A strong law of large numbers for branching processes: almost sure spine events
aa r X i v : . [ m a t h . P R ] F e b A strong law of large numbers for branchingprocesses: almost sure spine events
Simon C. Harris ∗ and Matthew I. Roberts † June 19, 2018
Abstract
We demonstrate a novel strong law of large numbers for branchingprocesses, with a simple proof via measure-theoretic manipulations andspine theory. Roughly speaking, any sequence of events that eventuallyoccurs almost surely for the spine entails the almost sure convergence ofa certain sum over particles in the population.
We shall work with a fairly general Markov branching process. To define thisprocess, we suppose that we are given three ingredients: • A Markov process ψ t , t ≥
0, in a measurable space ( J, B ); • A measurable function R : J → [0 , ∞ ); • A collection of random variables A ( x ), x ∈ J taking values in { , , , . . . } ,such that M ( x ) := E [ A ( x )] − < ∞ .Our branching process is then defined, under a probability measure P , as follows:we begin with one particle. This particle moves around in J like a copy of theprocess ψ t . When at position x , it dies at rate R ( x ), that is, if ∅ is our originalparticle, X ∅ ( t ) is its position at time t and τ ∅ is its time of death, then P ( τ ∅ > t | X ∅ ( s ) , s ≤ t ) = exp (cid:18) − Z t R ( X ∅ ( s )) ds (cid:19) . At its time of death, it is replaced in its position x by a random number ofchildren, the number being specified by a copy of A ( x ). These children theneach independently show the same stochastic behaviour as their parent, movingaround like independent copies of ψ t and branching at rate R ( x ) when at posi-tion x into a random number of particles that is an independent copy of A ( x ).We let N ( t ) be the set of all particles that are alive at time t ; if v ∈ N ( t ) then ∗ Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. email:
[email protected] † Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. email: [email protected]
1e let X v ( t ) be the position of particle v at time t ; and we let A v be the numberof children of particle v .We let F t , t ≥ P to a new probability measure ˜ P on a bigger space bychoosing one special line of descent which we call the spine . The initial particleis part of the spine, and when a spine particle dies the new spine particle ischosen uniformly from amongst its children. We let the natural filtration ofthe new process, in which there is a branching process with one marked line ofdescent, be ˜ F t , t ≥
0. Let ξ t be the position of the spine particle at time t , andlet spine( t ) be the set of particles that have been in the spine up to time t .For details of all of the above, see [1] or Chapter 2 of [3].Suppose that ζ ( t ) is a non-negative martingale with respect to the filtration G t := σ ( ξ s , s ≤ t ), such that ˜ E [ ζ ( t )] = 1. We may write ζ ( t ) = X v ∈ N ( t ) ζ v ( t ) { ξ t = v } where each ζ v ( t ) is an F t -measurable random variable (see page 24 of [3] for aproof). Then ˜ ζ ( t ) := e − R t M ( ξ s ) R ( ξ s ) ds ζ ( t ) Y v ∈ spine( t ) (1 + A v )is a martingale with respect to ˜ F t (see Theorem 2.4 of [3]). We define a newmeasure ˜ Q by setting d ˜ Q d ˜ P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ F t := ˜ ζ ( t ) . The measure ˜ Q has a nice description in terms of the spine, although this willnot be used in this article. Briefly, the motion of the spine is biased by themartingale ζ ( t ); branching events along the spine occur at an accelerated rate(1 + M ( ξ t )) R ( ξ t ) when the spine is at position ξ t ; and the number of childrenof the spine is size-biased. All other (non-spine) particles, once born, remainunaffected.We also let Q := ˜ Q | F t be a measure on F t , the natural filtration of theoriginal branching process. Then d Q d P (cid:12)(cid:12)(cid:12)(cid:12) F t = X v ∈ N ( t ) e − R t M ( X v ( s )) R ( X v ( s )) ds ζ v ( t ) =: Z ( t )and Z ( t ) is a P -martingale with respect to F t (again see Theorem 2.4 of [3]for details). Since Z ( t ) is a positive martingale, it converges P -almost surely to Z ( ∞ ) := lim inf Z ( t ).We now state our main result. Suppose that f ( t ) is ˜ F t -measurable for each t . Then, again, we may write each f ( t ) via the representation f ( t ) = X u ∈ N t f u ( t ) { ξ t = u } where f u ( t ) is F t -measurable for each t ≥ u ∈ N ( t ).2 heorem 1: Suppose that { f ( t ) : t ≥ } is ˜ Q -uniformly integrable. If f ( t ) → f ˜ Q -almostsurely as t → ∞ then X u ∈ N t f u ( t ) e − R t M ( X u ( s )) R ( X u ( s )) ds ζ u ( t ) Z ( t ) → ˜ Q [ f |F ∞ ] ( ⋆ ) Q -almost surely. Furthermore, P (cid:0) ( ⋆ ) (cid:12)(cid:12) Z ( ∞ ) > (cid:1) = 1 . Remarks:
1. Since 1 /Z ( t ) is a positive Q -supermartingale (and thus con-verges almost surely to an almost surely finite limit), Z ( t ) → Z ( ∞ ) Q -almost surely. Thus we may deduce from ( ⋆ ) that X u ∈ N t f u ( t ) e − R t M ( X u ( s )) R ( X u ( s )) ds ζ u ( t ) → ˜ Q [ f |F ∞ ] Z ( ∞ ) Q -almost surely.In fact under fairly mild conditions on the branching distributions A ( x ),we have Q ( Z ( ∞ ) < ∞ ) = 1, in which case we do not lose anything byrewriting ( ⋆ ) in this way.2. In many cases of interest the events { Z ( ∞ ) = 0 } and {∃ t ∈ [0 , ∞ ) : Z ( t ) =0 } agree to within a set of zero P -probability. Then, of course, X u ∈ N t f u ( t ) e − R t M ( X u ( s )) R ( X u ( s )) ds ζ u ( t ) → ˜ Q [ f |F ∞ ] Z ( ∞ ) P -almost surely. We outline here two examples showing how our strong law can be applied.The first example is folklore in branching processes, but we are not aware ofanother proof. Theorem 1 has also been used in [2] which considers a branchingBrownian motion with killing on the boundary of a strip near criticality.
In branching processes branching at rate β into on average m off-spring, most particles branch at rate mβ .Take a continuous-time branching process with constant birth rate R ( x ) ≡ β and birth distribution A ( x ) ≡ A satisfying E [ A log + A ] < ∞ with m := E [ A ].Let ζ ( t ) ≡
1. For any ε > f ( t ) = {| n t /t − mβ | <ε } , the indicatorthat birth rate along the spine up to time t is close to its expected value under ˜ Q , mβ . Then for any ε > f ( t ) converges ˜ Q -almost surely to 1. Thus Theorem1, together with some classical results on branching processes concerning themartingale e − ( m − βt | N ( t ) | , tells us that on the event that the process survives,1 | N ( t ) | X u ∈ N ( t ) f u ( t ) → P -almost surely.This may be interpreted as saying that if we choose a particle uniformly atrandom from those alive at a large time t , and look at its history, we are likelyto see that its average birth rate has been approximately mβ . In particular,with binary branching, we see an average birth rate of 2 β in typical particles(rather than β , which one might naively expect).3ur second example shows how the spatial behaviour of the spine can alsobe passed to other particles: if the spine shows ergodic behaviour, then so domany other particles. Occupation densities and ergodic spines.
Suppose that the motion of the spine ( ξ t , t ≥
0) is ergodic under ˜ Q with invari-ant probability density π in the sense that there exists some suitable class offunctions H such that for any h ∈ H ,1 t Z t h ( ξ s ) ds → L h := Z R h ( x ) π ( x ) dx ˜ Q -almost surely.Then for any continuous function g : R → R and any h ∈ H ,1 Z ( t ) X u ∈ N ( t ) g (cid:18) t Z t h ( X u ( s )) ds (cid:19) e − R t M ( X u ( s )) R ( X u ( s )) ds ζ u ( t ) → g ( L h ) Q -almost surely. The same holds under P on the event Z ( ∞ ) >
0, which isone exposition of the general principle that if forcing the spine to show certainbehaviour does not cause the corresponding martingale to disappear, then thatbehaviour appears in the original process.
