P. Biane and M. Yor, Valeurs principales associees aux temps locaux Browniens, Bull. Sci. Math. 111 (1987), 23-101.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....; x d ) E [ x 2 ) x d ) x 1 x 2 : x d , denoting the generalized covariances. In this paper we obtain explicit expressions for K(x 1 ; x 2 ; x d ) As an exercice, we revisit two well known processes: the area of the BE (g(x) x) and a result of Biane and Yor [2] related to g(x) 1=x. The paper is organized as follows. Sec. 2 gives the basic formula s we need in the sequel, Sec. 3 provides an ecient algorithm for the generalized covariances computation. In Sec. 4, we consider two typical applications: the Brownian excursion area and the Biane and Yor ....

....as expected. Similarly with G( given by (3) 3 0 x 1 dx 1 x 3 dx 3 G( 6:15 128 4 . Inverting, this leads to 2 15=128 as expected. An interesting open problem is to derive the recurrence (15) from the matrix representation given by Thm. 1. 4. 2 A formula of Biane and Yor In [2], Biane and Yor proved that Y : X(t) 2 ; where : sup X(t) As E [ x 1 ) e 4x 1 , we immediately check that the cost g(x) 1=x 2 ; 0 is the ultimate form of costs of type x . 2 leads to a singularity at the origin. It is well known that the distribution ....

P. Biane and M. Yor. Valeurs principales associees aux temps locaux browniens. Bulletin de la Societe Mathematique de France, 2e serie,111:23-101, 1987.

....as a stochastic process, to t # # #(t) 2. Aldous s conjecture was settled by Drmota and Gittenberger [9] As noted by these last authors, their result entails the weak convergence of W n # n to the maximum m of the Brownian excursion, as #(t) is itself a Brownian excursion changed of time [5]. Previously, the weak convergence of W n # n to m was proven directly by Takacs (1993) However weak convergence does not answer completely the question of Odlyzko Wilf, as it does not yield convergence of the first moment, and even less the speed of this convergence. The aim of our paper ....

....(see [16, 20] It also allows to analyze the size of parking blocks during the phase transition [7] Note that Aldous, or Drmota and Gittenberger s results are actually about general simple trees. Rooted labeled trees are a special case of simple trees, but an important one [16, 20] Recall [5, 8, 15] that the maximum m of the Brownian excursion satisfies Pr(m # x) # # k # (1 4k 2 x 2 )e 2k 2 x 2 , E(m) # # 2 , and, for r 1, E(m r ) 2 r 2 r(r 1)# # r 2 # #(r) We shall say that m is theta distributed by reference to Jacobi s Theta function. Incidentally, it ....

P. Biane, M. Yor, (1987) Valeurs principales associees aux temps locaux browniens. Bull. Sci. Maths 111, 23-101.

....weakly, as a stochastic pr# cess, to t # # #(t) 2. Aldous sconjectur# was settled byDr#]F] and Gittenber #fi] 9] As noted by these lastauthor#L their r#eir# entails the weak conver#Px]F of W # to the maximum m of the Br# wnianexcur#C] C as #(t) is itself aBr# wnian excur#8fi] changed of time [5].Pr#8Ffififi;8 , the weak conver#LL] of W # to m waspr# vendir#CL=8 by Takacs (1993) However weak conver#x[ fi does not answer completely the question of Odlyzko Wilf, as it does not yield conver#=CfiF of the fir#8 moment, and even less the speed of this conver#=PC=8 The aim ofour paper is ....

.... ondence (see [15, 18] It also allows to analyze the size of par#fi] blocksdur#]fi the phasetr#e8F=LFF [7] Note that Aldous,or Dr#ous Gittenber#; P] r##; P] ar# actually about general simple tr#ple Rooted labeled tr#d8 ar# a special case of simple tr#ple but an impor#8; t one [15, 18] Recall [5, 8, 14] that the maximum m of the Br# wnianexcur#;LL satisfies Pr# m # x) # k # (1 4k 2 x 2 )e 2k 2 x 2 , E (m) # 2 , and,for r 1,E (m r ) 2 r 2 r(r 1)# r 2 #(r) We shall say that m is th#j 6fDTVVjjfh d by r#;CC;P82 to Jacobi s Theta function. ....

P. Biane, M. Yor# (1987) Valeurs principales associees aux temps locaux browniens.

....for the space D[0; 1) in which step functions are allowed (see [4] and in fact, by using direct (but messy) extensions of the method presented in section 6 we are also able to prove the original conjecture. Since the distribution of sup t0 l(t) is the same as that of 2 sup 0t1 W (t) see [3] or [1] which has been shown to be P ae sup 0t1 W (t) x oe = 1 Gamma 2 X k1 (4x 2 k 2 Gamma 1)e Gamma2x 2 k 2 (1.4) by Kennedy [16] Theorem 1.1 immediately implies the following property for the width of trees: Corollary. Under the assumption of Theorem 1.1 we have sup ....

