Ancestor Problem for Branching Trees

Transcription

1 Mathematics Newsletter: Special Issue Commemorating ICM in India Vol. 9, Sp. No., August, pp. Ancestor Problem for Branching Trees K. B. Athreya Abstract Let T be a branching tree generated by a probability distribution {p j } j on the nonnegative integers N + {,, 3...}. For a random sample of two vertices at the nth level trace their paths back in time till they coalesce, i.e. meet. Call that level X n. This paper has results on the probability distribution of X n and its behavior as n under various conditions on the distribution {p j } j. Introduction Let {p j } j be a probability distribution on the nonnegative positive integers N + {,,, 3...}. Let {ξ n,i : i,n } be a doubly infinite family of independent random variables with distribution {p j } j.let{ n} n be a sequence of random variables defined by the stochastic recurrence relation: =, and for n, n+ = X n ξ n,i, if n > and if n =. i= Then { n} n is called the population size sequence of a Galton-Watson branching tree with offspring distribution {p j } j. (See Athreya and Ney [3]). Here =isthe ancestor of this population, n the size of the nth generation and ξ n,i the number of offspring of the ith individual in the nth generation and n+ is the sum of all these offspring of the nth generation individuals and constitutes the (n + )st generation. Let T denote the full tree from which all ξ n,i,i,n can be recovered. Now pick two individuals from the nth generation (provided n ) at random by simple random sampling, i.e., all the n C pairs have on equal chance of being chosen. Next trace the lines of descent of these two chosen individuals back in time till they coalesce, i.e. meet. Call that generation X n. This paper has results on the distribution of X n and its behavior as n under various conditions on the distribution {p j } j. Some preliminary results A crucial parameter for the behavior of the tree T as n is the mean m of {p j } j, i.e. m P jp j. The behavior of the four cases m<,m =, <m< and m = are very different. The case m< is called subcritical, m =critical and <m<, supercritical and m = explosive. The following results are well known in the literature. (See Athreya and Ney [3]) Departments of Mathematics and Statistics, Iowa State University, Ames, Iowa, 5, USA kbaiastate.edu; kbathreyagmail.com c Ramanujan Mathematical Society All rights reserved.

2 K. B. Athreya Theorem. (Subcritical case). Let Let <. Then: (i) <m jp j <. P ( n as n )= (Here and in what follows P (A) stands for the probability of the event A) (ii) For j, lim n P (n = j n > ) b j exists and b j =. j= (Here and in what follows P (A B) stands for the conditional probability of event A given that B has happened) Theorem. (Critical case). Let Let and Let <. Then: (i) (ii) (iii) m = σ = jp j =, p <. j p j < j p j. P ( n as n )= lim np (n > ) = n σ lim P n <x n > = e σ n n x for x<. Theorem.3 (Supercritical case). Let <m and p j for any j. Let <. jp j <

3 Ancestor Problem for Branching Trees 3 Then: (i) P ( n =)=q< and P ( n = k) =q k (ii) for k and q is the unique root in [, ) (i.e. q < ) of the equation s = P p js j P ( n = k) = q k for k. (i.e. P ( n or n = k) =)). (iii) There exist a sequence {C n} n of positive numbers such that (a) lim n n C n W exists w.p. (i.e. with probability one) and P ( <W < n )= (b) There exists a continuous positive function h(x) on (, ) such that for < a<b< b P (a <W <b)= h(x)dx a (c) P Cn+ C n m as n. (iv) If P j log jp j < then C n can be taken to be m n. (v) If j log jp j = then m Cn as n. n For the explosive case m = the results are not as definitive on the above three cases. For a number of interesting results in this case see Schuh and Barbour [8].. A deterministic example Consider a deterministic rooted tree with one ancestor and each individual in any generation giving birth to exactly b children where b is a finite positive integer. Then at the nth generation there are exactly b n individuals. For this tree, for any k, P (X n <k)= bk C b n k b n k b n C = bk (b k )b n k b n (b n ) = ( b k ) ( b n ) and lim n P (X n <k)= b k π k,say. Thus P (X n =)= ( b ) ( b n ) b. For b = this says that with probability approaching / from above the coalescence in a binary tree goes back to the root ancestor. Also since π k onk,x n converges to a proper probability distribution on N {,,,...} as n. It turns out that this carries over to the random supercritical case but not to the explosive case.

