positively recurrent case References

2 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

3 General linear-fractional generating functions E(s Z ) = h 0 + h 1 s 1 +h 2 s m m(g 1 s 1 +g 2 s ) Such a multivariate linear-fractional distribution can be viewed as a multivariate geometric distribution modified at zero: Z = X + (Y 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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 = , g n = P(L = n). 18

19 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 ( 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 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

20 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

21 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

22 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

23 References References [1] Jagers, P. and Sagitov, S. (2008) General branching processes in discrete time as random trees. Bernoulli 14, [2] Joffe A. and G. Letac (2006) Multitype linear fractional branching processes. J. Appl. Probab. 43, [3] Hoppe FM. (1997) Coupling and the Non-degeneracy of the Limit in Some Plasmid Reproduction Models. Theor. Pop. Biol. 52, [4] Kesten, H. (1989) Supercritical branching processes with countably many types and the sizes of random cantor sets. In T.W. Anderson, K.B. Athreya and D. Iglehart (eds), Probability,Statistics and Mathematics Papers in Honor of Samuel Karlin, pp New York:Academic Press. [5] Lambert, A. (2010) The contour of splitting trees is a Levy process. Ann. Probab. 38,

24 References [6] Moy,S.-T. C. (1967) Extensions of a limit theorem of Everett Ulam and Harrison multi-type branching processes to a branching process with countably many types. Ann. Math. Statist [7] Pollak, E. (1974) Survival probabilities and extinction times for some multitype branching processes. Adv. Appl. Prob. 6, [8] Seneta, E. (2006). Non-negative matrices and Markov chains. Springer Series in Statistics No. 21, Springer, New-York. [9] Seneta E, Tavare S. (1983) Some stochastic models for plasmid copy number. Theor. Pop. Biol. 23,