D. J. Aldous and J. Pitman, Brownian bridge asymptotics for random mappings, Random Struct. Alg. 5 (1994), 487-512.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 O ez uniformly for z in any compact region contained in a certain # 0 domain; and consequently 1 2n . Theorem 3 applies. An explicit expression for t n is easily obtained by the Lagrange inversion formula t n = n 0#j n . For further information on random mappings, see [2, 29, 34, 47, 60]. Profiles of increasing trees. A plane recursive tree (cf. 16] is a (plane) increasing tree, namely, a labeled rooted tree in which labels along any path from the root form an increasing sequence, with degree set 0, 1, 2, Let L n,m denote the expected number of nodes at depth m in ....

Aldous, D. J. and Pitman, J. (1994) Brownian bridge asymptotics for random mappings. Random Structures and Algorithms 5, 487--512.

....uniformly for z in any compact region contained in a certain Delta 0 domain; and consequently 1 2n Theorem 3 applies. An explicit expression for t n is easily obtained by the Lagrange inversion formula t n = n Gamma 1) 0j n : For further information on random mappings, see [2, 29, 34, 47, 60]. Profiles of increasing trees. A plane recursive tree (cf. 16] is a (plane) increasing tree, namely, a labeled rooted tree in which labels along any path from the root form an increasing sequence, with degree set f0; 1; 2; g. Let L n;m denote the expected number of nodes at depth m in a ....

Aldous, D. J. and Pitman, J. (1994) Brownian bridge asymptotics for random mappings. Random Structures and Algorithms 5, 487--512.

....local time of x (t) at level a, defined by (a) lim 0 1 Z 1 0 I [a;a ] x (t) dt: Several representations of the one dimensional density of this process are known. Though there is no direct study of this process, results on symmetric random walks or random mappings ([2, 7]) yield various density representations (see [21, 22, 7] Apart from the random mapping (see Sec. 2.3) applications of the rBB local time can be found in the analysis of Shellsort : see Louchard [17, Sec. 2] We need the distribution of the number I(2n) of inversions in a 2 ordered permutation ....

ALDOUS, D. and PITMAN, J. (1994), Brownian bridge asymptotics for random mappings. Random Struct. Alg. 5, 487--512.

....into itself equipped with the uniform distribution. These mappings can be represented by functional digraphs consisting of components which are cycles of trees. The set of points in distance r from a cycle is called the rth stratum of a random mapping. This parameter was previously studied in [2, 8, 11, 29, 32]. For general results on random mappings and literature see [24, 14] Let Mn (r) denote the number of nodes in the rth stratum of a random mapping of size n. Then in [11] we proved Theorem 6. Let B(t) denote reflecting Brownian bridge, i.e. a process on the interval [0; 1] which is identical in ....

....Flajolet and Odlyzko s [13] result on the moments of the height. Finally, we want to mention that it is also possible to obtain the moments of the height of a random mapping (this was done by Flajolet and Odlyzko [14] by our method. One has to use the weak limit theorem by Aldous and Pitman [2] and derive a tightness estimate in a similar fashion as has been done in [11] ....

D. J. Aldous and J. Pitman, Brownian bridge asymptotics for random mappings, Random Struct. Alg. 5 (1994), 487--512.

.... the limiting distribution for this case have been established by Proskurin [19] Finally, it should be mentioned that a survey of several related random mapping characteristics as well as the relations to branching processes and random trees are contained in Kolchin s book [17] Aldous and Pitman [4] studied the contour of a random mapping, i.e. the polygonal function obtained by traversing each tree of G successively. They showed that the suitably rescaled contour process weakly converges to reflecting Brownian bridge (rBB) i.e. the process identical in law to (jW (t) Gamma tW (1)j; 0 t ....

D. J. Aldous and J. Pitman, Brownian bridge asymptotics for random mappings, Random Struct. Alg. 5 (1994), 487--512.

....the convergence of widths of excursions of h n to widths of excursions of h . Its proof, given in Subsection 3. 3, requires some care, as the sequence of widths of excursions is not a continuous functional of h n : the proof relies on an extension of the invariance principle that we learned from [4, 7]. Further consequences of Theorem 3.1 are Theorem 1.10 and also some results about stochastic processes developped in [16] Theorem 1.3 is the consequence of Theorem 4.1, an extension of Theorem 3.1. The case = 0, 1 of Theorem 3.1 was developped in [15, Section 4] for the study of the width ....

