J. Pitman and M. Yor. Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab., 26:1683--1702, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....C ay e Gamma(1 Gamma )ay : Assembling these pieces yields lim sup y 1 y Gamma1 log P( 1 2 y) Gamma(1 Gamma )a. This completes the proof of the lemma by sending to 0 . tu Proof of Theorem 4.1. The Laplace transform of R 1 0 ae 2 (t) dt is well known, cf. for example, Pitman and Yor (1982, p. 432) for all 0, E h exp i Gamma Z 1 0 ae 2 (t) dt ji = i p 2 sinh p 2 j 3=2 : 4:4) 13 This, combined with an exponential type Tauberian theorem (cf. Bingham et al. 1987, Theorem 4.12.9) yields the following lower tail: log P i Z 1 0 ae 2 (t) dt y j Gamma 9 8 y ; y 0 ....

....j ; 8:1) where L 1 (jflj) def = sup x0 L 1 (jflj) and fm(t) 0 t 1g denotes a meander process. The joint law of sup 0t1 m(t) and m(1) is determined by the following Gauss transform : let N denote a Gaussian N (0; 1) variable, independent of the meander process m, then according to Pitman and Yor (1998b) for all y x 0, 2 x y P i jN j sup 0t1 m(t) y; jN j m(1) x j = sinh x (sinh y) 2 : 8:2) Unfortunately, we have not succeeded in obtaining accurate asymptotics of the joint tail of L 1 (jflj) and L 0 1 (jflj) from (8.1) 8.2) If we are only interested in the variable L 1 (jflj) ....

Pitman, J.W. and Yor, M. (1998a) Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab. (to appear) Pitman, J.W. and Yor, M. (1998b) Laws of homogeneous functionals of Brownian motion. (in preparation).

....at the times T k are more complicated than the corresponding decompositions for the times D V j expressed by Lemma 8. For the T partition, the pieces are not pure B bridges. Rather, when normalized they have density factors involving their local times at 0. Compare with similar constructions in [11, 13, 25, 33]. By the Poisson analysis of the previous section, conditionally given (T 1 ; L 1 ) the pieces of B before and after time T 1 are independent B bridges with prescribed lengths and local times at 0. The appearance of h k in formula (a) below shows that the right side does not factor into ....

....of pseudo bridge B [0; 1 ] are known. In particular, the occupation density of the reflected process is governed by the same stochastic differential equation governing the occupation density process of a reflecting Brownian bridge or Brownian excursion [28] According to Knight [19] see also [33] and papers cited there) the law of the maximum of the reflected pseudo bridge is identical to that of 1= 2 H 1 (R 3 ) where H 1 (R 3 ) is the hitting time of 1 by the three dimensional Bessel process, with transform E(exp( Gamma H 1 (R 3 ) sinh for real . Thus we deduce: ....

J. Pitman and M. Yor. Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab., 26:1683--1702, 1998.

....Brownian motion with reflection at 0, and T 1 (B Gamma B) is the first time that the range of the Brownian B up to time t is an interval of length 1. The result that 4T 1 (B Gamma B) has Laplace transform 1= cosh 2 is due to Imhof [30] See also Vallois [72, 73] Pitman [51] and Pitman Yor [56] for various refinements of this formula. Rows 4 and 5 These rows, which involve the distribution of the maximum of various processes over [0; 1] are discussed in Section 4.6. Row 6 Here m 1 is the maximum of the standard Brownian meander (m u ; 0 u 1) This entry is read from (60) Row 7. ....

.... R 1 = jBj and (L t (B) t 0; x 2 R) is the local time process of B defined by (63) The distribution of 1 =R 1; 1 was identified with that of 4T 1 (R 3 ) by Knight [42] while the distribution of 1 = B 1 Gamma B 1 ) was identified with that of T 1 (R 3 ) T 1 ( R 3 ) by Pitman Yor [56]. The result in the third column can be read from Hu Shi Yor [29, p. 188] 20 0) 1) Sigma 1 : 1 : 2 : 4;u du 3) T 1 (R 3 ) T 1 (R 3 ) T 1 ( R 3 ) T 1 (R 1 ) 4T 1 (B Gamma B) 4) Delta Gamma2 Gamma B 1 Gamma B 1 ....