4 4 K. B. Athreya 3Mainresults Theorem 3. (Supercritical case). Let {p j } j satisfy: p =, <m= jp j < Let T be a tree generated with this offspring distribution and =. Then, for almost all trees T,andforallk lim P n (Xn <k T )=π k(t ) exists and lim π k(t )=. k Call the above a quenched version. There is an annealed version. Corollary 3. (Annealed). Under the hypothesis of Theorem 3., lim P n (Xn <k)=π k exists and lim π k = k Theorem 3. (Explosive case). Let p =,m= jp j =. Let for some <α<, ψ(x) X j>x p j x α L(x) where L( ) is slowly varying at, i.e. x α ψ(x) L(x) as x and L(cx) L(x) as x for all <c<. Then: (i) For almost all trees T, X n in probability an n, i.e. P (X n <k T ) an n for each k< (ii) lim n P (n X n k) =π(k) exists and π(k) on k. Remark 3.. The results in Theorem 3. and 3. provide a sharp contrast. When < m <,X n converges in distribution to a proper probability distribution on {,,,...}. Thatis,coalescence takes place in the remote part, near the beginning of the tree. Based on this one would respect in a tree growing faster than this the coalescence should take place in even more back the remote past. But Theorem 3. asserts a surprising negation of this. If m = and ψ(x) X j x p j x α L(x)

5 Ancestor Problem for Branching Trees 5 for <α<andl( ) slowly varying at, thenthe coalescence does not take place in the remote past at all but does take place very close to the present. This somewhat counter-intuitive result depends on a surprising result due to Grey [5] that asserts that if { n} and {n} are two independent copies of the process with offspring distributing {p j } j satisfying the hypothesis of Theorem 3. then although both n and n go to as n their ratio n goes to either zero or w.p.. It also needs a nice result on n selfnormalised sums of a sequence of positive independent and identically distributed random variables due to LePage, Woodroffe and inn [6]. Theorem 3.3 (Critical case). Let m = jp j <, p <, j p j < Then, for <u<, lim P (Xn <nu n > ) = H(u) n exists and H(u) = Eϕ(N u) where φ(j) =E P P j Y i j ( Y i) {Y i} i being i.i.d. exp() random variables and N u is a positive integer valued random variable with P (N u = r) =( u)u r,r. Alsolim u H(u) =, lim u H u =. Remark 3.. In this case the coalescence time X n is neither in the remote past (see Theorem 3.) or very near the present (see Theorem 3.) but is of the order n, the current generation number. Theorem 3.4 (Subcritical case). Let <m= jp j < and X j log jpj <. Then, for k lim P (n Xn k n ) = π(k) n π(k) as k. exists and is positive Remark 3.3. In this case the coalescence time X n is not in the remote past but very close to the present time n (like in the explosive case Theorem 3.).

6 6 K. B. Athreya 4Proofs In this section we sketch the proofs of the four results stated in the previous section. For the full proofs see Athreya [,]. Proof of Theorem 3.. For k n, i k let (k) n k,i denote the number of individuals in the nth generation initiated by the ith member of the kth generation Then, for a given tree T, P (X n <k T )= B X k i j i,j= (k) n k,i (k) n k,j C A /n(n ). Since X k n = i= (k) n k,i, dividing both the numerator and the denominator on the rightside above by C n k where {C n} is an in Theorem.3. we get for k lim P (Xn <k T )= n P k i j i,j= P k W ki W kj i= W ki π k (T ), say where (k) n k,i W ki = lim, n C n k which exists w.p. (by Theorem.3). To finish the proof of Theorem 3. it remains only to show that for almost all trees T Since lim π k(t )= k α π k (T )= it suffices to show that the selfnormalised sum P k i= W ki P k i= W ki P k i= W ki P k i= W ki w.p.. Now conditioned on j,j k, the random variables {W ki, i k } are independent and identically distributed. It is shown in Athreya and Schuh [4] that the random variable W ki satisfies the condition that ϕ(x) E(W ki : W ki x) udp (W ki n) [,x]

7 Ancestor Problem for Branching Trees 7 is slowly varying at. It is known from O Brien [7] that if {η i } i are independent and identically distributed positive random variables such that P ( <η < ) = and udp (η u) [,x] is slowly varying at than the selfnormalised sum P n i= η i ( P n i= η i ) goes to zero in probability. Since {W ki : i k }, conditioned on k, are i.i.d. and since k w.p.. it follows that P P k i= W ki ( k i= W in probability. ki) Since π k (T )= P P k i= W ki ( k i= W ki), this implies that π k (T ) p as k But since π k (T ) is monotone nondecreasing in k and π k (T ) w.p.. it follows that π k (T ) w.p.. as k. The proof of Theorem 3. is now complete. Proof of Theorem 3.. As in the proof of Theorem 3., for any k P k i j (k) n k,i (k) n k,j i,j= P (X n <k T )= P k i= (k) n k,i P k i= n k,i By Grey s result [5], conditioned on k there exists an i, i k such that for all i i, i k (k) n k,i (k) n k,i w.p. as n Thus, for each k P (X n <k T ) as n proving (i) of Theorem 3.. Next, P (n X n k) =P (X n n k) = E P n k i= (n k) k,i ( (n k) P n k i= (n k) k,i P n k i= (n k) A k,i k,i ) It can be shown (see Athreya []) that if the offspring distribution {p j } j satisfies the conditions of Theorem 3. then for each k, the distribution of k given =, satisfies P ( k >x =) x αk L k (x) asx where L k ( ) in slowly varying at. Next, LePage, Woodroffe and inn [6] have established the following:

