J. Bertoin, and J. W. Pitman.Path transformation connecting Brownian bridge, excursion and meander. Bull. Sci. Math.,vol. 118 (2), p. 147--166, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....212, Boulevard du Triomphe, B 1050 Bruxelles, Belgium, email:louchard ulb.ac.be. where 1 and 2 are arbitrary positive constants. In order to nd the distribution of the BB local time at the origin we may use the results in Louchard [12] or the path transformation techniques in Bertoin and Pitman [1]. But for illustration we use use t (t; x) D t:t (1; x= t) in conjunction with Equ. 1) to get t (t;0) x(t) 0 dt = 1 Inverting on leads to x(t) 0] 2 db] dt = e 2 b=2 or x(t) 0] 2 db] 8t) 4t) which is nothing but the Rayleigh density. 1.4. ....

J. Bertoin and J. Pitman, Path transformations connecting Brownian bridge, excursion and meander, Bull. Sci. Math. 118 , no. 2, 147-166, 1994.

....also a tentative short biography. 1. Classical lattice paths A good reference for classical results on lattice paths is [35] more recent works show that lattice paths are still a subject of an intensive activity in combinatorics [24, 25, 27, 34, 42, 3, 38, 22, 26, 1] and in probability theory [2, 23]. Most of the classical number sequences or lattice paths have a name which is related to some famous mathematicians such as the Italian Leonardo Fibonacci ( 1170 1250) the French Blaise Pascal (1623 1662) the Swiss Jacob Bernoulli (1654 1705) the Scottish James Stirling (1692 1770) the ....

Jean Bertoin and Jim Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bulletin des Sciences Mathematiques, 118(2):147-166, 1994.

....some corollaries of Proposition 17 that relate to the Brownian bridge, the Brownian excursion, the Brownian meander, and the three dimensional Bessel process. We prove these results by applying well known path transformations that enable us to construct one of these processes from another. See [9] for a discussion of a large collection of such transformations. Lemma 22 below contains the path transformation results that we will use. Lemma 22 Let (B t ) t0 be a one dimensional Brownian motion started at zero, and let (R t ) t0 be a three dimensional Bessel process started at zero. Then, ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion, and meander. Bull. Sci. Math., 118:147--166, 1994.

....in the remaining sections. 2 Main results In this section we give precise descriptions of the shifts connecting the three processes X a , Y a and Z a , in all six possible directions. Let a 0 be fixed. First, assume that the Brownian bridge b is built from e using Vervaat s path transformation [10, 11, 33]: given a uniform random variable U , independent of e, b(t) e(U t) Gamma e(U) 2.1) Then Psi a b(t) Psi a e(U t) so that: Theorem 2.1 For U uniform and independent of Z a , Z a (U Delta) law = Y a : As a consequence, Y a is a stationary process on the line, or on the ....

....seen above in another way. A far less obvious result is: Theorem 2.2 For U uniform on [0; 1] and independent of X a , X a (U Delta) law = Y a : The proof will be given later. The case a = 0 of Theorem 2.2 is just Vervaat s path transformation, since, as remarked above, X 0 law = e. In [10], one can find a host of similar path transformations connecting the Brownian bridge, excursion and meander. 3 Corollary 2.3 The occupation measures of X a , Y a and Z a coincide, and have the distribution function 1 Gamma e Gamma2ax Gamma2x 2 : This is also the distribution function of Y ....

J. Bertoin & J. Pitman, Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994), no. 2, 147--166.

....the remaining sections. 3 2 Main results In this section we give precise descriptions of the shifts connecting the three processes X a , Y a and Z a , in all six possible directions. Let a 0 be xed. First, assume that the Brownian bridge b is built from e using Vervaat s path transformation [10, 11, 33]: given a uniform random variable U , independent of e, b(t) e(U t) e(U) 2.1) Then a b(t) a e(U t) so that: Theorem 2.1 For U uniform and independent of Z a , Z a (U ) law = Y a : As a consequence, Y a is a stationary process on the line, or on the circle R=Z, as was seen ....

