On methods for generating uniform random points on the surface of a sphere

Some methods for generating random points uniformly distributed on the surface of ann-sphere have been proposed to simulate spherical processes on computer. A standard method is to normalize random points inside of the sphere, see M. Muller [5]. Improved methods were given by J. M. Cook [1] and G. Marsaglia [4] in three and four dimensions, and computational methods in higher dimensions by J. S. Hicks and R. F. Wheeling [3] and M. Sibuya [6]. In this paper we shall offer direct methods for generating uniform random points on the surface of a unitn-sphere, which can be easily combined with Marsaglia's idea for getting more improved methods. Our method in even dimensions was obtained by M. Sibuya [6], but a differential-geometric view-point will make analyses simpler, even in odd dimensions.     

On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

In this paper, we use a combination of VDR theory and patchwork method to derive an efficient algorithm for generating uniform random points on a unit d-sphere. We first propose an algorithm to generate random vector with uniform distribution on a unit 2-sphere on the plane. Then we use VDR theory to reduce random vector X_d with uniform distribution on a unit d-sphere into , such that the random vector (X_d-1,X_d) is uniformly distributed on a unit 2-sphere and X_d-2 has conditional uniform distribution on a (d-2)-sphere of radius , given V=v with V having the p.d.f. . Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere.     

Poisson law for the number of lattice points in a random strip with finite area

Summary Let a smooth curve be given by a functionr=f() in polar coordinate system in the plane, and letR be a uniformly distributed random variable on the interval [a ₁ L, a ₂ L] with somea ₂>a ₁>0 and a largeL>0. Ya. G. Sinai has conjectured that given some real numbersc ₂>c ₁, the number of lattice points in the domain between the curves and is asymptotically Poisson distributed for good functionsf(·). We cannot prove this conjecture, but we show that if a probability measure with some nice properties is given on the space of smooth functions, then almost all functions with respect to this measure satisfy Sinai's conjecture. This is an improvement of an earlier result of Sinai [9], and actually the proof also contains many ideas of that paper.This article was processed by the author using the Springer-Verlag T_EX ProbTh macro package 1991.     

球面上的Poisson积分

证明了球面上的Poisson积分算子从Lp(Sn?1)到Lorentz空间Lq,1(B1)(q 1)有界,且从有界Borel测度集M(Sn?1)到Lq,1(B1)(q < nn?1)有界,推广了部分已知的结果.进一步构造了一个反例说明了球面上的Poisson积分算子不一定从M(Sn?1)到L n n?1(B1)有界.     

Brownian bridge asymptotics for random mappings

Uniform random mappings of an n-element set to itself have been much studied in the combinatorial literature. We introduce a new technique, which starts by specifying a coding of mappings as walks with ± 1 steps. The uniform random mapping is thereby coded as a nonuniform random walk, and our main result is that as n→∞ the random walk rescales to reflecting Brownian bridge. This result encompasses a large number of limit theorems for “global” characteristics of uniform random mappings. © 1994 John Wiley & Sons, Inc.     

Precise asymptotics for random matrices and random growth models

The author considers the largest eigenvaiues of random matrices from Gaussian unitary ensemble and Laguerre unitary ensemble, and the rightmost charge in certain random growth models. We obtain some precise asymptotics results, which are in a sense similar to the precise asymptotics for sums of independent random variables in the context of the law of large numbers and complete convergence. Our proofs depend heavily upon the upper and lower tail estimates for random matrices and random growth models. The Tracy-Widom distribution plays a central role as well.     

Frames of Poisson wavelets on the sphere

In this paper we show that for sufficiently dense grids Poisson wavelets on the sphere constitute a weighted frame. In the proof we will only use the localization properties of the reproducing kernel and its gradient. This indicates how this kind of theorem can be generalized to more general reproducing kernel Hilbert spaces. With the developed technique we prove a sampling theorem for weighted Bergman spaces.     

Limit theorems for projections of random walk on a hypersphere

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein–Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a “functional” generalisation of Poincaré’s observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli–Laplace central limit theorem. Given the relation of these two classic results to the central limit theorem for convex bodies, the modest results provided here would appear to motivate a functional generalisation.     

Steady spatial asymptotics for the Vlasov–Poisson system

A collisionless plasma is modelled by the Vlasov–Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as ∣x∣ tends to infinity is considered. Hence, the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behaviour were shown to exist locally in time in a previous work. This paper studies the time behaviour of the net charge and a natural quantity related to energy, and shows that neither is constant in time in general. Also, neither quantity is positive definite. When the background density is a decreasing function of ∣v∣, a positive definite quantity is constructed which remains bounded. A priori bounds are obtained from this. Copyright © 2003 John Wiley & Sons, Ltd.     

Number of lattice points in a sphere

A new estimate is obtained for the residue R in the asymptotics of the number of integer points in a ball of radius a. The estimate has the form R ? a ^{17/14 + ?} .     

Poisson approximation for non-backtracking random walks

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. The authors of [3] studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length n on a high-girth n-vertex regular expander, is typically (1+o(1)))log n/log log n, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun’s sieve with some extensions of the ideas in [3]. Let N _t denote the number of vertices visited precisely t times by a non-backtracking random walk of length n on a regular n-vertex expander of fixed degree and girth g. We prove that if g = ω(1), then for any fixed t, N _t /n is typically 1/et! + o(1). Furthermore, if g = Ω(log log n), then N _t /n is typically 1+o(1)/et! niformly on all t ≤ (1 − o(1)) log n/log log n and 0 for all t ≥ (1 + o(1)) log n/log log n. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.     

Rational points on the unit sphere

It is known that the unit sphere, centered at the origin in ℝ ⁿ , has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝ ⁿ , and every ν > 0; there is a point r = (r ₁; r ₂;…;r _n) such that: