M. Drmota and B. Gittenberger, On the profile of random trees, Random Structures and Algorithms, 10 (1997), 421--451. 27  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... scaled excursion X at a, denoted by (a) have been studied by several authors (note that for an excursion of length we have: a) a= See for instance Getoor and Sharpe [11] Knight [15] 16] Cohen and Hooghiemstra [4] Hooghiemstra [14] Drmota and Gittenberger [5], Louchard [19] Gittenberger and Louchard [13] Intuitively, the local time at a is the total time spent by the excursion in the neighbourhood of a. Applications of the BE are numerous: we will mention a few of them, emphasizing the meaning of the local time. For instance, consider a M=G=1 ....

M. Drmota and B. Gittenberger. On the pro le of random trees. Random Structures and Algorithms, 10:421-451, 1997.

....tree distribution. Section 2 of this paper is devoted to an alternative description of this limiting distribution P and to some explicit computations for the cases n = 1; 2 or 3 using generating functions. The case of general n relies on results for systems of algebraic equations due to Drmota [3], Lalley [7] or Woods [17] The combined approach by generating functions and branching processes also allows us to de ne a second probability distribution on boolean fonctions: starting from a critical branching process, we label at random its internal and external nodes We can also choose ....

.... i i 1 48z; z 48 Thus for z = 1=48 the system (9) has a unique solution ( 1 ; 14 ) Hence the xed point can be obtained by iteration, starting from a vector whose coordinates are all equal to zero. In order to compute the values i , i = 1; 14, observe that Drmota [3] showed that the vector ( i ) i=1; 14 is an eigenvector with eigenvalue 1 of the Jacobian Q t Q i t j i;j=1: 14 evaluated at z = 1=48. Since 1 is an eigenvalue of multiplicity 1 at z = 1=48, we can easily compute the eigenvector and normalize it to obtain the results presented ....

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M. Drmota. Systems of functional equations. Random Structures and Algorithms, 10:103-124, 1997.

....see Gardy [13] One can easily include other variables in (2) by simply thinking of the coe #cients of f , g,andw as involving the new variables. Furthermore, there are extensions of Lagrange inversion to several functions and other variables can be included in these as well. Recently Drmota [12] treated a system of functional equations using singularity analysis. His results can be applied to multifunction Lagrange inversion when g(w 1 ,w 2 , w d ) w i for some i. Not all cases of interest have this form, a prime example being map enumeration. The asymptotics of rooted convex ....

.... y is a Hayman admissible function [16] We are not aware of any multivariate singularity analysis that combines algebraic singularities and Hayman admissibility. We discussed multivariate Hayman admissibility in [8] We believe it should be possible to extend Drmota s functional equation results [12] to remove the limitation that g(w)bew i for some i. Also, one should be able to eliminate the conditioning on n in his Theorem 1, thereby obtaining a result like our Theorem 5.1, probably with his I F y and F y,y playingaroleakin to our L and B. We have not attempted to develop these ideas ....

M. Drmota, Systems of functional equations, Random Structures and Algorithms 10 (1997) 103--124.

....six NC configurations we are interested in, the generating function ( y z satisfies an algebraic equation. In the case of NC forests for example, we have: eq: y 3 ( z z 2 3) y 2 (z 3) y 1; eq y 3 ( z z 2 3 y 2 ( z 3 y 1 Like in many implicitely defined functions [Dr97, HaPa73], we expect a priori ( y z to have locally an expansion of the square root type, that is: y z c 0 c 1 1 z r c 2 1 z r O 1 z r 3 2 By a singularity analysis at the dominant singularity r and denoting = g c 1 2 , we get: z ....

M. Drmota, Systems of functional equations, Random Structures and Algorithms 10, 1-2, 1997.

....see Gardy [13] One can easily include other variables in (2) by simply thinking of the coe #cients of f , g,andw as involving the new variables. Furthermore, there are extensions of Lagrange inversion to several functions and other variables can be included in these as well. Recently Drmota [12] treated a system of functional equations using singularity analysis. His results can be applied to multifunction Lagrange inversion when g(w 1 ,w 2 , w d ) w i for some i. Not all cases of interest have this form, a prime example being map enumeration. The asymptotics of rooted convex ....

