Decomposition of supercritical branching processes with countably many types

Transcription

1 Pomorie 26 June 2012 Serik Sagitov and Altynay Shaimerdenova 20 slides Chalmers University and University of Gothenburg Al-Farabi Kazakh National University Decomposition of supercritical branching processes with countably many types Work in progress Bienayme-Galton-Watson processes The dual reproduction law Harris-Sevastyanov transformation BGW-process with countably many types Multi-type linear-fractional case Explicit formulae for the dual law and HS-decomposition Related projects 1

2 Bienayme-Galton-Watson processes f(q) = q Consider a single type BGW-process {Z (n) } n=0 with generating functions (usually we assume Z (0) = 1) f(s) = E(s Z(1) ) = p k s k, k=0 f (n) (s) = E(s Z(n) ) = f(... (f(s)...). The line of descent of the progenitor particle is either finite with probability q = P(Z (n) = 0 starting from some n) or infinite with probability 1 q = P(Z (n) > 0 for all n). 2

3 Bienayme-Galton-Watson processes Classification of the BGW-processes in terms of the mean offspring number M = f (1): subcritical M < 1 with q = 1, critical M = 1 with q = 1, supercritical M > 1 with q < 1. {q = 0} {p 0 = 0} One-type BGW-process = discrete time Markov CMJ-process. Particles vs individuals (generations vs years): individual = sequence of first-born particles. Here an individual has a Shifted Geometric (p 0 ) life length, yearly (except the moment of death) producing an independent number k of siblings with probability p k+1 (1 p 0 ) 1, k 0. 3

4 j=0 The dual reproduction law BGW-process {Z (n) } n=0 is a MC with transition probabilities ( i P (n) ij s j = f (s)) (n). Since j=0 P (n) ij q j = q i we get another set of transition probabilities ˆP (n) ij = P (n) ij q j i again possessing the branching property ( i ˆP (n) ij s j = ˆf (s)) (n), where j=0 ˆf (n) (s) = f (n) (sq). q 4

5 The dual reproduction law The dual BGW-process is subcritical with mean ˆM = ˆf (q) < 1 and the dual reproduction law ˆf(s) = f(sq), ˆp k = p k q k 1, k 0. q The dual subcritical BGW-process is distributed as the original supercritical BGW-process conditioned on extinction: P(Z (n) = j Z (0) = i, Z ( ) = 0) = P(Z(n) = j, Z ( ) = 0 Z (0) = i) P(Z ( ) = 0 Z (0) = i) = q i P(Z ( ) = 0 Z (n) = j)p (n) ij = q j i P (n) ij = ˆP (n) ij. 5

6 The dual reproduction law Elect. Comm. in Probab. 13 (2008), ELECTRONIC COMMUNICATIONS in PROBABILITY GENERAL BRANCHING PROCESSES CONDITIONED ON EXTINC- TION ARE STILL BRANCHING PROCESSES PETER JAGERS 1 Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE Gothenburg, Sweden jagers@chalmers.se ANDREAS N. LAGERÅS 2 Department of Mathematics, Stockholm University, SE Stockholm, Sweden andreas@math.su.se Submitted April 28, 2008, accepted in final form October 9, AMS 2000 Subject classification: 60J80 Keywords: Supercritical, Subcritical, Extinction, Multi-type branching process, General branching process, Crump-Mode-Jagers process. Abstract It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction. 1 Introduction The theory of branching processes was born out of Galton s famous family extinction problem. Later, interest turned to populations not dying out and their growth and stabilisation. In more recent years, extinction has retaken a place in the foreground, for reasons from both conservation and evolutionary biology. The time and path to extinction of subcritical general populations was studied in [4]. Here, time structure is crucial, and life spans and varying bearing ages cannot be condensed into simple, generation 6 counting Bienaymé-Galton-Watson processes. Thus, the question arises whether (non-critical) general branching populations (also known as Crump-

7 Harris-Sevastyanov transformation Two graphs use different scales to demonstrate duality between the subcritical and supercritical branching cases. Two parts of the curve - two transformations of the branching process. Lower left = dual process, upper right = HS-transformation. 7

8 Harris-Sevastyanov transformation Harris-Sevastyanov transformation for a supercritical BGW-process f(s(1 q) + q) q f(s) =, M = M, 1 q ( ) i p 0 = 0, p k = p i q i k (1 q) k 1, k i=k Looking into the future distinguish between two subtypes of particles: subtype 1 particles with infinite lines of descent building the skeleton of the genealogical tree, subtype 2 particles with finite lines of descent. These two subtypes form a decomposable two-type BGW-process with generating functions for the reproduction law f 1 (s, t) = f(s(1 q) + tq) f(tq) 1 q, f 2 (t) = f(tq). q 8

9 Harris-Sevastyanov transformation E(sZ(n) E(s Z(n) 1 t Z(n) 2 Z ( ) 1 t Z(n) 2 1 {Z >0}) > 0) = ( ) P (Z ( ) > 0) (n) = E(sZ 1 t Z(n) 2 ) E(t Z(n) 2 1 {Z =0}) ( ) 1 q = f (n) (s(1 q) + tq) f (n) (tq) 1 q Back to the original generating function: f(s) = (1 q)f 1 (s, s) + qf 2 (s). The marginal reproduction laws f(s) = f 1 (s, 1), ˆf(s) = f 2 (s). = f (n) 1 (s, t) 9

10 Single type linear-fractional case Linear-fractional reproduction law f(s) = h 0 + h 1 s 1 + m ms h 0 + h 1 = 1 h 0 = p 0 geometric number of offspring beyond the first one with mean m M = h 1 (1 + m), and if M > 1, then q = h 0 (1 + m 1 ). Dual reproduction law is again linear-fractional ˆf(s) = ĥ0 + ĥ 1 s 1 + ˆm ˆms, ĥ0 = m m + 1, ˆm = h 0/h 1, ˆM = 1/M. HS-transformation corresponds to a shifted geometric distribution s f(s) = 1 + m ms, m = m(1 q) = M 1, M = M. 10