D. J. Aldous & Jim Pitman, Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5, No. 4, 487-512 (1994).

....states the convergence of widths of excursions of h n to widths of excursions of h . Its proof, given in Subsection 3. 3, requires some care, as the sequence of widths of excursions is not a continuous functional of h n : we rely on an extension of the invariance principle that we learned from [4, 7]. Further consequences of Theorem 3.1 are Theorem 1.10 and also some results about stochastic processes developped in [16] Theorem 1.3 is the consequence of Theorem 4.1, an extension of Theorem 3.1. The case = 0, 1 of Theorem 3.1 was developped in [15, Section 4] for the study of the width ....

D. J. Aldous & Jim Pitman, Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5, No. 4, 487-512 (1994).

....local time of x (t) at level a, de ned by (a) lim 0 1 Z 1 0 I [a;a ] x (t) dt: Several representations of the one dimensional density of this process are known. Though there is no direct study of this process, results on symmetric random walks or random mappings ([2, 7]) yield various density representations (see [21, 22, 7] Apart from the random mapping (see Sec. 2.3) applications of the rBB local time can be found in the analysis of Shellsort : see Louchard [17, Sec. 2] We need the distribution of the number I(2n) of inversions in a 2 ordered permutation ....

ALDOUS, D. and PITMAN, J. (1994), Brownian bridge asymptotics for random mappings. Random Struct. Alg. 5, 487-512.

....fragmentation process of excursions of Z a appears in the study of coalescence models [8, 9, 13] an emergent topic in probability theory and an old one in physical chemistry, astronomy and a number of other domains [4, Section 1. 4] See [4] for background and an extensive bibliography, and also [3, 6, 16] among others. As explained later, Y a is tightly related to Z a through a path transformation, due to Vervaat [33] connecting e and b. Remark 1.1 For a = 0, we have X 0 law = e [25, Lemma 12] and, trivially, Y 0 = b Gamma min b and Z 0 = e, and the identity up to shift of these reduces to ....

....already used by Konheim Weiss. The two recent and beautiful papers by Flajolet, Poblete Viola [17] and Knuth [19] drew the attention of the authors to the connection between parking schemes and Brownian motion (see also [13, 14] For a similar connection between trees and Brownian motion, see [2, 6, 25, 30], among others. Let P n;m denote the set of all parking schemes of m cars on n places, and let CP n;m denote the subset of confined parking schemes, confined meaning that the last place is 6 assumed to be left empty. We have #P n;m = n m and #CP n;m = n m Gamma1 (n Gamma m) the last can ....

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D.J. Aldous & J. Pitman, Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 (1994), no. 4, 487--512.

....fragmentation process of excursions of Z a appears in the study of coalescence models [8, 9, 13] an emergent topic in probability theory and an old one in physical chemistry, astronomy and a number of other domains [4, Section 1. 4] See [4] for background and an extensive bibliography, and also [3, 6, 16] among others. As explained later, Y a is tightly related to Z a through a path transformation, due to Vervaat [33] connecting e and b. Remark 1.1 For a = 0, we have X 0 law = e [25, Lemma 12] and, trivially, Y 0 = b min b and Z 0 = e, and the identity up to shift of these reduces to the ....

....already used by Konheim Weiss. The two recent and beautiful papers by Flajolet, Poblete Viola [17] and Knuth [19] drew the attention of the authors to the connection between parking schemes and Brownian motion (see also [13, 14] For a similar connection between trees and Brownian motion, see [2, 6, 25, 30], among others. 6 14 1 2 4 3 6 6 7 8 9 9 10 11 11 11 12 13 5 12 13 13 14 14 15 16 16 17 18 19 20 21 22 23 24 20 21 21 21 21 21 14 1 2 4 3 6 6 7 8 9 9 10 11 11 11 12 13 5 12 13 13 14 14 15 16 16 17 18 19 20 21 21 21 21 21 21 14 1 2 4 3 6 6 7 8 9 9 10 ....