To prove Theorem 1 we need some simple measure theory. For this section weforget the branching setup and take any filtered probability space (Ω , F , F t , P )and define F ∞ := W t ≥ F t . Suppose that X t , t ≥ E [ X t |F t ] , t ≥
0) is almost surely c`adl`ag.
Proposition 2: If E [ X t |F ∞ ] → Y almost surely,then E [ X t |F t ] → Y almost surely. Proof:
Fix ε >
0. We show that there exists an almost surely finite random variable T such that sup t ≥ T E [ X t |F t ] ≤ Y + ε almost surely.By the c`adl`ag property, it is sufficient to take the supremum above over rationalsgreater than T ; from now on all our suprema will be over rationals.Since E [ X t |F ∞ ] → Y , there exists an almost surely finite random variable T such that sup t ≥ T E [ X t |F ∞ ] < Y + ε/ E [ Y |F t ] → E [ Y |F ∞ ] = Y ( Y is F ∞ -measurable since it is the limit of F ∞ -measurable random variables),there exists an almost surely finite random variable T such thatsup t ≥ T E [ Y |F t ] < Y + ε/ T = T ∨ T . Thensup t ≥ T E [ X t |F t ] ≤ sup t ≥ T sup s ≥ T E [ X s |F t ]= sup t ≥ T sup s ≥ T E [ E [ X s |F ∞ ] |F t ] ≤ sup t ≥ T E (cid:20) sup s ≥ T E [ X s |F ∞ ] (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≤ sup t ≥ T E [ Y + ε/ |F t ] ≤ Y + ε (all statements hold almost surely). Thus lim sup E [ X t |F t ] ≤ Y ; the proof thatlim inf E [ X t |F t ] ≥ Y is similar. Corollary 3:
Suppose that the collection of random variables { X t , t ≥ } is uniformly inte-grable. If X t → X almost surelythen E [ X t |F t ] → E [ X |F ∞ ] almost surely. Proof:
Let Y = E [ X |F ∞ ]; then by uniform integrability, E [ X t |F ∞ ] → Y almost surely.Proposition 1 now gives the result. We now return to the notation from Section 1.
Proof of Theorem 1.
We recall Theorem 8.2 of Hardy and Harris [1], which saysthat under the conditions above,˜ Q [ f ( t ) |F t ] = X u ∈ N t f u ( t ) e − R t M ( X u ( s )) R ( X u ( s )) ds ζ u ( t ) Z ( t ) . Now if f ( t ) converges ˜ Q -almost surely to f then by Corollary 3 we have˜ Q [ f ( t ) |F t ] → ˜ Q [ f |F ∞ ] ˜ Q -almost surelyand hence X u ∈ N t f u ( t ) e − R t M ( X u ( s )) R ( X u ( s )) ds ζ u ( t ) Z ( t ) → ˜ Q [ f |F ∞ ] ˜ Q -almost surely.5inally, for any ( F ∞ -measurable) event A such that Q ( A ) = 1, P ( A | Z ( ∞ ) >
0) = P ( A ∩ { Z ( ∞ ) > } ) P ( Z ( ∞ ) > Q h Z ( ∞ ) A ∩{ Z ( ∞ ) > } i P ( Z ( ∞ ) > Q h Z ( ∞ ) { Z ( ∞ ) > } i P ( Z ( ∞ ) > . References [1] R. Hardy and S. C. Harris. A spine approach to branching diffusions withapplications to L p -convergence of martingales. In S´eminaire de Probabilit´es,XLII , volume 1979 of
Lecture Notes in Math.
Springer, Berlin, 2009.[2] S.C. Harris, M. Hesse, and A.E. Kyprianou. Branching brown-ian motion in a strip: survival near criticality. 2012. Preprint: http://arxiv.org/abs/1212.1444v1 .[3] M.I. Roberts.
Spine changes of measure and branching diffu-sions . PhD thesis, University of Bath, 2010. Available online: http://people.bath.ac.uk/mir20/thesis.pdfhttp://people.bath.ac.uk/mir20/thesis.pdf