T. E. Harris. The Theory of Branching Processes. Springer Verlag, New York, 1963.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as follows. The background on branching processes and PCFGs in 1 D is provided in Section II. Although this section is mostly tutorial, we believe that our approach to defining a probability distribution on the set of trees associated with a branching process is simpler than previous approaches in [9, 11, 21]. In Section III, we introduce spatial random trees. Section IV discusses inference using SRTs. Specifically, Subsection IV A introduces exact recursive algorithms for calculating the probability of a multidimensional signal; Subsection IV B describes an exact recursive algorithm which computes ....

....just one object whose type is j with probability P root (j) where this initial probability distribution P root must also satisfy a normalization property: P root (j) 1. 4) This stochastic process of transformations of our objects is easily seen to be a branching process, defined in, e.g. [11, 15]. The term multiplicative process has also been used, e.g. in [9, 21] The corresponding context free grammar, equipped with probability distributions P root and P prod , is called in natural language processing [19] a probabilistic (or stochastic) context free grammar (PCFG or SCFG) We ....

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T.E. Harris. The Theory of Branching Processes. Springer-Verlag, 1963.

.... 0 we have c(T , r; s) PGW(s) 30 and this simple observation has several useful consequences. In particular, if q(s) denotes the probability that the component c(T , r; s) is infinite, then classical formulas for the extinction probability of a branching process (such as those of Harris [30], p.7) tell us that we have q(s) 0 when 1, and otherwise we find that q(s) is the unique strictly positive solution of q(s) exp( sq(s) when s 1. 4.32) By inverting this relationship, one also sees that the value of s for which we have extinction probability 0 q 1 is given by ....

Harris, T.H. (1989): The Theory of Branching Processes. Dover Publications, New York.

....1) given by g i = f i =i: g(s) i=1 g i s . The generating function of the minimal solution is then easily evaluated in terms of g. We nd that E i ( s g( g(s) b(s) jsj : Since 1=b(s) has a power series expansion near s = 0 with positive coecients (Lemma V. 12.1 of Harris [7]) we may write 1= b(s) j=0 e j s , jsj , where e j 0; note that e 0 = 1= a ) Thus, E i ( 1 if and only if g( 1, in which case E i ( g( e i 1 g i 1 j e j ; i 1: 8) Empty sums are taken to be 0. To illustrate this, take f i = where 0, so that g(s) ....

Harris, T. (1963). The Theory of Branching Processes. Springer-Verlag, Berlin.

....functions R(s) R n s , v(s) v i s and g(s) g i s , we nd that R(s) 1 v(s) 1 g(s) a. It is then easy to prove that b(s) a(1 s) 1 v(s) which implies that R(s) a g(s) 6) and hence, if b(s) 6= 0, a g(s) b(s) 7) We know, from Section V. 12 of Harris [6], that 1=b(s) has a power series expansion with positive coecients and with radius of convergence , where is the smallest zero of b on (0; 1] Furthermore, 1 or 1 according as D 0 or D 0, and b(s) 0 for all s 2 [0; Thus, if D 0, then letting s in (7) gives R( 1, ....

...., it is easy to see that i = 1 = ae i 1 =i, i 1. However, the coecients ( i ) form a stationary measure on f1; 2; g for the Markov Branching Process with o spring distribution (q i ; i 0) where q 0 = a, q 1 = 0 and q i = d i 1 for i 2 (refer to the corollary of Theorem V. 12.2 of [6]) Theorem 1(e) of [13] then gives i i a 1 = 1 d ( as i 1, whenever D 6= 0 (note that d ( 1 since b ( 0 when D 6= 0) Consequently, e i 1= 1 d ( Thus, by the monotonicity of this limit, e i 1 =e i and hence e i j =e i for j = 1; i. ....

T.H. Harris. The Theory of Branching Processes. Springer-Verlag, Berlin, 1963.

.... rooted ) we mean that we could be using either with the same conclusion. Before we prove this theorem, we should describe branching processes. These are discussed a bit in Section 5. 4 of [15] but we will follow the more concrete account of time independent Galton Watson branching processes in [16]. Imagine a process where we start with a single vertex v 0 . We get Z 1 arcs from v 0 to vertices v 1;1 ; v 1;Z 1 , where Z 1 is some random variable. For each v 1;i , we have Z v 1;i arcs from v 1;i to new vertices v 1;i 1 ;1 ; v 1;i 1 ;Zv 1;i . Here 9 each Z v 1;i is ....

....= 0, then for all m n, Zm = 0: once the process dies out, it is dead. Second, Pr[Z 1 = 0] 6= 0, so it is possible for a vertex to have no children (this is the only way that a process could die out) Then the following theorem is very useful (this is Theorem 5.4 (5) of [15] or Theorem (1) 6. 1 of [16]) Theorem 2 If (Z n : n 2 N) is a time independent branching process such that E[Z 1 ] 1, then a.s. there exists n such that Z n = 0. Recall from probability theory that a random variable X is Poisson distributed with parameter if, for each nonnegative integer m, Pr[X = m] 1= e m ) We ....