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D.J. Aldous & J. Pitman, Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 (1994), no. 4, 487-512.

....basic ideas are singularity analysis (introduced by Flajolet and Odlyzko [12] and saddle point approximation. Random mappings are widely and intensively discussed in literature; e.g. in Kolchin s book [15] a probabilistic approach via branching processes is presented, whereas Aldous and Pitman [3] use a completely di#erent probabilistic concept related to Aldous s continuum random trees [1, 2] Our concept of generating functions goes back to Arney and Bender [4] and to Flajolet and Odlyzko [13] They could identify many limiting distributions and provided asymptotic expansions for mean ....

D. J. Aldous and J. Pitman, Brownian bridge asymptotics for random mappings, Random Structures Algorithms, 5 (1994), pp. 487--512.

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D.J. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures Algorithms, 5:487--512, 1994.

....that mixtures over of (0; partitions could arise by suitable choice of v ffl so that j Pi n j= logn had a non degenerate limit distribution, but this phenomenon does not seem to arise naturally in combinatorial examples. Another case, treated by Pavlov [282, 283, 284] and Aldous Pitman [5] covers a large number of examples involving random forests, where the limit involves the stable subordinator of index 2 . A more general result, where the limit partition is derived from a stable subordinator of index ff for ff 2 (0; 1) can be formulated as follows: Theorem 13 Let w ffl = w j ....

....in distribution to Pi 1 as n 1, where Pi 1 has ranked frequencies distributed like the ranked jumps of (Tu ; 0 u cs) given T cs = 1; 143) where (Tu ; u 0) is the stable subordinator of index ff with E exp( GammaT u ) exp( Gammau ) Sketch of proof. This was argued in some detail in [5] for the particular weight sequence w j = j , corresponding to blocks with an internal structure specified by a rooted labeled tree. Then = e , ff = and the limiting partition can also be described in terms of the lengths of excursions of a Brownian motion or Brownian bridge, as ....

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D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....have a single basin . 33 6.2 Exchangeable interval partitions . 34 6.3 Intensity measures . 37 6.4 Two orderings of a bivariate Poisson process . 38 1 Introduction In a previous paper [2] we showed how features of a uniformly distributed random mapping M n , from [n] f1; 2; ng to itself, could be encoded as functionals of a particular non Markovian random walk on the non negative integers. This mapping walk, suitably rescaled, converges weakly in C[0; 1] as n 1 to the ....

....absolute value of a standard Brownian bridge B with B 0 = B 1 = 0 obtained by conditioning a standard Brownian motion B on B 1 = 0. Two important features of a mapping are the vector of sizes of connected components of its digraph, and the vector of sizes of cycles in its digraph. Results of [2] imply that for a uniform random mapping, as n 1, the component sizes rescaled by n, jointly with corresponding cycle sizes rescaled by n, converge in distribution to a limiting bivariate sequence of random variables ( I j ; L I j ) j=1;2; where (I j ) j=1;2; is a random interval ....

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D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....family to the lengths of excursions of a Markov process whose zero set is the range of a stable subordinator of index #. Section 8 provides further detail in the case # = which corresponds to partitioning a time interval by the lengths of excursions of a Brownian motion. As shown in [2, 3], it is this stable( model which governs the asymptotic distribution of partitions derived in various ways from random forests, random mappings, and the additive coalescent. See also [5, 9] for further developments in terms of Brownian paths, and [10, 25] for applications to hashing and ....

D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....loi limite, g en eralisant la formule pour sa moyenne due a Flajolet et Odlyzko (1990) Research supported in part by N.S.F. Grants DMS 9970901 and DMS 0071448 1 Introduction Let F n be a uniformly distributed random mapping from the set [n] f1; 2; ng to itself, as studied in [10, 8, 1] and papers cited there. The diameter of F n is the random variable Delta n : max i2[n] T n (i) where T n (i) is the number of iterations of F n starting from i until some value is repeated: T n (i) minfj 1 : F n (i) F n (i) for some 0 k jg where F n (i) i and F n (i) F ....

