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The law of the maximum of a Bessel bridge
 Electronic J. Probability
, 1998
"... Let M ffi be the maximum of a standard Bessel bridge of dimension ffi . A series formula for P (M ffi a) due to Gikhman and Kiefer for ffi = 1; 2; : : : is shown to be valid for all real ffi ? 0. Various other characterizations of the distribution of M ffi are given, including formulae for its Mel ..."
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Cited by 11 (7 self)
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Let M ffi be the maximum of a standard Bessel bridge of dimension ffi . A series formula for P (M ffi a) due to Gikhman and Kiefer for ffi = 1; 2; : : : is shown to be valid for all real ffi ? 0. Various other characterizations of the distribution of M ffi are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of M ffi as is described both as ffi !1 and as ffi # 0. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, Bessel process, zeros of Bessel functions, Riemann zeta function Contents 1 Introduction 3 2 The maximum of a diffusion bridge 8 3 The GikhmanKiefer Formula 9 4 The law of T ffi and the agreement formula 11 5 The first passage transform and its derivatives 13 6 Moments 16 7 Dimensions one and three 20 8 Limits as ffi !1 22 9 Limits as ffi # 0 24 10 Relation to last exit times 27 11 A series involving the zeros of J 30 A Some Useful Formulae 33 A.1 Bessel Functions : : : : : : : : : : :...
Random Brownian Scaling Identities and Splicing of Bessel Processes
 ANN. PROBAB
, 1997
"... An identity in distribution due to F. Knight for Brownian motion is extended in two different ways: firstly by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion, and secondly by replacing the reflecting Brownian motion by a recurrent Bessel process ..."
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Cited by 7 (5 self)
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An identity in distribution due to F. Knight for Brownian motion is extended in two different ways: firstly by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion, and secondly by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Ito's law of Brownian excursions, due to D. Williams and J.M. Bismut, each involving backtoback splicing of fragments of two independent threedimensional Bessel processes. Generalizations of both splicing constructions are described which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.
The Asymptotic Distribution of the Diameter of a Random Mapping
, 2002
"... The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an nelement set to itself is represented as the distribution of a functional of a reflecting Brownian bridge. This yields a formula for the Mellin transform of the asymptotic distribution, ge ..."
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Cited by 5 (3 self)
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The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an nelement set to itself is represented as the distribution of a functional of a reflecting Brownian bridge. This yields a formula for the Mellin transform of the asymptotic distribution, generalizing the evaluation of its mean by Flajolet and Odlyzko (1990).
ON EXACT SIMULATION ALGORITHMS FOR SOME DISTRIBUTIONS RELATED TO JACOBI THETA FUNCTIONS
"... Abstract. We develop exact random variate generators for several distributions related to the Jacobi theta function. These include the distributions of the maximum of a Brownian bridge, a Brownian meander and a Brownian excursion, and distributions of certain first passage times of Bessel processes. ..."
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Cited by 2 (0 self)
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Abstract. We develop exact random variate generators for several distributions related to the Jacobi theta function. These include the distributions of the maximum of a Brownian bridge, a Brownian meander and a Brownian excursion, and distributions of certain first passage times of Bessel processes. The algorithms are based on the alternating series method. Furthermore, we survey various distributional identities and point out ways of dealing with generalizations of these basic distributions. Keywords and phrases. Random variate generation. Stable distribution. First passage time. Brownian motion. Rejection method. Simulation. Monte Carlo method. Expected time analysis. Probability inequalities.
Some explicit Krein
, 2005
"... representations of certain subordinators, including the Gamma process ..."
unknown title
, 2009
"... The distribution of the maximal difference between Brownian bridge and its concave majorant ..."
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The distribution of the maximal difference between Brownian bridge and its concave majorant
Gibbs Partitions . . . Meijer G Transforms
, 2007
"... This paper derives explicit results for the infinite Gibbs partitions generated by the jumps of an α−stable subordinator, derived in Pitman (39; 38). We first show that for general α the conditional EPPF can be represented as ratios of FoxH functions, and in the case of rational α, MeijerG functi ..."
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This paper derives explicit results for the infinite Gibbs partitions generated by the jumps of an α−stable subordinator, derived in Pitman (39; 38). We first show that for general α the conditional EPPF can be represented as ratios of FoxH functions, and in the case of rational α, MeijerG functions. This extends results for the known case of α = 1/2, which can be expressed in terms of Hermite functions, hence answering an open question. Furthermore the results show that the resulting unconditional EPPF’s, can be expressed in terms of H and G transforms indexed by a function h. Hence when h is itself a H or G function the EPPF is also an H or G function. An implication, in the case of rational α, is that one can compute explicitly thousands of EPPF’s derived from possibly exotic special functions. This would also apply to all α except that computations for general Fox functions are not yet available. However, moving away from special functions, we demonstrate how results from probability theory may be used to obtain calculations. We show that a forward recursion can be