11 Single type linear-fractional case Reproduction law of the subtype with infinite lines of descent f 1 (s, t) = f(s(1 q) + tq) f(tq) 1 q ( = h 1 1 q = s(1 q) + tq 1 + m m(s(1 q) + tq) tq s 1 + m m(s(1 q) + tq) where we see three independent components ˆm ˆmt one particle of type 1 (the infinite lineage), a bivariate geometric distribution, a geometric distribution for type 2 particles. 1 + m mtq Even though both marginal distributions f 1 (s, 1) and f 2 (s) are linear-fractional, the two type process is not a LF process. ) 11

12 BGW-process with countably many types BGW-process with countably many types Z (n) = (Z (n) 1, Z (n) 2,...), n = 0, 1, 2,..., where Z (n) i is the number of particles of type i existing at time n. We use the following vector notation: x = (x 1, x 2...), 1 = (1, 1...), e i = (1 {i=1}, 1 {i=2},...), xy = (x 1 y 1, x 2 y 2,...), x/y = (x 1 /y 1, x 2 /y 2,...), x y = x y 1 1 xy Multivariate generating functions are iterations of f (n) i (s) = E(s Z(n) Z (0) = e i ), i = 1, 2,... f i (s) = E(s Z(1) Z (0) = e i ), i = 1, 2,... 12

13 BGW-process with countably many types Matrix of means M = (M ij ) i,j=1 and extinction probabilities q M ij = E(Z (1) j Z (0) = e i ), q i = P(Ext Z (0) = e i ). Decomposition of a supercritical BGW-process with countably many types: each type is decomposed in two subtypes infinite lines of descent f 1i (s, t) = f i(s(1 q) + tq) f i (tq) 1 q i, finite lines of descent f 2i (t) = f i(tq) q i. Positive recurrent case: there exists a Perron-Frobenius eigenvalue for M with positive eigenvectors v = stable type distribution, and u = vector of reproductive values of types, such that vm = ρv, Mu t = ρu t, vu t = v1 t = 1, ρ n M n u t v, n. 13

14 Multi-type linear-fractional case Linear-fractional BGW-process with countably many types j=1 f i (s) = h i0 + h ijs j 1 + m m j=1 g, i = 1, 2,... js j can be treated as a single-type CMJ-process. Individual life length L has a phase-type distribution governed by transitions H = (h ij ) i,j=1 and initial distribution g = (g 1, g 2,...): P(L > k) = gh k 1 t, k 0. Perron-Frobenius eigenvalue ρ satisfies mφ(ρ 1 ) = 1 where Φ(s) = s k P(L > k). k=1 Mean offspring number per individual µ = m(e(l) 1) = mφ(1). 14

15 Mean age at childbearing Multi-type linear-fractional case β = m k=1 kρ k P(L > k) = m k=1 kρ k gh k 1 t. Positive recurrent case when β <. Perron-Frobenius eigenvectors Mu t = ρu t, vm = ρv are given by u t = (1 + m)β 1 v = m 1 + m k=1 ρ k gh k. k=0 ρ k H k 1 t, Besides vu t = v1 t = 1 we have gu t = 1+m mβ. Moreover, if ρ > 1 q = 1 (ρ 1)(1 + m) 1 βu, gq t = 1 + m ρ. m 15

16 If ρ > 1, the dual reproduction law is linear-fractional ˆf i (s) = f i(sq) = q ĥi0 j=1 + ĥijs j i 1 + ˆm ˆm j=1 ĝjs, j where ĥ i0 = h i0 q i, ĥ ij = h ijq j q i ρ, Explicit formulae ˆm = 1 + m ρ, ĝ j = g jq j m ρ 1 + m ρ. Moreover, one can show that ˆρ = ρ 1, ˆβ = µ 1 ρ 1 and ˆP(L = n) = ˆβû = βu/q, ˆv = m 1 + m gh k q, k=0 m 1 + m β(ρ 1) P(L = n), ˆµ = (1 + m ρ)ρn m ρ. 16

17 Explicit formulae The reproduction law of the subtype with infinite lines of descent f 1i (s, t) = f i(s(1 q) + tq) f i (tq) 1 q i h ij s j = 1 + m m j=1 k=1 g k(s k (1 q k ) + t k q k ) ( ρh ij + where 1 ρ(h ij + mg j (q i h i0 )) h ij = 1 q j 1 q i (h ij + mg j (q i h i0 )). mg j k=1 h ikq k t k 1 + ˆm ˆm k=1 ĝkt k Again, as in the single-type case, there are three components but now with dependence. ) 17

18 Explicit formulae Harris-Sevastyanov transformation results in multivariate geometric distributions f h j=1 ij s j i (s) = 1 + m m j=1 g, js j where m = ρ 1, g j = m ρ 1 g j(1 q j ). Moreover, one can show that ρ = ρ, β = ρ ρ 1 and ũ = 1, ṽ = m ρ 1 k g(h + m(h1 t 1 t + q t )g) k (1 q). k=0 Mean age at childbearing is finite even if P(L = ) = 1. 18

19 Related projects With M.C.Serra Decomposed branching processes modeling sequences of mutations transforming subcritical (near critical) types to (slightly) supercritical. With V.A.Vatutin, E.Dyakonova, P.Jagers Decomposed branching processes in random environment exhibit unusual asymptotics of the annealed probability of survival. 19

20 Acknowledgements This work was supported by the Swedish Research Council grant Thank you Captain Nick! Goodbye Pomorie... 20