J. Pitman and M. Yor (1999). The law of the maximum of a Bessel bridge. Electron. J. Probab. 4, no. 15, 35 pp.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....22 of [1] T n has the same distribution as y n . By Theorem 13 of [1] the length of any path between two vertices of T n is at most 2 max t2[0;1] f(t) 4M 1 , where M 1 = max t2[0;1] B t . Therefore, E[d(x n ) 4A Gamma1 2 E[M 1 ]n 1=2 . Since E[M 1 ] 1, as shown, for example, in [8], it follows from Markov s inequality that the conclusion of Lemma 5 holds with A 1 = 8A Gamma1 2 E[M 1 ] Now, we work towards using Corollary 3 to obtain an O(n 2 ) bound for the relaxation time n of the Markov chain on cladograms defined in the introduction. We use the notation from ....

J. Pitman and M. Yor. The law of the maximum of a Bessel bridge. Electron. J. Probab., 4:1--35, 1999.

....2 : Multiplying on both sides of (4. 1) by y a , and integrating with respect to y over R , we can determine all the positive moments of T (3) 1 : for a 0, 4:1 0 ) E hi T (3) 1 j a i = 2 a Gamma2a 1=2 Gamma(a 1=2) Z 1 0 x 2a (cosh x) 2 dx: See also Pitman and Yor [13]) We note that the negative moments of T (3) 1 , which are calculated in Yor [18, Chap. XI] are in formal agreement with (4:1 0 ) see also, for similar computations related to the Riemann zeta function in terms of the sum T (3) 1 b T (3) 1 of two independent copies of T (3) 1 , D. ....

....1] R, E [F (R br ) c ffi E h F ( e R br ) f M br ) 2 Gammaffi i ; 10 where f M br def = sup 0u1 e R br u = T b T ) Gamma1=2 ; c ffi def = 2 (ffi Gamma2) 2 Gamma(ffi=2) Remark. A detailed study of the law of sup 0u1 R br u is made in Pitman and Yor [13] with the help of the agreement formula. We now consider the particular case ffi = 1; we obtain the following relationship between sup u1 jb(u)j and T b T : 4:3) r 2 E h f Gamma sup u1 jb(u)j Delta i = E h (T b T ) Gamma1=2 f Gamma (T b T ) Gamma1=2 Delta i : ....

Pitman, J.W. and Yor, M.: The law of the maximum of a Bessel bridge. Technical Report No. 534, Department of Statistics, University of California, Berkeley, September 1998.

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J. Pitman and M. Yor (1999). The law of the maximum of a Bessel bridge. Electron. J. Probab. 4, no. 15, 35 pp.

....3, can be viewed in the Brownian case as asymptotic counterparts (under weak convergence of mapping walks) of some combinatorial symmetries of random mappings, discussed in Section 2.2. Other results in the Brownian case, especially those involving the method of Poissonization by random scaling [34, 35], are not obvious from the combinatorial perspective, but provide explicit limit distributions for functionals of uniform random mappings. See also [4] where we apply this method to characterize the asymptotic distribution of the diameter of the digraph of a uniform mapping. Sections 4 and 5 ....

.... : 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; 2) and fi = 2 . Other examples are provided by recurrent stable L evy processes [8] symmetrized or skew Bessel processes [42] and Walsh processes [6, ....

J. Pitman and M. Yor. The law of the maximum of a Bessel bridge. Electron. J. Probab., 4:Paper 15, 1--35, 1999.

....distribution, is the known result that for B 1 standard Gaussian independent of M 1 , and y 0 P (jB 1 jM 1 y) tanh y = 1 1 2= e Gamma 1) 20) As observed in [6] formula (20) allows the Mellin transform of M 1 to be expressed in terms of the Riemann zeta function. See also [15, 19, 20] for closely related Mellin transforms obtained by the technique of multiplication by a suitable independent random factor to introduce Poisson or Markovian structure. In [2] and [4] we show that Brownian bridge asymptotics apply to models of random mappings more general than the uniform model, ....