J. Pitman and M. Yor. Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab., 26:1683--1702, 1998.

.... 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 : R(t) 0 t ....

....t 1) q 2 E h F (b # t ; 0 t 1)L # i (54) where L # = 1= p 1 is the local time at 0 of (b # t ; 0 t 1) up to time 1. In terms of the Brownian motion (B t ) define I t : Gamma inf 0ut B u ; M t : sup 0ut B u ; A t : I t M t ; Q t : I t A t : It was shown in [21] that Q 1 has uniform distribution on (0; 1) In view of (54) and Cs aki s formula (8) this implies E h L Gamma1 g(Q) i = r 2 Z 1 0 dv g(v) for all non negative Borel functions g. This formula can also be obtained quite easily from Theorem 2. From this formula we deduce that E[L ....

J. Pitman and M. Yor. Random Brownian Scaling Identities and Splicing of Bessel Bridges. Technical Report 490, Dept. Statistics, U.C. Berkeley, 1997. To appear in Ann. Probab.. Available via http://www.stat.berkeley.edu/users/pitman.

....BES 0 (ffi) processes run till their hitting times of x, where the first process starts at 0 and the second starts at 1. The identity in distribution (110) follows from this decomposition by Brownian scaling, along with (108) and the consequence of (114) that V 2 ffi d = U Gamma1= See also [40] where we discuss the variation of the above construction with j = L ffi ; b j = b L ffi , which does not yield the same law of e r, but one with a different density relative to the law of r ffi . A last exit result for ffi 2 (0; 2) For ffi 2 (0; 2) formula (95) has an interpretation explained ....

J. Pitman and M. Yor. Random Brownian Scaling Identities and Splicing of Bessel Bridges. Technical Report 490, Dept. Statistics, U.C. Berkeley, 1997. To appear in Ann. Probab.. Available via http://www.stat.berkeley.edu/users/pitman.

....motion with reflection at 0, and T 1 (B Gamma B) is the first time that the range of the Brownian B up to time t is an interval of length 1. The result that 4T 1 (B Gamma B) has Laplace transform 1= cosh 2 p 2 is due to Imhof [30] See also Vallois [72, 73] Pitman [51] and Pitman Yor [56] for various refinements of this formula. Rows 4 and 5 These rows, which involve the distribution of the maximum of various processes over [0; 1] are discussed in Section 4.6. Row 6 Here m 1 is the maximum of the standard Brownian meander (m u ; 0 u 1) This entry is read from (60) Row 7. ....

.... 1 = jBj and (L x t (B) t 0; x 2 R) is the local time process of B defined by (63) The distribution of 1 =R 2 1; 1 was identified with that of 4T 1 (R 3 ) by Knight [42] while the distribution of 1 = B 1 Gamma B 1 ) 2 was identified with that of T 1 (R 3 ) T 1 ( R 3 ) by Pitman Yor [56]. The result in the third column can be read from Hu Shi Yor [29, p. 188] 20 Table 2 0) p 2 sinh p 2 p 2 sinh p 2 2 1 cosh p 2 1 cosh p 2 2 1) Sigma 1 : 2 2 1 X n=1 n n 2 Sigma 2 : 2 2 1 X n=1 n n n 2 Sigma # 1 : 2 2 1 X n=1 ....

J. Pitman and M. Yor. Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab., 26:1683--1702, 1998.

.... 2 ; OE( 1 2 2 ) sinh e Gamma ; 1 2 2 ) tanh : 87) Formula (79) in this instance simplifies to E exp Gamma 1 2 2 1 M 2 n # = 2 sinh(2) 2 e 2 1 n Gamma1 : 88) For n = 1 this identity is due to Knight [21] See also [31] for a number of other extensions of Knight s identity. As shown by Knight [20] random variables with Laplace transforms OE and 1= can be constructed in this case as follows. Let T 1 be the first hitting time of 1 by jBj, and let (G 1 ; D 1 ) be the excursion interval of B straddling time T 1 . ....

J. Pitman and M. Yor. Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab., 26:1683--1702, 1998.

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