.... y is a Hayman admissible function [16] We are not aware of any multivariate singularity analysis that combines algebraic singularities and Hayman admissibility. We discussed multivariate Hayman admissibility in [8] We believe it should be possible to extend Drmota s functional equation results [12] to remove the limitation that g(w)bew i for some i. Also, one should be able to eliminate the conditioning on n in his Theorem 1, thereby obtaining a result like our Theorem 5.1, probably with his I F y and F y,y playingaroleakin to our L and B. We have not attempted to develop these ideas ....

M. Drmota, Systems of functional equations, Random Structures and Algorithms 10 (1997) 103--124.

....23. This implies an integrated form of joint convergence, as follows: 1 2 m Gamma1=2 f Wm ; 2m Gamma1=2 I m ) d (B; I) 15) where I m (s) R s 0 e Qm (y) dy and I(s) R s 0 L(y) dy. The stronger assertion 2m Gamma1=2 e Qm d L was proved by Drmota and Gittenberger [8]. Convergence of the marginal processes in Theorem 3 implies tightness of the joint processes, and then (15) identifies the limit and hence establishes joint convergence. In Theorem 3 the convergence in distribution was for random elements of C[0; 1] Theta D[0; 1) Then because L is continuous ....

M. Drmota and B. Gittenberger. On the profile of random trees. Random Structures and Algorithms, 10:421--451, 1997.

....six NC configurations we are interested in, the generating function ( y z satisfies an algebraic equation. In the case of NC forests for example, we have: eq: y 3 ( z z 2 3) y 2 (z 3) y 1; eq y 3 ( z z 2 3 y 2 ( z 3 y 1 Like in many implicitely defined functions [Dr97, HaPa73], we expect a priori ( y z to have locally an expansion of the square root type, that is: y z c 0 c 1 1 z r c 2 1 z r O 1 z r 3 2 By a singularity analysis at the dominant singularity r and ....

M. Drmota, Systems of functional equations, Random Structures and Algorithms 10, 1-2, 1997.

....see Gardy [13] One can easily include other variables in (2) by simply thinking of the coefficients of f , g, and w as involving the new variables. Furthermore, there are extensions of Lagrange inversion to several functions and other variables can be included in these as well. Recently Drmota [12] treated a system of functional equations using singularity analysis. His results can be applied to multifunction Lagrange inversion when g(w 1 ; w 2 ; w d ) w i for some i. Not all cases of interest have this form, a prime example being map enumeration. The asymptotics of rooted convex ....

....is a Hayman admissible function [16] We are not aware of any multivariate singularity analysis that combines algebraic singularities and Hayman admissibility. We discussed multivariate Hayman admissibility in [8] We believe it should be possible to extend Drmota s functional equation results [12] to remove the limitation that g(w) be w i for some i. Also, one should be able to eliminate the conditioning on n in his Theorem 1, thereby obtaining a result like our Theorem 5.1, probably with his I Gamma Fy and Fy;y playing a role akin to our L and B. We have not attempted to develop these ....

M. Drmota, Systems of functional equations, Random Structures and Algorithms 10 (1997) 103--124.

....distributions in a combinatorial setting which is equivalent to the conditioned branching process with a Poisson offspring distribution, as discussed in Section 3. The result of Theorem 7 for k = 1 and = 0 was anticipated by Aldous [3, Conjecture 4] and proved by Drmota Gittenberger [16], with the limiting process (X 0;1;v ; v 0) defined by the process of local times of a Brownian excursion rather than by an SDE. Corollary 5 follows by comparison of this case of Theorem 7 with the result of Drmota Gittenberger [16] or with the weaker integrated form of this result found by ....

....by Aldous [3, Conjecture 4] and proved by Drmota Gittenberger [16] with the limiting process (X 0;1;v ; v 0) defined by the process of local times of a Brownian excursion rather than by an SDE. Corollary 5 follows by comparison of this case of Theorem 7 with the result of Drmota Gittenberger [16], or with the weaker integrated form of this result found by Aldous [3, Cor. 3] which is enough to characterize the limit process, assuming it exists, as the process of local times of a Brownian excursion. The sentence following (27) below gives an alternative proof of Corollary 5. The key to the ....

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M. Drmota and B. Gittenberger. On the profile of random trees. Random Structures and Algorithms, 10:421--451, 1997.

