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115
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 98 (9 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Allfrequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation
 ACM Trans. Graph
, 2006
"... This paper introduces a new data representation and compression technique for precomputed radiance transfer (PRT). The light transfer functions and light sources are modeled with spherical radial basis functions (SRBFs). A SRBF is a rotationinvariant function that depends on the geodesic distance b ..."
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Cited by 37 (0 self)
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This paper introduces a new data representation and compression technique for precomputed radiance transfer (PRT). The light transfer functions and light sources are modeled with spherical radial basis functions (SRBFs). A SRBF is a rotationinvariant function that depends on the geodesic distance between two points on the unit sphere. Rotating functions in SRBF representation is as straightforward as rotating the centers of SRBFs. Moreover, highfrequency signals are handled by adjusting the bandwidth parameters of SRBFs. To exploit intervertex coherence, the light transfer functions are further classified iteratively into disjoint clusters, and tensor approximation is applied within each cluster. Compared with previous methods, the proposed approach enables realtime rendering with comparable quality under highfrequency lighting environments. The data storage is also more compact than previous allfrequency PRT algorithms. CR Categories: I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism—Color, shading, shadowing, and texture; G.1.2 [Numerical Analysis]: Approximation—Special function approximations; E.4 [Coding and Information Theory]: Data compaction and compression
The noncoherent Rician fading channel – Part I : Structure of the capacityachieving input
 IEEE Trans. Wireless Commun
, 2005
"... Abstract—Transmission of information over a discretetime memoryless Rician fading channel is considered, where neither the receiver nor the transmitter knows the fading coefficients. First, the structure of the capacityachieving input signals is investigated when the input is constrained to have l ..."
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Cited by 28 (5 self)
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Abstract—Transmission of information over a discretetime memoryless Rician fading channel is considered, where neither the receiver nor the transmitter knows the fading coefficients. First, the structure of the capacityachieving input signals is investigated when the input is constrained to have limited peakedness by imposing either a fourth moment or a peak constraint. When the input is subject to second and fourth moment limitations, it is shown that the capacityachieving input amplitude distribution is discrete with a finite number of mass points in the lowpower regime. A similar discrete structure for the optimal amplitude is proven over the entire signaltonoise ratio (SNR) range when there is only a peakpower constraint. The Rician fading with the phasenoise channel model, where there is phase uncertainty in the specular component, is analyzed. For this model, it is shown that, with only an average power constraint, the capacityachieving input amplitude is discrete with a finite number of levels. For the classical averagepowerlimited Rician fading channel, it is proven that the optimal input amplitude distribution has bounded support. Index Terms—Capacityachieving input, channel capacity, fading channels, memoryless fading, peak constraints, phase noise, Rician fading. I.
Convolution operator and maximal function for Dunkl transform
 J. Anal. Math
"... Abstract. For a family of weight functions, hκ, invariant under a finite reflection group on R d, analysis related to the Dunkl transform is carried out for the weighted L p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inver ..."
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Cited by 23 (2 self)
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Abstract. For a family of weight functions, hκ, invariant under a finite reflection group on R d, analysis related to the Dunkl transform is carried out for the weighted L p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the BochnerRiesz means. We also define a maximal function and use it to prove the almost everywhere convergence. 1.
A Continuous Metric Scaling Solution for a Random Variable
 Journal of Multivariate Analysis
, 1994
"... As a generalization of the classical Metric Scaling solution for a finite set of points, a countable set of uncorrelated random variables is obtained from an arbitrary continuous random variable X. The properties of these variables allow us to regard them as Principal Axes for X with respect to the ..."
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Cited by 19 (15 self)
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As a generalization of the classical Metric Scaling solution for a finite set of points, a countable set of uncorrelated random variables is obtained from an arbitrary continuous random variable X. The properties of these variables allow us to regard them as Principal Axes for X with respect to the distance function d(u; v) = p ju \Gamma vj. Explicit results are obtained for uniform and negative exponential random variables. Keywords and Phrases Principal components of a stochastic process, Principal Coordinate Analysis. AMS Subject classification: 62H25 1 Introduction Metric Scaling or Principal Coordinate Analysis, introduced by Torgerson [14] and especially Gower [9], is a method of ordination aiming to provide a graphical representation of a finite set of n elements. The method obtains a n \Theta m matrix X from an n \Theta n Euclidean distance matrix \Delta = (ffi ij ) . The set of n rows of X, considered as points in R m , has interdistances which reproduce those in \Delta ...
Mercer’s Theorem, Feature Maps, and Smoothing
"... Abstract. We study Mercer’s theorem and feature maps for several positive definite kernels that are widely used in practice. The smoothing properties of these kernels will also be explored. 1 ..."
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Cited by 12 (0 self)
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Abstract. We study Mercer’s theorem and feature maps for several positive definite kernels that are widely used in practice. The smoothing properties of these kernels will also be explored. 1
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 : : : : : : : : : : :...
On Fourier coefficients of Maass waveforms for PSL(2
 Z), Minnesota Supercomputer Institute Research Report
"... Dedicated to the memory ofD. H. Lehmer Abstract. In this paper, we use machine experiments to test the validity of the SatoTate conjecture for Maass waveforms on PSL(2, Z)\H. We also elaborate on Stark's iterative method for calculating the Fourier coefficients of such forms. 1. Introductory remark ..."
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Cited by 8 (2 self)
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Dedicated to the memory ofD. H. Lehmer Abstract. In this paper, we use machine experiments to test the validity of the SatoTate conjecture for Maass waveforms on PSL(2, Z)\H. We also elaborate on Stark's iterative method for calculating the Fourier coefficients of such forms. 1. Introductory remarks Around 20 years ago, D. H. Lehmer [19] empirically investigated the extent to which the numbers Çp = r(p)p~xx^2 obey SatoTate statistics as p> oo. Here, t(«) is the usual Ramanujan taufunction1 and the proposed statistics assert that m n r N[p<x:6peE] 2 f. 2 (1.1) lim ————^ = sin Odd x^oo n(x) n JE for ¡tp = 2cos(f9p) and any Jordan measurable E ç [0, n]. [n(x) is the usual counting function for the primes.] The corresponding assertion for é,p itself will then read: (1.2) lim *lPg*:p/n = 1 t ^^ x»oo n(x) In JE for E C [2, 2]. In this form, the proposed distribution coincides with the socalled Wigner semicircle law familiar from the study of spectra of random