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Tree-Valued Markov Chains Derived from Galton

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Tree-Valued Markov Chains Derived from Galton TREE-VALUED MARKOV CHAINS DERIVED FROM GALTON-WATSON PROCESSES. David Aldous and Jim Pitman Technical Rep ort No. 481 Department of Statistics University of California 367 Evans Hall 3860 Berkeley, CA 94720-3860 February 1997 Abstract Let G b e a Galton-Watson tree, and for 0 u 1letG be u the subtree of G obtained by retaining each edge with probability u. We study the tree-valued Markov pro cess G ; 0 u 1 and u an analogous pro cess G ; 0 u 1 in which G is a critical or u 1 sub critical Galton-Watson tree conditioned to b e in nite. Results simplify and are further develop ed in the sp ecial case of Poisson o spring distribution. Running head. Tree-valued Markovchains. Key words. Borel distribution, branching pro cess, conditioning, Galton- Watson pro cess, generalized Poisson distribution, h-transform, pruning, ran- dom tree, size-biasing, spinal decomp osition, thinning. AMS Subject classi cations 05C80, 60C05, 60J27, 60J80 Research supp orted in part by N.S.F. Grants DMS9404345 and 9622859 1 Contents 1 Intro duction 2 1.1 Related topics : :: :: :: ::: :: :: :: :: ::: :: :: :: 4 2 Background and technical set-up 5 2.1 Notation and terminology for trees :: :: :: ::: :: :: :: 5 2.2 Galton-Watson trees : :: ::: :: :: :: :: ::: :: :: :: 9 2.3 Poisson-Galton-Watson trees :: :: :: :: :: ::: :: :: :: 10 2.4 Uniform Random Trees :: ::: :: :: :: :: ::: :: :: :: 11 2.5 Conditioning on non-extinction :: :: :: :: ::: :: :: :: 12 3 Pruning Random Trees 17 3.1 Transition rates :: :: :: ::: :: :: :: :: ::: :: :: :: 19 3.2 Pruning a Galton-Watson tree : :: :: :: :: ::: :: :: :: 22 3.3 Pruning a GW tree conditioned on non-extinction ::::::: 26 3.4 The sup ercritical case : :: ::: :: :: :: :: ::: :: :: :: 30 4 The PGW pruning pro cess 32 4.1 The jointlawofG ; G : ::: :: :: :: :: ::: :: :: :: 32 4.2 Transition rates and the ascension pro cess :: ::: :: :: :: 35 1 4.3 The PGW 1 distribution :: :: :: :: :: ::: :: :: :: 37 4.4 The pro cess G ; 0 1 :: :: :: :: :: ::: :: :: :: 38 4.5 A representation of the ascension pro cess : :: ::: :: :: :: 40 :: :: :: :: ::: :: :: :: 43 4.6 The spinal decomp osition of G 4.7 Some distributional identities : :: :: :: :: ::: :: :: :: 45 4.8 Size-Mo di ed PGW-trees : ::: :: :: :: :: ::: :: :: :: 48 1 Intro duction This pap er develops some theory for Galton-Watson trees G i.e. family trees asso ciated with Galton-Watson branching pro cesses, starting from the fol- lowing twoknown facts. i [Lemma 10] For xed 0 u 1letG b e the \pruned" tree obtained by u cutting edges of G and discarding the attached branch indep endently with probability1 u. Then G is another Galton-Watson tree. u 1 , ii [Prop osition 2] For critical or sub critical G one can de ne a tree G 2 1 interpretable as G conditioned on non-extinction. Qualitatively, G consists of a single in nite \spine" to which nite subtrees are attached. Weinterpret i as de ning a pruning process G ; 0 u 1, whichis u a tree-valued continuous-time inhomogeneous Markovchain such that G is 0 the trivial tree consisting only of the ro ot vertex, and G = G . An analogous 1 1 pruning pro cess G ; 0 u 1 with G = G is constructed from the con- u 1 1 ditioned tree G of ii. Section 3 gives a careful description of the transition rates and transition probabilities for these pro cesses. The two pro cesses are qualitatively di erent, in the following sense. If G is sup ercritical then on the event G is in nite there is a random ascension time A suchthatG is nite A but G is in nite: the chain \jumps to in nite size" at time A.Incontrast, A the pro cess G \grows to in nity" at time 1, meaning that G is nite for u u u<1 but G = G is in nite. A connection b etween the two pro cesses is 1 1 made Section 3.4 by conditioning G ; 0 u< Aontheeventthat A u equals the critical time, i.e. the a for which G has mean o spring equal 1. a By rescaling the time parameter wemaytake a = 1, and the conditioned pro cess is then identi ed with G ; 0 u< 1. u These results simplify, and further connections app ear, in the sp ecial case of Poisson o spring distribution, which is the sub ject of Section 4. There we consider G ; 0 <1, where G is the family tree of the Galton-Watson branching pro cess with Poisson o spring, and the asso ciated pruned con- ; 0 1. To highlight four prop erties: ditioned pro cess G The distribution of