....that each class of non crossing configurations leads to an asymptotic estimate of the form fn fl n p n 3=2 ; 25) where fn is the number of objects of size n, and fl; are context dependent algebraic numbers. Such estimates are for instance familiar in the theory of tree enumerations [8, 19, 24, 26]. Roughly, each of the six counting generating functions is an algebraic function, as seen in Sections 1,2,3. It is known that the singularities of GFs determine the asymptotics of their coefficients. Here, we a priori expect local singular expansions in the form of Puiseux expansions, that is to ....

....the asymptotics of their coefficients. Here, we a priori expect local singular expansions in the form of Puiseux expansions, that is to say expansions involving fractional exponents. Generically, singularities of the square root type are expected, like in many implicitly defined functions [8, 19]. All our GFs appear to be of this type, with a local expansion near the dominant singularity ae being f(z) c 0 c 1 p 1 Gamma z=ae: 26) Then singularity analysis [13] is used to achieve the transfer of (26) to coefficients leading to estimates of the form (25) Rather than examining each ....

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Drmota, M. Systems of functional equations. Random Structures and Algorithms 10, 1--2 (1997), 103--124.

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M. Drmota and B. Gittenberger, On the profile of random trees, Random Structures and Algorithms, 10 (1997), 421--451. 27

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M. Drmota and B. Gittenberger, On the pro le of random trees, Random Structures and Algorithms 10, 421-451, 1997.

.... 16y 2 (4y) 8y 2 (2y) higher moments can be integrated with Maple only numerically. 2.2. Second approach. Let b k;m;n;N denote the (weighted) number of forests in F (n; N) with m nodes in stratum k. Then standard methods on generating functions (see [9] for a general introduction and [7] for the treatment of the particular case of random trees) give P fL n;N (k) mg = b k;m;n;N u ]y k (z; ua(z) y 0 (z; u) u; y i 1 (z; u) z (y i (z; u) i 0: Consequently, we have E L n;N (k) u y k (z; ua(z) j u=1 and moreover E L n;N (k) L n;N (k) 1) ....

M. Drmota and B. Gittenberger, On the pro le of random trees, Random Structures and Algorithms 10, 421-451, 1997.

.... 16y 2 (4y) 8y 2 (2y) higher moments can be integrated with Maple only numerically. 2.2. Second approach. Let b k;m;n;N denote the (weighted) number of forests in F (n; N) with m nodes in stratum k. Then standard methods on generating functions (see [5] for a general introduction and [4] for the treatment of the particular case of random trees) give P fL n;N (k) mg = b k;m;n;N u ]y k (z; ua(z) y 0 (z; u) u; y i 1 (z; u) z (y i (z; u) i 0: Consequently, by (3) we have E L n;N (k) u y k (z; ua(z) j u=1 and moreover E L n;N (k) L n;N (k) 1) ....

M. Drmota and B. Gittenberger, On the pro le of random trees, Random Structures and Algorithms 10, 421-451, 1997.

....problem. Remark 3. The average extinction time of a branching process conditioned on the total progeny to be n is proportional to p n. Thus the behavior changes if we choose c n = p n as scaling factor. In this case Brownian excursion local time is obtained as limit process as was shown in [4]. The proof of Theorem 1.1 is divided into two parts: First, we have to show that the finite dimensional distributions (fdd s) of l n (t) converge weakly to those of l(t) which is done in the next section. The one dimensional limit theorem has been established by Kennedy [13, Theorem 1] and ....

....[15] proved finite dimensional convergence results of a similar type for branching processes conditioned to have infinite total progeny. Perhaps it is possible to use their ideas to obtain a different way of attacking this problem. Finally, note that recently Pitman [17] reproved the results in [4, 5] by means of an approach via stochastic differential equations. CONVERGENCE OF BRANCHING PROCESSES TO THE LOCAL TIME OF A BESSEL PROCESS 3 2. Finite Dimensional Convergence 2.1. The limiting distributions of l n . We will prove the convergence of the fdd s to those of local time by computing ....

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M. Drmota and B. Gittenberger, On the profile of random trees, Random Structures and Algorithms 10 (1997), 421--451.