8 8 K. B. Athreya Theorem 4.. Let {X i } i be i.i.d. r.v. such that P ( <X < ) =and P (X > x) x α L(x) as x where < < and L( ) is slowly varying at. Then lim n P P n Xi n Yα exists ( X i ) in distribution and in mean. Further, Y α is a random variable with P ( <Y α < ) =. Since for fixed k, n k w.p. and conditioned an n k, the random variables { (n k) k,i : i n k } are i.i.d. with (n k) k,i satisfying the hypothesis of Theorem 4. with α replaced by α k it follows that lim n P (n Xn k) =E(Y α k π(k)), say. It has been shown by Athreya [] that as α,y α implies that EY α asα. Thus lim k π(k) = lim k E(Y α k)=. Thus the proof of Theorem 3. is complete. d and being bounded this Proof of Theorem 3.3. We need the following result proved in Athreya []. Theorem 4.. Let the hypothesis of Theorem 3.3 hold. On the event A n { n > }, for k<n, consider the point process V n,k ( k n k,i (n k) = i J k where J k {,,..., k } is the set of all i such that (k) n k,i > where {(k) j,i : j } is the branching process initiated by the ith parent in the kth generation. Let n, k and n k u, <u<. Then, conditioned on An, the distribution of the point process V n,k converges to that of V {Y i : i N u} where {Y i } i are i.i.d. exponential random variables with mean σ and Nu is independent of {Y i} i with distribution P (N u = k) =( u)u k,k. Now returning to the proof of Theorem 3. we note that for k<n, P (X n <k n > ) = E ) P k i= (k) n k,i fi fififi P A n ( k i= (k) n k i ) Now let n,k such that u k n, <u<. By the continuous mapping theorem and Theorem 4. the above expression converges as n to = E ψ P Nu! P i= Y i N ( u i= Y i) where {Y i } i and N u are as in Theorem 4.. Since { σ α Y i} i are i.i.d. exp() it follows that in the above expression we may assume that {Y i } i are i.i.d. exp() random variables by canceling the scale factor. Thus Theorem 3.3 is proved. A

9 Ancestor Problem for Branching Trees 9 Proof of Theorem 3.4. As in the proof of Theorem 3.3 P (n X n k n ) = E ψ P n k i= (n k) k,i ( P n k i= (n k) k,i )( P n k ( (n k) k,i ) i= (n k) k,i P ( P n k i= (n k) k,i ) ) I(P i k i= (n k) k,i ) where { (n k) j,i j }, i n k are as in the proof of Theorem 3.3. The right side above equals = E(ϕ k( n k ) n k > ) E(ψ k ( n k ) n k > ) where for j ϕ k (j) =E ψ k (j) =P P j i= k,i( k,i ) i= k,i ) I ( P j i= k,i) ( P j X j i= k,i fi fi jx fi i= k,i > A X j i= k,i A fi fi jx fi i= k,i > where { k,i i =,...} are i.i.d. with distribution same as k with =. By Theorem., (ii) the above expression converges to π(k) E ϕ k (Y )/E ψ k (Y ) where Y is a random variables with distribution {b j } j where ϕ k (j) =E ψ k (j) =P P j i= k,i( k,i ) i= k,i ) I P P j ( i= j k,i)( X j i= k,i A. Thus, we have shown that for k X j i= k,i AA A! lim P (n Xn k n ) = π(k) exists. n Clearly π(k) is >. This completes the proof of Theorem Concluding remarks The author would like to thank Professor C. S. Aravinda, TIFR, Bangalore Centre for the invitation to contribute this paper to this journal in connection with the ICM,, in India.

10 K. B. Athreya References [] K. B. Athreya, Coalesence in recent past in rapidly growing populations (submitted) (). [] K. B. Athreya, Coalesence in critical and subcritical Galton-Watson trees (submitted) (). [3] K. B. Athreya and P. Ney, Branching Processes, Dover, Mineola, N.Y. (4). [4] K. B. Athreya and H. J. Schuh, On the Supercritical Bellman-Harris process with finite mean, Sankhya 65() (3) [5] D. R. Grey, On regular branching processes with infinite mean, Stoch. Proc. & Appl. 8 (979) [6] R. Lepage, M. Woodroffe and J. inn, Convergence to a stable distribution via order statistics, Ann. Prob. 9(4) (98) [7] O Brien, A limit theorem for sample maxima and heavy branches in Galton-Watson trees, J. Appl. Prob. 7 (98) [8] H. J. Schuh and A. D. Barbour, On the asymptotic behavior of branching processes with infinite mean, Adv. Appl. Prob. 9 (977)