....as was seen above in another way. A far less obvious result is: Theorem 2.2 For U uniform on [0; 1] and independent of X a , X a (U ) law = Y a : The proof will be given later. The case a = 0 of Theorem 2.2 is just Vervaat s path transformation, since, as remarked above, X 0 law = e. In [10], one can nd a host of similar path transformations connecting the Brownian bridge, excursion and meander. Corollary 2.3 The occupation measures of X a , Y a and Z a coincide, and have the distribution function 1 e 2ax 2x 2 : This is also the distribution function of Y a (t) for any xed ....

J. Bertoin & J. Pitman, Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994), no. 2, 147-166.

....of the law of L as a random time change of the Brownian excursion: 1 2 L s=2 ; s 0) d = B Gamma1 (s) s 0) for (t) Z t 0 1=B s ds where d = indicates equality in law. Tak acs [14] gives a combinatorial approach to formulas for the marginal law of L s . Bertoin Pitman [3] discuss transformations between Brownian excursion and other Brownian type processes. References to further papers on standard Brownian excursion can be found in those references. Consider the question Given a function = s) 0 s 1) can we define a process B = B u ; 0 u 1) whose ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math., 118:147--166, 1994.

....normalized excursion has the same distribution as the range of the Brownian bridge, which suggests that there would exist a close relationship between the two processes. The answer is in affirmative, as is revealed by the following theorem. For detailed surveys of Brownian path decompositions, cf. Bertoin and Pitman (1994), Biane (1993) Yor (1995, Lecture 4) Theorem C (Vervaat 1979) Let ffl(t) 0 t 1g be a standard Brownian bridge process, and U the almost surely unique location of the minimum of fl, i.e. such that fl(U) inf 0t1 fl(t) Then U is uniformly distributed in (0; 1) Furthermore, ae(t) def = ....

....1 o is referred to by Chung (1976) as the Brownian meander process. It is observed by Kennedy (1976) that the supremum of the meander is distributed as 2 sup 0t1 jfl(t)j, where fl is a Brownian bridge. A pathwise explanation to this ( a la Vervaat) is provided by Biane and Yor (1987) cf. also Bertoin and Pitman (1994). The following analogue of Jeulin s theorem (Theorem D) for the local time of the reflecting Brownian bridge is known. Theorem E (Biane and Yor 1987) Let K(s) def = R s 0 L x 1 (jflj) dx for all s 0, then n 1 2 L K Gamma1 (t) 1 (jflj) 0 t 1 o 21 is distributed as a ....

Bertoin, J. and Pitman, J.W. (1994) Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math., 118, 147--166.

....0 a b 1, define X [a;b] def = i 1 p b Gamma a X a t(b Gammaa) 0 t 1 j : Then, b def = B [0;g] is a standard Brownian bridge; m def = jBj [g;1] is a Brownian meander; b, m and g are independent. Another representation of the meander process is given by ( 3] [1]) jb u j u ; u 1) where ( u ) is the local time at 0 of the bridge b. We also recall the following representation of the meander ( 1] 14, Exercise XII.4.25] 2 sup su b s Gamma b u ; u 1) We now study the joint law of (g; fl) on two disjoint events: fg flg and fg flg. 2.1. ....

....bridge; m def = jBj [g;1] is a Brownian meander; b, m and g are independent. Another representation of the meander process is given by ( 3] 1] jb u j u ; u 1) where ( u ) is the local time at 0 of the bridge b. We also recall the following representation of the meander ([1], 14, Exercise XII.4.25] 2 sup su b s Gamma b u ; u 1) We now study the joint law of (g; fl) on two disjoint events: fg flg and fg flg. 2.1. First situation: g fl 4 Observe that on fg flg, fl = supft g : jB t j = L t g = g supfu 1 : jb u j = u g; where ( u ) denotes ....

[Article contains additional citation context not shown here]

Bertoin, J. and Pitman, J.W.: Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994) 147--166.

