Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton--Watson processes. Ann. Math. Statist. 37, 1211--1233 (1966)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....large but finite populations as well, i.e. for the underlying (multitype) branching process. For finite state spaces, such as in the mutation selection models discussed in [6] the results by Ethier and Kurtz [4, Thm. 11.2.1] and the generalization [2, Thm. V.7. 2] of the Kesten Stigum theorem [10,11] guarantee that in the infinite population limit the relative genotype frequencies of the branching process converge almost surely to the deterministic solution (if the population does not go to extinction) Since for the UC models considered here the equilibrium distributions are exponentially ....

Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton--Watson processes. Ann. Math. Statist. 37, 1211--1233 (1966)

....furthermore the sequence (W n ) is almost surely converging to a finite random variable W . 167 Refinements. The following theorems give more insight on the behavior on the sequence (Z n ) There are three theorems, one for each of the three cases m 1, m = 1 and m 1. Theorem 2 (Kesten Stigum [7]) If m 1, the following conditions are equivalent 1. Z n =m n ) converge to W in L 1 (P ) 2. E(N log N) P 1 i=2 k log(k)q k 1; 3. P (W = 0) q. The above result is mainly a strengthening of Theorem 1. It can be proved in an elegant way [8] with the formalism we described in the ....

Kesten (Harry) and Stigum (B.). -- A limit theorem for multidimensional Galton-Watson processes. Annals of Mathematical Statistics, vol. 37, 1966, pp. 1211--1223.

....P. Sloan Foundation Research Fellowships (Lyons and Pemantle) by NSF Grants DMS 9306954 (Lyons) DMS 9300191 (Pemantle) and DMS 9213595 (Peres) by a Presidential Faculty Fellowship (Pemantle) and by the hospitality of Washington University (Pemantle) 1 Theorem A: Supercritical Processes (Kesten and Stigum (1966)) Suppose that 1 m 1 and let W be the limit of the martingale Z n =m n . The following are equivalent: i) P[W = 0] q; ii) E[W ] 1; iii) E[L log L] 1. Theorem B: Subcritical Processes (Heathcote, Seneta and Vere Jones (1967) The sequence fP[Z n 0] m n g is ....

Kesten, H. and Stigum, B. (1966). A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Statist. 37, 1211--1223.

....differences will become clear later(see for instance Theorem 5 and Theorem 7) We now describe the fundamental limit theorems associated with supercritical branching processes. The first limit theorem describes the behavior of the branching process in the supercritical case. Kesten and Stigum [47] and Athreya [11] Theorem 2. The sequence W n j Z n =m n (2.7) is a nonnegative martingale and hence converges with probability 1 (w.p.1) to a limit W . Further, i) P (W = 0jZ 0 = 1) is one or q according as 1 X 0 j(log j)p j = 1 or 1: 2.8) ii) If P 1 0 j(log j)p j 1 then E(W ....

....1; 2; k if and only if ae 1 where ae is the Perron Frobenius eigenvalue of M . 2 The cases ae 1, ae = 1, ae 1 are referred to respectively as subcritical,critical and supercritical. The next theorem describes the behavior of the process in the supercritical case (Kesten and Stigum [47], Athreya and Ney [23] Theorem 9. Let u and v be as in (2.3) and ae 1. Then W n j u Delta Z n ae n (3.8) is a nonnegative martingale and hence converges w.p.1. to a limit W . Further, P (W = 0jZ 0 = e i ) q i for all i if and only if E(Z 1j log Z 1j jZ 0 = e i ) 1 for all i; j ....

Kesten,H. and Stigum,B.P.(1966): A limit theorem for multidimensional GaltonWatson processes, Annals of Mathematical Statistics, 37, 1211-1223.

....k leavesg) N i (t) max 1jn fProbfN j (l) r k gg) 2 t : 4.4) We choose t = ffk for a small ff and wish to bound the probability that a tree of depth l = 1 Gamma ff)k has no more than r k leaves. Since M is positively regular and second moments exist, the Kesten Stigum theorem [Kesten and Stigum 1966, Theorem 1] applies to give positive constants u i such that E[N i (l) u i o(1) ae l as l 1: 4.5) Furthermore, by the finite second moment assumption, there is a finite upper bound on the second moment of N i (l) ae l valid for all l 1 [Harris 1963, Theorem 9.2] Hence by ....

....are no jumps (all q i = 0) and each conditional distribution w i is strictly positive on R , by [Athreiya and Ney 1972, x V. 6, Theorem 2(iv) A detailed proof of the positivity of w for the single type Galton Watson process appears in [Athreiya and Ney 1972, x II.5, Theorem 2] See also [Kesten and Stigum 1966, Lemma 7] Now the random variables ( 4 3 ) Gammak N Gamma (k) and ( 4 3 ) Gammak N (k) sample values in the tails of the distributions w k , that is, values that lie outside any fixed region ( 1 Gamma ) in the cumulative distribution for large enough k. Since the w ....

H. Kesten and B. P. Stigum, "A limit theorem for multidimensional Galton-- Watson processes", Ann. Math. Stat. 37 (1966), 1211--1223.

....by Kolmogorov (1938) As is common, we prove this theorem by consideration of the law of Zn conditioned on Zn 0, which we denote by n . It is interesting that this sequence f n g always converges in a strong sense, even when its means are unbounded; see Section 6. Theorem 3: Critical Processes (Kesten, Ney and Spitzer (1966)) Suppose that m = 1 and let oe 2 = Var(L) E[L 2 ] Gamma 1 1. Then we have: i) Kolmogorov s estimate: lim n 1 nP[Zn 0] 2 oe 2 : ii) Yaglom s limit law: If oe 1, then the conditional distribution of Zn=n given Zn 0 converges as n 1 to an exponential law with mean oe 2 ....

Kesten, H. and Stigum, B. (1966). A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Statist. 37, 1211--1223.

.... that the process is supercritical (i.e. ae 1) and positive regular (i.e. some power of M has all entries positive) Let Z (j) n be the number of particles of type j in generation n and Zn : Z (1) n ; Z (J) n ) The Kesten Stigum theorem says the following (Kesten and Stigum [3], Athreya and Ney [1] p. 192) Theorem 1. There is a scalar random variable W such that lim n 1 Zn ae n = Wb a.s. 1) and P[W 0] 0 iff E 2 4 J X i;j=1 L (i;j) log L (i;j) 3 5 1 : 2) We shall give a proof of this theorem that avoids much analysis, extending the proof of ....

Kesten, H. and Stigum, B. (1966) A limit theorem for multidimensional GaltonWatson processes. Ann. Math. Statist. 37, 1211--1223.

....jbj = 1. We assume that the process is supercritical (i.e. ae 1) and positive regular (i.e. some power of M has all entries positive) Let Z (j) n be the number of particles of type j in generation n and Z n : Z (1) n ; Z (J) n ) The Kesten Stigum theorem says the following (Kesten and Stigum (1966), Athreya and Ney (1972) p. 192) Theorem 1. There is a scalar random variable W such that (1) lim n 1 Z n ae n = Wb a.s. and P[W 0] 0 iff (2) E 2 4 J X i;j=1 L (i;j) log L (i;j) 3 5 1 : 1991 Mathematics Subject Classification. Primary 60J80. Key words and phrases. ....

Kesten, H. and Stigum, B. (1966). A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1211--1223.

No context found.

Kesten, H. and Stigum, B.P. (1966) A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1211-1223.

No context found.

H. Kesten and B. P. Stigum (1966a). A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat. 37, 1211--1223.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC