Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hyperg
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Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series J. Roderick McCrorie University of St Andrews, Scotland, UK Abstract 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 Monthly 127(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. Keywords. Four-step uniform random walk in the plane, Dickey-Fuller distribution, very well-poised generalized hypergeometric series, elliptic integral, universal parabolic constant, moments AMS (2000) subject classification. Primary 33C20, 60G50, 62M10, Secondary 11Y60
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Introduction
In his assessment of the impact of Karl Pearson's work in the development of Statistics in India, Nayak (2009) outlined the nascent role it played in attracting Prasanta Chandra Mahalanobis to the discipline, and in inspiring
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J. R. McCrorie
him towards his eventual leading role in its development across the Indian subcontinent. See also Ghosh (1994). C.R. Rao (1973) saw Mahalanobis existentially as “. . . a physicist by training, a statistician by instinct and a planner by conviction.” He argued that “He did not consider statistics as a narrow subject confined to the mathematical theory of probability, or routine analysis of data in applied research, or collection of data as an aid to administrative decisions . . . But he took a wider view of statistics as ‘a new technology for increasing the efficiency of human efforts in the widest sense'. This has naturally aroused his interest in various fields and enabled him to enrich the science of statistics with a practical base of great depth and spread. (ibid., p. 463) This paper is also influenced by an early Pearson contribution: a statement of the problem on the random walk. W
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