No context found.

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994. 36

.... derived by Brownian scaling: B [0; T ] B[0; 1] B [0; GT ] 13) jBj [G T ; T ] jBj [GT ; DT ] 14) It is also known that B ; B can be constructed by various other operations on the paths of B, and transformed from one to another by further operations [42]. For 0 t 1 let B be a Brownian bridge of length t, which may be regarded as a random element of C[0; t] or of C[0; 1] as convenient: s) tB ( s=t) 1) s 0) 15) Let B me;t denote a Brownian meander of length t, and B be a Brownian excursion of length t, defined ....

....like B with B 0 = a conditioned to hit b before 0. R 3 (t) B(t) Gamma 2B(t) t 0) 18) where B is a standard Brownian motion with past minimum process B(t) B[0; t] GammaR 3 [t; 1) L evy s identity The identity in distribution (18) admits numerous variations and conditioned forms [292, 42, 44] by virtue of L evy s identity of joint distributions of paths [331] B Gamma B; GammaB ) jBj; L) 19) where L : L t ; t 0) is local time process of B at 0. L evy Ito Williams theory of Brownian excursions Due to (19) the process of excursions of jBj away from 0 is equivalent in ....

[Article contains additional citation context not shown here]

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

.... defined by the following identities in distribution, valid for all t 0: B [0; G t ] B [G t ; D t ] B [G t ; t] 14) where G t : G t (B) D t : D t (B) and for any process X we use the notation G t (X) supfu t : X u = 0g (15) D t (X) inffu t : X u = 0g: 16) See [9] for a review of properties of B , B in the Brownian case when B is Brownian motion with state space R, and fi = ff = See [25, x3] and [34] for some treatment of B in the Bessel case when B with state space R0 is a recurrent Bessel process of dimension ffi = 2 Gamma 2ff 2 (0; ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....: of uniform r.v. s on [0; 1] independent of the excursions, and then let Y s = i A if l j s U j U i l j : It is well known that such a process is a reflecting Brownian bridge on [0; 1] conditioned to have local time 2A at level 0 and time 1. To conclude, we recall (see e.g. [5]) that 2A = 2B U has Rayleigh law, which is that of L 1 , half the occupation measure of B at level 0 and time 1. So J d = B . 3.5 Completing the proof of Theorem 1 As in Proposition 3, we may take the p mapping M n in its representation M n = J(T n ; X 1;n ) where T n is a ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....t ; 0 t 1) 61) The surprising consequence of (54) and (60) that max 0t1 m t = 2 max 0t1 jb t j, was explained in [8] by a transformation of bridge b into a process distributed like the meander m. For a review of various transformations relating Brownian bridge, excursion and the meander see [6]. 17 4.2 Bessel processes The work of Williams [77, 78, 79] shows how the study of excursions of one dimensional Brownian motion leads inevitably to descriptions of these excursions involving higher dimensional Bessel processes. For d = 1; 2; let R d : R d;t ; t 0) be the d dimensional ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

.... the standard Brownian excursion.Thestandard Brownian meander is recovered as B me : B me,# ,where# = B me 1 is independent of the three bridges with the Rayleigh density P (# # dx) dx = xe 1 2 x 2 (x 0) The above descriptions of B me,r and B me are read from [20, 12] See also [8, 3, 6, 17, 18] for further background. Many other path transformations relating these processes are known. For instance, the transformation of Vervaat [19] see also Biane [4] and Imhof [13] shows that the standard Brownian excursion can be obtained by transposing the pre minimum and the post minimum parts of ....

....and the post minimum parts of a standard Brownian bridge. Analogously, reversing the pre minimum part and then tacking on the post minimum part of a standard Brownian bridge from 0 to 0 yields a standard Brownian meander, as shown by Bertoin [2] We refer to Biane and Yor [5] Bertoin and Pitman [3], Chaumont [7] and Yor [22] for many further results in this vein. The work of Williams [21] and Denisov [10] shows how the path of B over [0, 1] decomposes at the a.s. unique time of its minimum on [0, 1] into two path fragments, which given are are two independent Brownian meanders of lengths ....