P. Biane and M. Yor, Valeurs principales associees aux temps locaux Browniens, Bull. Sci. Math. 111 (1987), 23--101.

....t 0 (t) t) has a minimal positive solution R. Then we have (l n (t) t 0) w Gamma (l(t) t 0) in C[0; 1) as n 1. Thus we have Theorem 2. Under the assumptions of Theorem 1 we have sup t0 l n (t) w Gamma sup t0 l(t) The maximum of local time is well studied (cf. [20, 5, 12, 3, 26]) We have sup t0 l(t) d = 2 sup 0t1 W (t) P ae sup 0t1 W (t) x oe = 1 Gamma 2 X k1 (4x 2 k 2 Gamma 1)e Gamma2x 2 k 2 ; and E sup t0 l(t) p = 2 Gammap=2 p(p Gamma 1) Gamma i p 2 j i(p) Recently, Marckert and Chassaing [27] used the relation of ....

P. Biane and M. Yor, Valeurs principales associees aux temps locaux Browniens, Bull. Sci. Math. 111 (1987), 23--101.

....which step functions are allowed (see [4] and in fact, by using direct (but messy) extensions of the method presented in section 4 we are also able to prove the same assertion for the step function process. Since the distribution of sup t 0 l(t) is the same as that of 2 sup 0 t 1 W (t) see [3] or [1] which is indeed a beta distribution Theorem 1, immediately implies the following property for the maximal width of trees. Corollary . Under the assumption of Theorem 1 we have sup t 0 l (d) n (t) w c d sup 0 t 1 W (t) as n 1. ON NODES OF GIVEN DEGREE IN RANDOM TREES 3 ....

P. Biane and M. Yor, Valeurs principales associees aux temps locaux Browniens, Bull. Sci. Math. 111, (1987), 23-101.

....the pre minimum 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 ....

....B s ,thenM 1 = UR 1 where U is uniform on [0, 1] independent of R. There is an exact analog for lattice walks, which can be given a bijective proof and then passed to the limit as in [15] and [14] Theorem 4 can also be deduced from excursion theory, or by the techniques developed by Biane and Yor [5]. Acknowledgement. We thank Jean Francois Le Gall for posing the problem of finding a simple construction of Brownian motion conditioned on its minimum. ....

Ph. Biane and M. Yor. Valeurs principales associees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....the pre minimum 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 ....

....s , then M 1 = UR 1 where U is uniform on [0; 1] independent of R. There is an exact analog for lattice walks, which can be given a bijective proof and then passed to the limit as in [15] and [14] Theorem 4 can also be deduced from excursion theory, or by the techniques developed by Biane and Yor [5]. Acknowledgement. We thank Jean Francois Le Gall for posing the problem of finding a simple construction of Brownian motion conditioned on its minimum. ....

Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....= inffs 0 : L 0 s Deltag, the right continuous inverse of the local time at level 0. Fitzsimmons and Getoor [9] have observed that when X is recurrent, i C oe(t) t 0 j is a symmetric Cauchy process with parameter : 1) More precisely, 1) has first been shown by Biane and Yor [6] in the special case when X is a Brownian motion. Then (1) has been proven in [9] under the additional condition that X is symmetric; and that the symmetry condition can be dropped has been noted in [4] The argument of [9] for establishing (1) is based on a combinatorial lemma on Euler numbers, ....

....that no known description of the excursion measure n is adequate to determine the law of C T . In the 1 By a slight common abuse of notation, we denote both the L evy process and its generic excursion by X. The distinction will always be clear from the context. Brownian case, Biane and Yor [6] have been able to established (2) using decompositions of the excursion measure of Bessel processes and a connection between two Bessel processes of different dimensions based on a simple time change. The main purpose of this work is to point out that there is another situation where this can be ....

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Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux browniens. Bull. sc. Math. 111 (1987), 23-101.

....and phrases: Brownian excursion, continuum random tree, Kingman s coalescent, local time. 1 Introduction Let (Bu ; 0 u 1) be standard Brownian excursion and (L s ; 0 s 1) its local time, more precisely its local time at time 1: Z h 0 L s ds = Z 1 0 1 (Buh) du; h 0: Biane Yor [4] give an extensive treatment, including an elegant description 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 ....