T. E. Harris, The Theory of Branching Processes (Dover, 1963).

....depth d. The penetration probability depends on the probability of success in each trial. We study one such trial and show how vertex degree R affects this probability. For simplicity, we only consider the case with perfect resource recoveries. Mathematically, this problem is a branching process [13]. Consider any pair of adjacent proxies, for example A and B in Figure 9. Let q be the conditional probability of B being eventually compromised if A is compromised. Without retrials, it is straightforward to prove that q l = l and r defined in Table 2 1) Using this equation and results ....

....Consider any pair of adjacent proxies, for example A and B in Figure 9. Let q be the conditional probability of B being eventually compromised if A is compromised. Without retrials, it is straightforward to prove that q l = l and r defined in Table 2 1) Using this equation and results in [13], we can compute the penetration probability. plots how vertex degree affects the penetration probability. It shows that proxy networks with higher vertex degrees are easier to be penetrated. 2 4 6 8 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Vertex Degree of ....

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Harris, T.E., The Theory of Branching Processes. 1963: Prentice-Hall Inc.

....branching process with reproduction distribution Bin(N; 1 ) 2n) N; n; are fixed in this section and are defined as in the rest of the paper. S will denote the total size of the process, which can be infinite. An introduction and further results about such processes can be found in [3, 7]. Lemma 1 Consider a Galton Watson branching process defined as above. Then for any function w(n) tending to infinity with n, Conditionally on jSj w(n) 2n,we have E[jSj] Proof : The probability that such a process doesn t reach the size 2n can be lower bounded by the probability ....

T. E. Harris. The Theory of Branching Processes. Springer Verlag, New York, 1963. 6

....branching process with family size X is a discrete Markov chain (X t ) 1 t=0 with the following properties: at time t = 0 we have one ancestor , so we set X 0 = 1. For t 1 de ne X t to be the sum of X t 1 independent random variables, each having the same distribution as X (see e.g. [8] for details) As we shall only be concerned with the case when X has Poisson distribution with parameter c, in what follows (X t ) 1 t=0 will always refer to a branching process with this property. Let be the extinction probability of this process: c) P(X t = 0 for some t) 23) A ....

....to a branching process with this property. Let be the extinction probability of this process: c) P(X t = 0 for some t) 23) A simple result in branching process theory says that if c 1 then is the (unique) root of s = E (s X ) e c(s 1) in 0 s 1. Another basic result (see [8] again) is that for any constant k 0, we have lim t 1 P(X t = k) 0: 24) We now de ne branching processes in P(n) Consider any x 2 P(n) at level m, say. An upward x process (Z n t ) n m t=0 = Z n t (x) n m t=0 is de ned by letting Z n t denote the number of z 2 Pm t (n; p) that ....

T.E. Harris, The theory of branching processes, Springer Verlag, Berlin, 1963.

....based. Thanks to Lauri Grier and Fanny Mak for proofreading the paper. 19 A Proofs to the Analytic Results This appendix contains the proofs to the analytical results given in Section 4. The analysis is based on the observation that the growth of a random tree corresponds to branching processes [17]. To make our discussion self contained, we first briefly describe branching processes. A.1 A brief description of branching processes Branching processes [17] describe phenomena in which objects generate additional objects of the same kind, and different objects independently reproduce offspring ....

....results given in Section 4. The analysis is based on the observation that the growth of a random tree corresponds to branching processes [17] To make our discussion self contained, we first briefly describe branching processes. A. 1 A brief description of branching processes Branching processes [17] describe phenomena in which objects generate additional objects of the same kind, and different objects independently reproduce offspring of the same kind. An initial set of objects, called the 0 th generation, have children that are called the first generation; their children are the second ....

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T. Harris. The Theory of Branching Processes. Springer, Berlin, Germany, 1963.

....stationary point (i.e. common xed point) if it exists. For semigroups of holomorphic mappings this problem has been considered by many mathematicians for more than one hundred years in connection with classical geometric function theory (see, for example, 20] stochastic branching processes [29, 43], the theory of composition operators on Hardy spaces [5, 7] optimization and control theory [31] and the theory of linear operators in inde nite metric spaces (Krein and Pontryagin spaces) 51] 53] In the one dimensional case, the asymptotic behavior of a discrete time semigroup de ned by ....

T. E. Harris, The Theory of Branching Processes, Springer, Berlin, 1963.

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T. E. Harris. The Theory of Branching Processes. Springer Verlag, New York, 1963.

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T. E. Harris. The Theory of Branching Processes. Springer-Verlag, 1963.

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T.E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).

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T.E. Harris, Theory of branching processes, Dover NY 1989.

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T. Harris. The Theory of Branching Processes. Springer, 1963.

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T.E. Harris, Theory of branching processes, Dover, New York 1989.

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T.E. Harris. The Theory of Branching Processes. Springer-Verlag, 1963.

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T.E. Harris, Theory of branching processes, Dover NY 1989.

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Harris, TE (1963). The theory of branching processes. Springer-Verlag, Berlin.

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Theodore E. Harris. The theory of branching processes. Springer-Verlag, Berlin, 1963.

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Harris, T.E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.

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T.E. Harris. The Theory of Branching Processes. Springer-Verlag (Berlin and New York), 1963.

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Harris, T.E., The Theory of Branching Processes. 1963: Prentice-Hall Inc.

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Harris, T.E. (1963). The Theory of Branching Processes, Springer-Verlag, Berlin.

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Harris, T. E. 1963. The theory of branching processes. Springer, Berlin

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