P. Chassaing and S. Janson (2001). A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab. 29, 1755-1779. 11  Home/Search   Document Details and Download   Summary   Related Articles   Check

....index ff instead of index 2 remains open. A natural guess is that such a construction might be made with one of the self similar fragmentation processes of Bertoin [40] but Miermont and Schweinsberg [260] have recently shown that a construction of this form is possible only for ff = See [76] for more about conditioning B on its local time at 0, and [18] 370] regarding the more difficult problem of conditioning a Brownian path fragment on its entire local time process. 91 5 Random walks and random forests The main point of this lecture is summarized by the following paragraph, ....

....the limit, using almost sure instead of weak convergence. 3. Vervaat s transformation of a L evy bridge) 80, 259] Generalize the result of the previous exercise to bridges and excursions derived from a suitable L evy process with no negative jumps instead of Brownian motion. See also [81] and [76] for further variations of Vervaat s transformation. 5.2 Galton Watson forests It is well known to combinatorialists that the enumerations of lattice paths (275) 276) related to the Lagrange inversion formula can also be expressed, by suitable bijections, as enumerations of various sets of ....

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P. Chassaing and S. Janson. A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab., 29(4):1755--1779, 2001.

....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 parking algorithms. This paper is a revision of the earlier preprint [42] See [48] for a broader context and further developments. 2 Preliminaries This section recalls some basic ideas from ....

P. Chassaing and S. Janson. A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab., 29(4):1755-- 1779, 2001.

....the equality in (7) combined with scaling arguments is sufficient to establish that the fragmentation kernels for these two processes must be the same. See section 7 of [13] for the necessary scaling arguments. In section 4, we will show how (7) follows from a path transformation result in [12] (see Lemma 18 and Remark 19) 3 The work in [6] raises further questions pertaining to the processes (F ( 0 and ( Psi e t ) 0tl . The main purpose of this paper is to answer three such questions using the continuous time ballot theorem. We introduce the three questions in subsections 1.1, ....

.... of Proposition 10 of [6] is that the length of the first excursion interval of Psi e has the same distribution as a size biased pick from the interval lengths in the sequence V 1 ( Psi e) Using the continuous time ballot theorem combined with a path transformation identity proved in [12], we show that the length of the first excursion interval of Psi e is indeed a size biased pick from V 1 ( Psi e) We state this result as Proposition 3 below. Proposition 3 Let e = e t ) 0t1 be a Brownian excursion of length 1, and define Psi e as in (2) Let H = infft : t Gamma e t 0g. ....

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P. Chassaing and S. Janson. A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Available via http://altair.iecn.unancy. fr/~chassain, 1999.

....a random shift. The shift does not affect the integral, and thus we can take W 0 = R 1 0 e(t) the Brownian excursion area, as found by [9] More generally, it follows from Vervaat s result that we can take W a : Z 1 0 max 0st Gamma e(t) Gamma e(s) Gamma a(t Gamma s) Delta dt too [5]. This can also be derived by arguing as above with confined hashing instead of the unconfined version, but the details become technically more complicated, cf. 5, 6, 7] Furthermore, it follows from [5, Theorem 2.2] that W a also can be defined as the integral of a reflecting Brownian bridge ....

.... from Vervaat s result that we can take W a : Z 1 0 max 0st Gamma e(t) Gamma e(s) Gamma a(t Gamma s) Delta dt too [5] This can also be derived by arguing as above with confined hashing instead of the unconfined version, but the details become technically more complicated, cf. [5, 6, 7]. Furthermore, it follows from [5, Theorem 2.2] that W a also can be defined as the integral of a reflecting Brownian bridge jbj conditioned on having local time at 0 equal to a. 3. The dense case: moments Our second proof of Theorem 1.1(iii) is based on expressions for generating functions ....

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P. Chassaing & S. Janson, A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Preprint, 1999. Available at http://www.math.uu.se/svante/papers/

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P. Chassaing and S. Janson (2001). A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab. 29, 1755-1779. 11

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