G has several di erentinterpretations as a limit 1 Section 4.3. is the distribution of G ,size- For xed <1, the distribution of G biased by the total size of G Section 4.4. The pro cess G rununtil its ascension time A> 1 has a representation in terms of G as Section 4.5 d ; 0 < log U =1 U G ; 0 <A =G U where U is uniform 0; 1, indep endentofG ; 0 1. , a certain vertex b ecomes distin- by pruning G In constructing G 1 guished, i.e. the highest vertex of the spine of G retained in G . This 1 3 vertex turns out to b e distributed uniformly on G , and a simple spinal decomposition of G into indep endent tree comp onents is obtained by cutting the edges of G along the path from the ro ot to the distinguished vertex Section 4.6. Other topics include consequent distributional indentities relating Borel and size-biased Borel distributions Sections 4.5 and 4.7 and the interpretation of trees conditioned to b e in nite as explicit Do ob h-transforms, with the related identi cation of the Martin b oundary of ; G Section 4.4. None of the individual results is esp ecially hard; the length of the pap er is due partly to our development of a precise formalism for writing rigorous pro ofs of such results. Section 2 contains this formalism and discussion of known results. 1.1 Related topics Of course, branching pro cesses form a classical part of probability theory. Various \probability on trees" topics of contemp orary interest are treated in the forthcoming monograph byLyons [30], which explores several asp ects of Galton-Watson trees but touches only tangentially on the sp eci c topics of this pap er. Our motivation came from the following considerations, whichwillbe elab orated in a more wide-ranging but less detailed companion pap er [7]. Supp ose that for each N there is a Markovchain taking values in the set of forests on N vertices. Lo oking at the tree containing a given vertex gives a tree-valued pro cess, and taking N !1limits may give a tree-valued Markov chain. The prototyp e example not exactly forest-valued, of course is the random graph pro cess GN; P edge = =N ; 0 N for whichthe limit tree-valued Markovchain is our pruned Poisson-Galton-Watson pro- cess G ; 0 <1 [1]. The pruned conditioned Poisson-Galton-Watson pro cess G ; 0 1 arises in a more subtle way as a limit of the Marcus- Lushnikov discrete coalescent pro cess with additivekernel see [6] for back- ground on the general Marcus-Lushnikov pro cess and [42, 43 , 44, 38 ] for recent results on the additive case. More exotic variations of G , e.g. a stationary Markov pro cess in which branches grow and are cut down up on b ecoming in nite, arise as other N !1limits and are studied in [7]. Finally, we remark that the unconditioned and conditioned critical Poisson-Galton- 4 Watson distributions arise as N !1limits in several other contexts as \fringes" in random tree mo dels [3], in particular in random spanning trees [2, 37 ]; in the Wright-Fisher mo del where there is no natural pruning struc- ture. 2 Background and technical set-up Here we set up our general notation for random trees, and presentsome background material ab out Galton-Watson trees. 2.1 Notation and terminology for trees Except where otherwise indicated, bya tree t wemeana rooted labeledtree, that is a set V =vertst, called the set of vertices or labels of t, equipp ed t with a directededge relation ! such that for some obviously unique element ro ot t 2 V there is for eachvertex v 2 V a unique path from the root to v , that is a nite sequence of vertices v = roott;v ;:::;v = v suchthat 0 1 h t v ! v for each1 i h.Thenh = hv; t is the height of vertex v in i1 i the tree t.Formally, t is identi ed by its vertex set V and its set of directed t edges, thatisthesetfv; w 2 V V : v ! w g. If a subset S of vertst t is such that the restriction of the relation ! to S S de nes a tree s with vertss= S , then either S or s may b e called called a subtree of t. Let V 2f0; 1; 2;:::;1g be the number of elements of a set V , and for a tree t let t =vertst. The numb er of edges of a tree t is t 1. For a tree t t and v 2 vertst let childrenv; t:=fw 2 vertst: v ! w g denote the set of children of v in t, and let c t := childrenv; t, the number of children v of v in t.Each non-ro ot vertex w of t is a child of some unique vertex v of t,say v =parentw; t. Let s and t be two trees. Call s a relabeling of t if s there exists a bijection ` :vertss ! vertstsuchthat v ! w if and only if t `v ! `w .
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