Branching, smoothing and endogeny

Transcription

1 Branching, smoothing and endogeny John D. Biggins, School of Mathematics and Statistics, University of Sheffield, Sheffield, S7 3RH, UK September 2011, Paris Joint work with Gerold Alsmeyer and Matthias Meiners To appear: Ann Probab

2 Aldous and Bandyopadhyay Ann Probab 2005 Recursive distributional equations Given: (ξ, N) where ξ = (ξ i, i = 1,..., N). (X i, i = 1,..., N) independent of these. A function g of these Get a new distribution by Y d = g (ξ, (X i, i = 1,..., N)) Transformation of distributions; could be iterated. Fixed point when Y has same distribution as the X s. Why? Fractals, algorithms,...

3 Aldous and Bandyopadhyay Ann Probab 2005 Recursive tree framework Use N as family size in a Galton-Watson process. Attach an independent copy of ξ to each person. Gives ((ξ i, N i ) i T). F (n) = σ{(ξ i, N i ) i n 1}; F = σ{(ξ i, N i ) i T}. Recursive tree process (RTP) {X i : i = n}, i.i.d given F (n) X i = g (ξ i, (X ii, i = 1,..., N i )) N.B. the distribution of X can change with generation RTP is invariant if (X i, i T) are identically distributed. Fixed point of RDE invariant RPT (with that marginal)

4 Aldous and Bandyopadhyay Ann Probab 2005 Definition (Endogony) Suppose you have an invariant RTP: (X i, i T) identically distributed, with then it is endogenous if X i = g (ξ i, (X ii, i = 1,..., N i )) i T. X F (= σ{(ξ i, N i ) i T}). no extra randomness a kind of stability

5 Aldous and Bandyopadhyay Ann Probab 2005 The linear case g (ξ, (X i, i = 1,..., N)) = Also called the smoothing transform Durrett and Liggett ZW 1983 and... N ξ i X i i=1 Now write T = ξ, and assume T i 0 i. Fixed point of smoothing transform gives invariant RTP X i = i T i,i X ii i

6 Example Simple example: P(N 1) = 1; EN (0, ) T i = 1/EN for i = 1,..., N; T i = 0 otherwise W the limit of the normed population size more generally number in generation (n + i ) born to i W i = lim n (EN) n Endogenous W i = i W ii EN

7 Questions When are there (non-zero) fixed points? [Literature on this] What do the fixed points look like? [Much literature on this: including Alsmeyer, B and Meiners] When does a fixed point correspond to an (invariant) endogenous RTP? [Simple case discussed in Aldous and Bandyopadhyay: their Open Problem 18].

8 Non-negative fixed points Confine attention to X 0 solutions: X d = i T i X i. Let then φ(t) = E exp( tx ) φ(t) = E i 1 φ(tt i ) When there is a solution say the associated invariant RTP is based on T.

9 Non-negative fixed points X d = i T i X i. Assume throughout 1 (for good reasons) EN > 1 (and, usually, P(N 1) = 1; ) (supercritical) P(T i {0, 1} i) < 1 (not degenerate) E T i = 1. (conservative) E (T i ) β > 1 for β < 1 ( drift down ) 1 conservative / drift down changed at end

10 Non-negative fixed points Let L i be the product of the values of T down the line of descent from to i. Then W (n) = L i = L i W (1) i i =n i =n 1 is a martingale. Its limit (W ) satisfies W i = i T i,i W ii. An endogenous (invariant) RTP and an interesting one if also P(W > 0) > 0.

11 Martingale, and other, non-trivial fixed points EW = 1 exactly when E i T i log T i (, 0) and EW (1) log + W (1) < or... or... Alsmeyer and Iksanov, Electron. J. Probab Now, by approximation, we can now get non-trivial fixed points when P(N < ) = 1 Liu Adv Appl Probab 1998 [General conditions for existence when P(N = ) > 0 is an open problem.]