....is the image of i under j fold iteration of F n for j 1. Flajolet Odlyzko [8, Theorem 7] showed by singularity analysis of generating functions that E ( Delta n = n) dv (1) E 1 (v) du I(v) Z v 1 Gamma exp Gamma2u v Gammau du: According to our analysis [1] of the asymptotic distributions of various functionals of random mappings, there is the convergence in distribution P ( Delta n = n x) P ( Delta x) 2) for a limiting random variable Delta which can be constructed as a function of a standard Brownian bridge and a sequence of ....

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D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....and DMS 0071448 1 Introduction A mapping m : n] n] is just a function, identified with its digraph D(m) f(i; m(i) i 2 [n]g. Exact and asymptotic properties of random mappings have been studied extensively in the combinatorial literature since the 1960s [8, 10] Aldous and Pitman [3] introduced the method of associating a mapping walk with a mapping, and showed that (for a uniform random mapping) rescaled mapping walks converge in law to reflecting Brownian bridge. The underlying idea that to rooted trees one can associate tree walks in such a way that random tree walks ....

.... has been developed in many directions over the last 15 years, and this paper, together with a companion paper [4] takes another look at invariance principles for random mappings with better tools. As is well known, the digraph D(m) decomposes into trees attached to cycles. The argument of [3] was that the walk segments corresponding to different cycles, considered separately, converge to Brownian excursions, and that the process of combining these walk segments into the mapping walk turned out (by calculation) to be the same as the way that excursions of reflecting Brownian bridge are ....

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D.J. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures Algorithms, 5:487--512, 1994.

....of Poisson Dirichlet models derived in [36] from the Poisson process of jumps of a stable(ff) subordinator for 0 ff 1. Section 5.8 provides further detail in the case ff = 1 2 which corresponds to partitioning a time interval by the lengths of excursions of a Brownian motion. As shown in [2, 3], it is this stable( 1 2 ) model which governs the asymptotic distribution of partitions derived in various ways from random forests, random mappings, and the additive coalescent. Appendix A collects together some facts about the Hermite function h , which is involved in the description of the ....

....(z) 97) which combined with (96) and h 0 (x) 1 yields h Gamma2 (x) 1 Gamma xh Gamma1 (x) 98) 2 h Gamma3 (x) Gammax (1 x 2 )h Gamma1 (x) 99) 3 h Gamma4 (x) 2 x 2 Gamma (3x x 3 )h Gamma1 (x) 100) and so on, as discussed further in Section A. Recall the notation [ 1 2 ] n Gamma1 = Gamma( 1 2 n) Gamma( 1 2 ) 2n) 2 2n n : Proposition 11 now yields: Corollary 16 The distribution of Pi( a Brownian excursion partition conditioned on L 1 = is determined by the EPPF p 1 2 (n 1 ; n k jj ) 2 n Gammak k Gamma1 h k 1 Gamma2n ....

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D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....natural probabilistic problems discussed here. Probabilists may find motivation in the application of some of the results of this paper to the theory of measure valued and partition valued coalescent processes and the asymptotic structure of large random trees and random mappings, as considered in [3, 4, 5, 24, 33, 71, 69, 14, 66, 6, 7]. This paper is organized as follows. Section 2 explains how a simple formula of Burtin [23] for random mappings is equivalent to a very useful form of Cayley s multinomial expansion over trees, called here the forest volume formula. This formula yields some enumerations of rooted labeled forests ....

....to say, the M s ; s 2 S are independent random variables with common probability distribution p, defined on some probability space( Omega ; P ) See [27, 63, 35, 36] for combinatorial background. There is a large probabilistic literature on the stucture of random mappings for uniform p. See e.g. [56, 3, 41, 65, 7] and papers cited there. The case when all of the p s but one are equal is studied in [83, 64, 18] Random p mappings for general p are studied in [23, 79, 48, 66, 6] See also [23, 31, 44, 45, 16, 17, 19, 50, 13, 75, 40] regarding various other models for random mappings. For each subset B of S ....