J. Pitman and M. Yor. The law of the maximum of a Bessel bridge. Electron. J. Probab., 4:Paper 15, 1--35, 1999.

....Brownian bridge. 18 4.3 A table of identities in distribution We now discuss the meaning of Table 2, which presents a number of known identities in distribution. The results are collected from the work of numerous authors, including Gikhman [25] Kiefer [39] Chung [14] Biane Yor [8] See also [80, 55, 58]. In the following sections we review briefly the main arguments underlying the results presented in the table. Each column of the table displays a list of random variables with the distribution determined by the Laplace transform in Row 0. Each variable in the second column is distributed as the ....

.... Row 5, with r 1;1 : max 0u1 jb u j, is read from (54) The second entry of Row 5, involving the maximum r 3;1 of a three dimensional Bessel bridge (r 3;u ; 0 u 1) is read from the work of Gikhman [25] and Kiefer [39] who found a formula for P (r d;1 x) for arbitrary d = 1; 2; See also [58]. This result involving r 3;1 may be regarded as a consequence of the previous identification (57) of the law of e 1 , the maximum of a standard Brownian excursion, and the identity in law e 1 = r 3;1 implied by the remarkable result of L evy Williams [47, 77] that (e t ; 0 t 1) r 3;t ....

J. Pitman and M. Yor. The law of the maximum of a Bessel bridge. Electronic J. Probability, 4:Paper 15, 1--35, 1999.

....Brownian bridge. 18 4.3 A table of identities in distribution We now discuss the meaning of Table 2, which presents a number of known identities in distribution. The results are collected from the work of numerous authors, including Gikhman [25] Kiefer [39] Chung [14] Biane Yor [8] See also [80, 55, 58]. In the following sections we review briefly the main arguments underlying the results presented in the table. Each column of the table displays a list of random variables with the distribution determined by the Laplace transform in Row 0. Each variable in the second column is distributed as the ....

.... 5, with r 1;1 : max 0u1 jb u j, is read from (54) The second entry of Row 5, involving the maximum r 3;1 of a three dimensional Bessel bridge (r 3;u ; 0 u 1) is read from the work of Gikhman [25] and Kiefer [39] who found a formula for P (r d;1 x) for arbitrary d = 1; 2; See also [58]. This result involving r 3;1 may be regarded as a consequence of the previous identification (57) of the law of e 1 , the maximum of a standard Brownian excursion, and the identity in law e 1 d = r 3;1 implied by the remarkable result of L evy Williams [47, 77] that (e t ; 0 t 1) d = r ....

J. Pitman and M. Yor. The law of the maximum of a Bessel bridge. Electronic J. Probability, 4:Paper 15, 1--35, 1999.

....complicated. This is yet another instance where the introduction of a suitable random multiplier provides a substantial simplification, as we have recognized in a number of other studies of homogeneous functionals of Brownian motion and self similar Markov processes ( 35, Ch. XII Ex. 4. 24) [6, 34]) See also Perman Wellner [26] and Jansons [17] for further applications of this device. The rest of this paper is organized as follows. In Section 2 we present a general characterization of the distribution of ranked values of a homogeneous functional F of excursions of the standardized bridge ....

....(52) determines the distribution of the maximum of a standard BES(ffi) bridge for 0 ffi 2. Kiefer [19] found an explicit formula for the distribution of the maximum of a standard BES(ffi) bridge for all positive integer dimensions ffi. See [6] for an alternative approach to formula (52) and [34] for further developments. 5 Evaluations at time 1 Recall that ( 0) is the inverse of the local time process at 0 for the self similar Markov process B, with the local time process normalized so that (14) holds for some constants ff 2 (0; 1) and K 2 (0; 1) Thus the L evy measure of ( ....

J. Pitman and M. Yor. The law of the maximum of a Bessel bridge. Electronic J. Probability, 4:Paper 15, 1--35, 1999.

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J. Pitman and M. Yor, The law of the maximum of a Bessel bridge, Electron. J. Probab. 4 (1999) n. 15; MR1701890 (2000j:60101)

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