[Article contains additional citation context not shown here]

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull.Sci.Math.(2), 118:147--166, 1994.

....of a standard Brownian excursion found by Kennedy [16] and Chung [9] is identical to the law of I M , known as the amplitude or range of the bridge, whose distribution is given by the formula [12] P (I M b) 2 1 X k=1 (4k 2 b 2 Gamma 1) exp( Gamma2k 2 b 2 ) 3) for b 0. See also [2] for a survey of transformations related to Vervaat s construction. As observed by Chung [9] the distribution of I M is characterized by the Laplace transform E exp( Gamma 1 2 2 (I M) 2 ) 2 sinh( 2 ) 4) while that of I M is characterized by the companion formula E ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....Brownian excursion. The standard Brownian meander is recovered as B me : B me;ae , where ae = B me 1 is independent of the three bridges with the Rayleigh density P (ae 2 dx) dx = xe Gamma 1 2 x 2 (x 0) The above descriptions of B me;r and B me are read from [20, 12] See also [8, 3, 6, 17, 18] for further background. Many other path transformations relating these processes are known. For instance, the transformation of Vervaat [19] see also Biane [4] and Imhof [13] shows that the standard Brownian excursion can be obtained by transposing the pre minimum and the post minimum parts of ....

....and the post minimum parts of a standard Brownian bridge. Analogously, reversing the pre minimum part and then tacking on the post minimum part of a standard Brownian bridge from 0 to 0 yields a standard Brownian meander, as shown by Bertoin [2] We refer to Biane and Yor [5] Bertoin and Pitman [3], Chaumont [7] and Yor [22] for many further results in this vein. The work of Williams [21] and Denisov [10] shows how the path of B over [0; 1] decomposes at the a.s. unique time of its minimum on [0; 1] into two path fragments, which given are are two independent Brownian meanders of lengths ....

[Article contains additional citation context not shown here]

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....; 0 t 1) 61) The surprising consequence of (54) and (60) that max 0t1 m t d = 2 max 0t1 jb t j, was explained in [8] by a transformation of bridge b into a process distributed like the meander m. For a review of various transformations relating Brownian bridge, excursion and the meander see [6]. 17 4.2 Bessel processes The work of Williams [77, 78, 79] shows how the study of excursions of one dimensional Brownian motion leads inevitably to descriptions of these excursions involving higher dimensional Bessel processes. For d = 1; 2; let R d : R d;t ; t 0) be the ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

.... s ; 0 s 1) by the following path transformation: x B 0 b s : 8 : B 0 b oe x s Gamma x if 0 s 1 Gamma oe x b Gamma B 0 b 1 Gammas if 1 Gamma oe x s 1 with oe x the first hitting time of x by (B 0 b s ; 0 s 1) The invariance (7) can be checked by a standard technique [5, 20]: the corresponding transformation on lattice paths is a bijection, which gives simple random walk analogs of (7) 6) and (5) the results for the Brownian bridge then follow by weak convergence. Formula (7) implies also that the process (oe x ; 0 x b) derived from (B 0 b s ; 0 s 1) has ....

....classical identity is equivalent to the duplication formula for the gamma function [21] Another Brownian representation of (20) is jB 0 2 j d = q 2 Gamma fl 0 2 M 1 (21) where M 1 is the final value of a standard Brownian meander. Compare with [19, Ch. XII, Ex s (3.8) and (3. 9) and [5]. See also [3, p. 681] for an appearance of (20) in the study of random trees. Acknowledgment Thanks to Marc Yor for several stimulating conversations related to the subject of this note. ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....does not depend on the choice of T , each rescaled process is independent of T if T is a random variable independent of B. It is also known that these processes can be constructed by various other operations on the paths of B, and transformed from one to another by further operations. See [12] for a review of results of this kind. One well known construction of B br,x from B is B br,x (u) B(u) uB(1) ux (0 # u # 1) 4) Then B me,r can be constructed from three independent standard Brownian bridges B br i , i = 1, 2, 3 as B me,r (u) # (ru B br 1 (u) 2 ....

