A Closed Form for the Density Functions of Random Walks in Odd Dimensions

Borwein, Jonathan M.; Sinnamon, Corwin

doi:10.1017/s0004972715001112

Cited by 6 publications

(2 citation statements)

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“…. , N − 1) therein; this latter property is well known for the probability density function of fixed step length uniform random walks [4], and of that for certain pattern avoiding permutations [5] as examples from the recent literature. However, there are some features in common.…”

Section: Introductionmentioning

confidence: 83%

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

Forrester,

Kumar

2020

Preprint

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Examples of the β-Jacobi ensemble specify the joint distribution of the transmission eigenvalues in scattering problems. In this context, there has been interest in the distribution of the trace, as the trace corresponds to the conductance. Earlier, in the case β = 1, the trace statistic was isolated in studies of covariance matrices in multivariate statistics, where it is referred to as Pillai's V statistic. In this context, Davis showed that for β = 1 the trace statistic, and its Fourier-Laplace transform, can be characterised by (N + 1) × (N + 1) matrix differential equations. For the Fourier-Laplace transform, this leads to a vector recurrence for the moments. However, for the distribution itself the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameter b and Dyson index β non-negative integers. For the other Jacobi parameter a also a non-negative integer, the power series portion of each Frobenius solution terminates to a polynomial, and the matrix differential equation gives a recurrence for their computation.

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Section: Introductionmentioning

confidence: 83%

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

Forrester,

Kumar

2020

Preprint

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“…General approaches to the small-N problem are Borwein, Nuyens, Straub and Wan (2011), Borwein, Straub, Wan and Zudilin (2012), Borwein, Straub and Vignat (2016) and Joyce (2017). Related papers, including some of a more technical nature, are Borwein, Straub and Wan (2013), , Borwein and Sinnamon (2016) and Zhou (2019a). Borwein (2016) offers an introduction and perspective on recent work.…”

Section: Introductionmentioning

confidence: 99%

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

McCrorie

2020

Sankhya B

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This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis.

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Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

Silva

Quilodrán²

2021

Journal of Functional Analysis

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A Closed Form for the Density Functions of Random Walks in Odd Dimensions

Abstract: We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.

Cited by 6 publications

References 6 publications

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

Differential recurrences for the distribution of the trace of the $β$-Jacobi ensemble

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

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