P. Biane and M. Yor. Valeurs principales associees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....of the range of a Brownian bridge (cf. 2.17) and hence also that of L 1 (fl) cf. 2.13) is related to the Jacobi theta function. For probabilistic interpretations of this famous function (and of the Riemann zeta function) in terms of Brownian motion, we refer to Biane et al. 1998) Biane and Yor (1987), Chung (1976) Cs aki (1979) Cs aki and Mohanty (1981; 1986) Csorgo and Horv ath (1997, p. 102) Deheuvels (1985) Smith and Diaconis (1988) Williams (1990) and Yor (1997, Chap. 11) Remark 2.3.2. i) We can choose various functions f and g in Theorem 2.1 to obtain many identities in law, ....

....I; sup x2R L x 1 (fl) This kind of independence is explained and extended by Chaumont (1998) ii) From (2.10) we deduce: S Gamma I law = 1 2 Z 1 0 dt ae(t) Therefore, 2.13) and (2. 14) also express the distribution function of R 1 0 dt=ae(t) For further discussions on this, cf. Biane and Yor (1987), Chung (1976) Pitman and Yor (1996) We also mention Chung s identity in law: if e fl denotes an independent copy of fl, sup 0t1 fl 2 (t) sup 0t1 e fl 2 (t) law = sup 0t1 ae 2 (t) cf. Chung (1976) Yor (1997, p. 16) 9 (iii) The identities (2.13) and (2.14) are ....

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Biane, P. and Yor, M. (1987) Valeurs principales associ'ees aux temps locaux browniens.

....b with 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 a 1 Gamma2a (1 Gamma 2 1 Gamma2a ) Gamma(a 1) i(2a) if a 6= 1=2, p 2= ln 2; if a = 1=2. 9 We now give two confirmations of (4. 1) i) Recall that the Fourier transform (on R) of the function y 7 =4 (cosh( y=2) 2 is = sinh , see e.g. the Table in Biane and Yor [3]. Thus, 4.1) is equivalent to: 1 p 2 Z 1 Gamma1 dy exp(i y) E h S (3) 1 exp i Gamma y 2 (S (3) 1 ) 2 2 ji = sinh ; and it is now easily shown that the left hand side is equal to E h exp i Gamma 2 2 T (3) 1 ji ; which as we already recalled is equal to = ....

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Biane, P. and Yor, M.: Valeurs principales associ'ees aux temps locaux browniens. Bull. Sci. Math. 111 (1987) 23--101.

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P. Biane and M. Yor. Valeurs principales associees aux temps locaux browniens. Bull. Sci. Math., 111:23--101, 1987.

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Ph. Biane, and M. Yor.Valeurs principales associees aux temps locaux browniens. Bull. Sci. Maths.,2eme serie, vol. 111, p. 23--101, 1987.

....next lemma, which follows from the preceding discussion and formulas tabulated in Section 3, gathers different characterizations of a random variable Y with this density. Here Y is assumed to be defined on some probability space( Omega ; F ; P ) with expectation operator E. 6 Proposition 1 ([14, 8]) For a non negative random variable Y , each of the following conditions (i) iv) is equivalent to Y having density y H(y) for y 0: i) 2(s) s 2 C ) 21) ii) for y 0 P (Y y) G(y) yG (y) Gammay (y ) 22) P (Y y) 1 Gamma 2n = 4y ....

.... of [35, 21] that fi fi m t where (m t ; 0 t 1) denotes a standard Brownian meander, defined by the limit in distribution on C[0; 1] fi fi (m 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 ....

[Article contains additional citation context not shown here]

Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....C and S belong to the class of generalized z processes [29] whose definition in recalled in Section 4. The distributions of C 1 , S 1 and T 1 arise in connection with L evy s stochastic area formula [38] and in the study of the Hilbert transform of the local time of a symmetric L evy process [8, 24]. The laws of C t and S t arise naturally in many contexts, especially in the study of Brownian motion and Bessel processes [68, x18.6] For instance, the distribution of C 1 is that of the hitting time of Sigma1 by the one dimensional Brownian motion fi. The distribution of S 1 is that of the ....

....of the hitting time of Sigma1 by the one dimensional Brownian motion fi. The distribution of S 1 is that of the hitting time of the unit sphere by a Brownian motion in R 3 started at the origin [16] while ( 2) p S 2 has the same distribution as the maximum of a standard Brownian excursion [14, 8]. This distribution also appears as an asymptotic distribution in the study of conditioned random walks and random trees [58, 1] The distributions of C t and S t for t = 1; 2 are also of significance in analytic number theory, due to the Mellin representations of the entire function (s) 1 ....

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Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....we know there is still no satisfying explanation in terms of Brownian paths for these remarkable identities found by Chung. For further discussion of these results, their relation to the functional equations satisfied by the Jacobi theta and Riemann theta functions, and various applications, see [5, 4, 30]. Let Q : I= I M ) Cs aki [10, Theorem 2] deduced from (1) a fairly complicated expression for P (I M u; Q v) from which he obtained by letting u 1 the remarkable formula [10, 2.12) P (Q v) 2v 2 (1 Gamma v) 1 X n=1 1 n 2 Gamma v 2 = 1 Gamma v) 1 Gamma v cot(v) ....