12 Multiplicative martingales If a non-trivial solution exists, so there is a φ with φ(t) = E φ(tt i ) = E φ(tl i ), i then M (n) (t) = φ(tl i ) = i =n i =n i =n 1 is a bounded martingale with the limit: M (t) = M i (tl i ) i =n A disintegration of the fixed point equation. M (1) i (tl i )

13 Slow variation If E i T i log T i 0, or..., then (it can be shown that) 1 φ(t)/t is slowly varying at t = 0 and so log M (n) (t) = log φ(tl i ) i =n i =n(1 φ(tl i )) = i =n i =n (1 φ(tl i )) tl i tl i (1 φ(l i )) tl i = t log M (n) (1) L i i.e. log M (t) = t log M (1) = t lim n (1 φ(l i )) i =n

14 Endogenous RTP, identifying the variables Let Then and X i = log M i (1) = lim (1 φ(l i )). n i =n log M i (t) = t log M i (1) = tx i φ(t) = EM (t) = E exp(log M (t)) = E exp( tx ). Thus X i have the right distribution.

15 Endogenous RTP Recall X i = log M i (1). Recall M (t) = M i (tl i ) i =n X = log M (1) = i =n log M i (L i ) = i =n L i log M i (1) = i =n L i X i an endogenous RTP (should be for a general node, not just ).

16 No choice once endogenous Let Z i produce an endogenous invariant RTP for φ. ( E exp( tz ) F (n)) exp( tz ) ( endogenous) but also ( E exp( tz ) F (n)) = E exp t L i Z i F (n) i =n Hence Z = X. = E φ(tl i ) i =n = M (n) (t) exp( tx )

17 Endogeny is forced Let Z i produce an invariant RTP for φ. Let Y i be independent variables with Laplace transform φ. Let Y (n) = i =n L i Y i [ ( E exp ) F (n)] sz ty (n) = E exp s L i Z i t L i Y i F (n) i =n i =n = M (n) (t)m (n) (s) exp( (s + t)x ) i.e. (Z, Y (n) d ) (X, X )

18 Endogeny is forced (Aldous and Bandyopadhyay) Recall: R 1, R 2 i.i.d. 2var(R 1 ) = E(R 1 R 2 ) 2. Λ bounded function. Z, Y (n) indep given F (n). [ ] E 2 var(λ(z ) F (n) ) = E [ ( ) ] E(Λ(Z ) F (n) ) E(Λ(Y (n) 2 ) F (n) ) ( = E E(Λ(Z ) Λ(Y (n) ) F (n) ) ( ) E Λ(Z ) Λ(Y (n) 2 ) E (Λ(X ) Λ(X )) 2 = 0 ) 2 That is [ E Λ(Z ) F (n)] Λ(Z ) i.e. for every Λ, Λ(Z ) is F-measurable, as required.

19 Fixed points that are not necessarily Laplace transforms Decreasing f with f (t) = E i 1 f (tt i), f (0) = f (0+) = 1 E T α i = 1; E T β i 1 as β α (*) (*) replaces (conservative) 2 and ( drift down ) M(t) = lim f (tl i ) n Then i =n M(t) = exp( Wh(t)t α ) where W is an endogenous fixed point w.r.t. T (α) and h(t) is constant in the non-lattice case (and multiplicatively periodic otherwise). When f is a Laplace transform, this corresponds to an invariant RTP that is not endogenous with X i = i T i,ix ii. 2 N.B (conservative) might still hold, without ( drift down )

20 ... and now for something (completely?) different Not from Alsmeyer B and Meiners; arxiv Supercritical surviving BRW first generation {z i } B (n) rightmost in generation n exp κ(θ) = E i exp(θz i ) B (n) n { } κ(θ) Γ = inf θ>0 θ

21 Reducible Multitype Types 1, 2,..., k; a type can only have children of its own type and those later in the order (and 1 eventually can give k) i.e. genealogies increase in type; type i has κ i ; start with 1, B (n) k rightmost of type k; i j: i can have descendants of type j. B (n) k n max i j { inf max κi (ϕ) 0<ϕ θ ϕ, κ } j(θ) θ Speed can be faster than max{γ 1,..., Γ k } Speed can be slower than expectation speed.