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We investigate the properties of the (complex) coefficients obtained in a spherical harmonic representation of temperature maps of the cosmic microwave background (CMB). We study the effect of the coefficient phase only, as well as the combined effects of phase and amplitude. The method used to check for anomalies is to construct a ``random walk trajectory in the complex plane where the step length and direction are given by the amplitude and phase (respectively) of the harmonic coefficient. If the fluctuations comprise a homogeneous and isotropic Gaussian random field on the sky, the path so obtained should be a classical ``Rayleigh flight with very well known statistical properties. We illustrate the use of this random-walk representation by using the net walk length as a test statistic, and apply the method to the coefficients obtained from a Wilkinson Microwave Anisotropy Probe (WMAP) preliminary sky temperature map.
We present a fast algorithm for generating full sky, high resolution ($sim 5$) simulations of the CMB anisotropy pattern. We also discuss the inverse problem, that of evaluating from such a map the full set of $a_{ell m}$s and the spectral coefficien
We study the Doobs $h$-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula
This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a s
Graphs are often used to organize data because of their simple topological structure, and therefore play a key role in machine learning. And it turns out that the low-dimensional embedded representation obtained by graph representation learning are e
In this paper we consider a particular version of the random walk with restarts: random reset events which bring suddenly the system to the starting value. We analyze its relevant statistical properties like the transition probability and show how an