....value k n and covariance matrix Sigman. Remark . The same theorem holds if only planted or rooted labeled trees are considered instead of labeled (unrooted) trees or forests. 3. Analytic Background The basic property which will be used in the sequel is the following observation (compare with [2, 3]) Proposition 3.1. Set u = u 1 ; uM ) and suppose that F (x; u; y) is an analytic function around (x 0 ; u 0 ; y 0 ) such that F (x 0 ; u 0 ; y 0 ) y 0 ; F y (x 0 ; u 0 ; y 0 ) 1; F yy (x 0 ; u 0 ; y 0 ) 6= 0; F x (x 0 ; u 0 ; y 0 ) 6= 0: Then there exist a neighborhood U of (x ....

....u) and f(u) which are defined on U such that the only solutions y 2 V with y = F (x; u; y) x; u) 2 U) are given by y = g(x; u) Sigma h(x; u) r 1 Gamma x f(u) 3.1) Furthermore g(x 0 ; u 0 ) y 0 and h(x 0 ; u 0 ) p 2f(u 0 )F x (x 0 ; u 0 ; y 0 ) F yy (x 0 ; u 0 ; y 0 ) Proof. See [3]. 3.1. Unlabeled, Nonplane Trees. As a first application of Proposition 3.1 we show that representation (1.1) follows just from the facts that the radius of convergence ae satisfies 0 ae 1 and that t (r) ae) lim x ae Gamma t (r) x) is finite. By using this representation corresponding ....

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M. Drmota, Systems of functional equations, Random Structures and Algorithms, 10, (1997), 103--124.

....exist positive constants c 1 and c 2 such that the estimate c 1 p n Ewn c 2 p n log n (3) holds. The exact order of magnitude was left as an open problem. Aldous conjectured [1, Conj. 4] that Ln (suitably normalized) converges to Brownian excursion local time. This was first proved in [10], later by different methods by Kersting [21] and Pitman [31] Partial results go back to [6, 16, 19, 26, 34] This result implies that wn = p n weakly converges to the maximum of Brownian excursion local time, which was proved directly by Tak acs [33] Thus this suggests (but does not imply) ....

....there exist constants c 1 ; c 2 such that for every s; t 0 E jl n (t s) Gamma l n (t)j 2d c 1 e Gammac 2 t s d : 5) Proof(Sketch) 5) is equivalent to E jLn (r) Gamma Ln (r h)j 2d c 1 e Gammac 2 r= p n h d n d=2 (6) which is quite similar to [10, Theorem 6. 1] From [10] it follows that E jLn (r) Gamma Ln (r h)j 2d = 1 an [z n ]H rh (z) in which H rh (z) u u 2d y r (z; uy h (z; u Gamma1 a(z) fi fi fi fi fi u=1 : with y 0 (z; u) u y i 1 (z; u) z (y i (z; u) i 0: 7) In order to evaluate this we use a transfer lemma of ....

M. Drmota and B. Gittenberger, On the profile of random trees, Random Structures and Algorithms 10 (1997), 421--451.

....l (d) n (t) w c d 2 l 2 t in C[0; 1) as n 1, in which c d = d 1 d 1 = Remark 1. It is well known that the expected value of the number of nodes of degree d in trees of size n is approximately c d n. This property is re ected by the processes l (d) n (t) In [6] it is shown that the process l n (t) 1 p n Ln (t p n) converges to 2 l 2 t , where Ln (k) denotes the total number of nodes at distance k to the root (in trees of size n) Remark 2. The case g 1 can be treated analogously. All limit theorems throughout this paper remain ....

....d 1 a(z) d 1 a(z) In order to extract the desired coecient asymptotically we will use Cauchy s integral formula with a suitably chosen integration contour and approximate the integrand there. Therefore we need a detailed knowledge of the behaviour of the recursion (2. 1) see [6, 7]: Lemma 1. Let z 0 be the point on the circle of convergence of a(z) which lies on the positive real axis. Set z = z 0 1 x n and = z 0 (a(z) Furthermore assume that ju a(z)j = O 1 p n and x n 0 in such a way that j arg( x)j and 1 r x n 1 ....

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M. Drmota and B. Gittenberger, On the prole of random trees, Random Structures and Algorithms, to appear.

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M. Drmota and B. Gittenberger. On the profile of random trees. Random Structures and Algorithms, 10:421--451, 1997.

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M. Drmota. Systems of functional equations. Random Structures and Algorithms, 10:103{ 124, 1997.

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