....is independent of the three bridges B br i , with the Rayleigh density P (# # dx) dx = xe 1 2 x 2 for x 0. Then by construction B me (1) # = # 2# 1 (6) where # 1 is a standard exponential variable. The above descriptions of B me,r and B me are read from [78, 41] See also [21, 12, 15, 68] for further background. This paper characterizes each member X of the Brownian quartet B, B br , B me , B ex in a very di#erent way, in terms of the distribution of values obtained by sampling X at independent random times with uniform distribution on (0, 1) The characterization of ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....u = 1 p t Gamma g t jB g t u(t Gamma g t ) j ; 0 u 1 ; known as a Brownian meander [10] has a law which does not depend on t, and the meander is independent of oefB u ; 0 u g t g. Details of this decomposition can be found in Revuz Yor [33] and Yor [48] See also Bertoin Pitman [4] for a survey of various transformations relating the bridge, excursion and meander derived as above from Brownian motion. d. Finally, as a consequence of the strong Markov property of BM, the process (B d t u ; u 0) is a BM independent of (B u ; 0 u d t ) 2.2 A decomposition of (B u ; 0 u ....

J. Bertoin and J. Pitman, Path transformations connecting Brownian bridge, excursion and meander, Bull. Sci. Math. (2), 118 (1994), pp. 147--166.

....a Brownian meander of length t. It is well known that these Brownian bridges, excursions and meanders of length t can be constructed by Brownian scaling from the corresponding standard processes of length 1. If the length of one of these processes is not mentioned, it is assumed to be 1. See [65, 5] for background and further references to these processes. For a suitable continuous function f with domain containing [0; t] let L t;v (f) denote the local time of f up to time t at level v as defined by the occupation density formula Z t 0 g(f u )du = Z 1 Gamma1 g(v)L t;v (f)dv (1) for ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

....does not depend on the choice of T , each rescaled process is independent of T if T is a random variable independent of B. It is also known that these processes can be constructed by various other operations on the paths of B, and transformed from one to another by further operations. See [12] for a review of results of this kind. One well known construction of B br;x from B is B br;x (u) B(u) Gamma uB(1) ux (0 u 1) 4) Then B me;r can be constructed from three independent standard Brownian bridges B br i ; i = 1; 2; 3 as B me;r (u) q (ru B br 1 (u) 2 ....

....of the three bridges B br i , with the Rayleigh density P (ae 2 dx) dx = xe Gamma 1 2 x 2 for x 0. Then by construction B me (1) ae = q 2 Gamma 1 (6) where Gamma 1 is a standard exponential variable. The above descriptions of B me;r and B me are read from [77, 41] See also [21, 12, 15, 67] for further background. This paper characterizes each member X of the Brownian quartet fB; B br ; B me ; B ex g in a very different way, in terms of the distribution of values obtained by sampling X at independent random times with uniform distribution on (0; 1) The characterization of ....

J. Bertoin and J. Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. (2), 118:147--166, 1994.

No context found.

J. Bertoin, and J. W. Pitman.Path transformation connecting Brownian bridge, excursion and meander. Bull. Sci. Math.,vol. 118 (2), p. 147--166, 1994.

No context found.

J. Bertoin and J. Pitman, Path transformations connecting Brownian bridge, excursion and meander, Bull. Sci. Math. 118 , no. 2, 147-166, 1994.

No context found.

Jean Bertoin and Jim Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bulletin des Sciences Mathematiques, 118(2):147-166, 1994.

No context found.

Bertoin, J. and Pitman, J. (1994). Path transformations connecting Brownian bridge, excursion, and meander. Bull. des Sciences Mathematiques 118, 147-166.

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