D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....subsection) is his j Gamma1 (n Gamma k; k) According to Proposition 7, after these translations each of Pavlov s results describes some asymptotic feature of the partition of n generated by Pi k as in Proposition 6, in various limiting regimes as both n and k tend to 1. Results of [40] and [2] imply that the random sequence N n;k (1) N n;k (2) Delta Delta Delta obtained by ranking the sequence of k component sizes of the random partition Pi k of [n] is such that the normalized sequence N n;k (1) n ; N n;k (2) n ; has a non degenerate limiting distribution ....

....defined by formula (13) Random Mappings. To give one more application of formula (17) let OE be a random mapping from [n] to [n] with uniform distribution on the set of all n n such mappings. Let D be the asociated random digraph with the set of n edges fi OE(i) 1 i ng. 14 See [28, 2] for background. Let C be the set of vertices in cycles of D, let K = #C, and let R be the rooted forest with roots in C obtained by first cutting the edges of D between points in C, then reversing all edge directions. So the edges of R are directed away from C. It is known [21] that P (K = k) ....

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D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....factor (u k Gamma u k Gamma1 ) by ffi Gamma1 sinh(u k Gamma u k Gamma1 )ffi. The existence of a cyclically stationary Brownian local time process was suggested by a problem about random mappings posed by Steve Evans, and the Brownian bridge asymptotics for random mappings of Aldous Pitman [1]. See [2] for details. The process that arises in this setting is ffi L T Gamma1 where T Gamma1 is the hitting time of Gamma1 by B governed by P 0 . Section 5 considers the distribution of ffi L T for various random times T including T Gamma1 . It appears that none of these local time ....

D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....Brownian motion [18, 22, 6] See [14, 17] for various characterizations of the local time processes of Brownian bridge and Brownian excursion, and further references. These local time processes arise naturally both in combinatorial limit theorems involving the height profiles of trees and forests [1, 17] and in the study of level crossings of empirical processes [20, 4] Section 2 of this note presents an elementary derivation of formula (1) based on L evy s identity in distribution [15] 19, Ch. VI, Theorem (2.3) L 0 t ; jB t j) d = M t ; M t Gamma B t ) where M t : sup 0st B s ; ....

....appearing in (13) is the probability density function of B 0 b U for U a uniform[0; 1] variable independent of the bridge. In particular, for b = 0, formula (13) yields jB 0 0 U j d = 1 2 UR (15) where the Rayleigh variable R is independent of U . This and related identities were found in [1], where the reflecting bridge (jB 0 0 s j; 0 s 1) was used to describe the asymptotic distribution of a path derived from a random mapping. Recall that oe x is the first hitting time of x by the Brownian motion B. Let fl x 1 denote last time B is at x before time 1, with the convention fl ....

D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....br at time U , and the minimum of B br after time U is specified by (B br (0, U) B br (U) B br (U, 1) d = 1 2 # 2# 2 ( V, U V, U 1) 18) In particular, B br (U) d = 1 2 # 2# 2 U V . It is easily checked that this agrees the observation of AldousPitman [2] that B br (U) d = 1 2 # 2# 1 U , corresponding to the formula [71, p. 400] P (B br (U) # dx) dx = # # 2 x e 1 2 y 2 dy (x # R ) 19) See also [62] for various related identities. Corollary 16 in Section 8 gives a sampling identity for B br,x which reduces for x = 0 to ....

D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5:487--512, 1994.

....obscures symmetry properties (e.g. size biasing, section 5.3) of the underlying random graph. This paper continues a line of work which identifies limits in probabilistic combinatorics with Brownian type processes (random trees with Brownian excursion [1, 2] random mappings with Brownian bridge [5]) But these previous results (based on depth first search ) are not used here, and indeed the weak convergence arguments in this paper are technically rather easier. Pitman [23] discusses a similar kind of problem numbers of excursions containing j marks for a different process (Brownian ....

D.J. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Struct. Alg., 5:487--512, 1994.

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D. J. Aldous and J. Pitman, Brownian bridge asymptotics for random mappings, Random Struct. Alg. 5 (1994), 487-512.

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