....Bernoulli number, and [13, 1. 411.7] 1 X n=1 2 2n jB 2n j (2n) x 2n = 1 Gamma x cot x (jxj ) 20) 5 3 Path decomposition at the maximum We start by formulating the path decomposition of the Brownian bridge at its maximum in terms of the following construction, which we adapt from [28, 29, 19, 5, 20]. See also [21] for variations of this construction and [14, 15, 26] for other decompositions of the Brownian path involving the range process and BES (3) pieces. Construction 1 Given two continuous path processes with random finite lifetimes, each with initial value 0 and final value z, say R ....

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Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....where c M 1 is an independent copy of M 1 and b T 3 an independent copy of T 3 : T 3;1 . As far as we know, no proof of these identities (8) and (10) has ever been given in probabilistic terms, without involving some analysis related to the functional equation for Jacobi s theta function. See [5, 38, 56, 4] for further discussion of this circle of results. Stimulated by these remarkable results involving M ffi and T ffi for ffi = 1 and 3, we were led to study the distributions of M ffi and T ffi for a general ffi 0 and to look for relations involving the distributions of these random variables. ....

....sup 0ut R ffi (u) 1) 5 and Brownian scaling. An analog of (11) for M ffi is provided by the following absolute continuity relation between the law of M Gamma2 ffi and the law of e T ffi : T ffi b T ffi for b T ffi an independent copy of T ffi : Theorem 2 (The agreement formula. [5, 37, 38] For each ffi 0 there is the identity E h g(M Gamma2 ffi j ] C ffi E h e T ffi g( e T ffi ) i (12) for every non negative measurable function g, where C ffi : 2 (ffi Gamma2) 2 Gamma(ffi=2) and : ffi Gamma 2) 2 as in (6) In particular, M Gamma2 ffi d = e T ffi ....

[Article contains additional citation context not shown here]

Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

....next lemma, which follows from the preceding discussion and formulas tabulated in Section 3, gathers different characterizations of a random variable Y with this density. Here Y is assumed to be defined on some probability space( Omega ; F ; P ) with expectation operator E. 6 Proposition 1 ([14, 8]) For a non negative random variable Y , each of the following conditions (i) iv) is equivalent to Y having density y Gamma1 H(y) for y 0: i) E(Y s ) 2(s) s 2 C ) 21) ii) for y 0 P (Y y) G(y) yG 0 (y) Gammay Gamma2 G 0 (y Gamma1 ) 22) that is P (Y y) ....

.... 1 (60) where (m t ; 0 t 1) denotes a standard Brownian meander, defined by the limit in distribution on C[0; 1] jS 2nt j p 2n ; 0 t 1 fi fi fi fi R 2n d (m t ; 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 ....

[Article contains additional citation context not shown here]

Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

.... Gammas=2 Gamma(s=2) 1 X n=1 1 n s ( s 1) 9) and deduced from it and the classical functional equation (t) t Gamma1=2 (t Gamma1 ) t 0) that (9) defines a unique entire function which satisfies the functional equation (s) 1 Gamma s) s 2 C ) As shown by Biane Yor [4], Chung s formula (8) for P (M x) is equivalent to the following expression of the Mellin transform of M : E[M s ] 2 s 2 2 (s) s 2 C ) 10) See also [45, 3] for reviews of this circle of ideas and other interpretations of (t) in the context of Brownian motion. These ....

....bridge may be generalized to the bridge of a recurrent Bessel process. In Sections 5, 6 and 7 we return to the general setting of Section 2 to consider excursions of the basic Markov process up to an inverse local time. In particular Section 6 presents some generalizations of results of Biane Yor [4] regarding the maximum of a Brownian or Bessel excursion, and Section 7 generalizes some of the results of Knight [21] and Pitman Yor [30] See also [8, 9] for some further applications of the results of this paper. 2 Bridges and Excursions of a self similar Markov process. Recall that for fi 2 R ....

Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

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P. Biane and M. Yor, Valeurs principales associees aux temps locaux Browniens, Bull. Sci. Math. 111 (1987), 23-101.

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P. Biane and M. Yor, Valeurs principales associees aux temps locaux browniens, Bull. Sci. Math. 111 (1987) 23---101; MR0886959 (88g:60188).

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P. Biane and M. Yor. Valeurs principales associees aux temps locaux browniens. Bull. Sci. Math., 111:23--101, 1987.

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P. Biane and M. Yor. Valeurs principales associees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

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P. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23--101, 1987.

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