60 results found
Bingham NH, Krzanowski WJ, 2022, Linear algebra and multivariate analysis in statistics: development and interconnections in the twentieth century, British Journal for the History of Mathematics, Vol: 37, Pages: 43-63, ISSN: 2637-5451
The most obvious points of contact between linear and matrix algebra and statistics are in the area of multivariate analysis. We review the way that, as both developed during the last century, the two influenced each other by examining a number of key areas. We begin with matrix and linear algebra, its emergence in the nineteenth century, and its eventual penetration into the undergraduate curriculum in the twentieth century. We continue with a similar account for multivariate analysis in statistics. We pick out the year 1936 for three key developments, and the early post-war period for three more. We then turn to some special results in linear algebra that we need. We briefly discuss four of the main contributors, and close with thirteen ‘case studies’, showing in a range of specific cases how these general algebraic methods have been put to good use and changed the face of statistics.
Bingham NH, 2022, Prediction theory for stationary functional time series, Probability Surveys, Vol: 19, Pages: 160-184, ISSN: 1549-5787
We survey aspects of prediction theory in infinitely many dimensions, with a view to the theory and applications of functional time series.
Barlow MT, Bingham NH, 2021, Samuel James Taylor, 1929-2020 OBITUARY, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 53, Pages: 1528-1545, ISSN: 0024-6093
Bingham NH, 2021, THE LIFE, WORK, AND LEGACY OF P. L. CHEBYSHEV, THEORY OF PROBABILITY AND ITS APPLICATIONS, Vol: 66, Pages: 506-521, ISSN: 0040-585X
Bingham NH, Ostaszewski AJ, 2020, SEQUENTIAL REGULAR VARIATION: EXTENSIONS OF KENDALL'S THEOREM, QUARTERLY JOURNAL OF MATHEMATICS, Vol: 71, Pages: 1171-1200, ISSN: 0033-5606
Bingham NH, Jablonska E, Jablonski W, et al., 2020, On Subadditive Functions Bounded Above on a "Large" Set, RESULTS IN MATHEMATICS, Vol: 75, ISSN: 1422-6383
Bingham NH, Ostaszewski AJ, 2020, General regular variation, Popa groups and quantifier weakening, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 483, ISSN: 0022-247X
Bingham N, Ostaszewski A, 2019, Beyond Haar and Cameron-Martin: the Steinhaus support, Topology and its Applications, Vol: 260, Pages: 23-56, ISSN: 0166-8641
Motivated by a Steinhaus-like interior-point property involving the Cameron-Martin space of Gaussian measure theory, we study a group-theoretic analogue, the Steinhaus triple , and construct a Steinhaus support, a Cameron-Martin-like subset, in any Polish group G corresponding to ‘sufficiently subcontinuous’ measures μ, in particular for ‘Solecki-type’ reference measures.
Bingham NH, Symons TL, 2019, Dimension walks on Sd×R, Statistics & Probability Letters, Vol: 147, Pages: 12-17, ISSN: 0167-7152
We verify that the established one- and two-step recurrences for positive definite functions on spheres extend to the spatio-temporal case of Sd X R.
Bingham N, Ostaszewski A, 2019, Variants on the Berz sublinearity theorem, Aequationes Mathematicae, Vol: 93, Pages: 351-369, ISSN: 0001-9054
We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice, which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function—results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.
Bingham NH, Ostaszewski AJ, 2019, Set theory and the analyst, European Journal of Mathematics, Vol: 5, Pages: 2-48, ISSN: 2199-675X
This survey is motivated by specific questions arising in thesimilarities and contrasts between (Baire) category and (Lebesgue) measure– category-measure duality and non-duality, as it were. Thebulk of the textis devoted to a summary, intended for the working analyst, ofthe extensivebackground in set theory and logic needed to discuss such matters: to quotefrom the Preface of Kelley [Kel]: ”what every young analyst should know”.
Kasahara Y, Bingham NH, 2018, Matricial Baxter's theorem with a Nehari sequence, Mathematical News / Mathematische Nachrichten, Vol: 291, Pages: 2590-2598, ISSN: 0025-584X
In the theory of orthogonal polynomials, (non‐trivial) probability measures on the unit circle are parametrized by the Verblunsky coefficients. Baxter's theorem asserts that such a measure is absolutely continuous and has positive density with summable Fourier coefficients if and only if its Verblusnky coefficients are summable. This note presents a version of Baxter's theorem in the matrix case from a viewpoint of the Nehari problem.
Bingham NH, Ostaszewski AJ, 2018, Beyond Lebesgue and Baire IV: density topologies and a converse Steinhaus-Weil theorem, Topology and its Applications, Vol: 239, Pages: 274-292, ISSN: 0166-8641
The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the original topology and a suitable refinement (a density topology). This prompts a systematic study of such density topologies, and the corresponding σ-ideals of negligibles. Such ideas go back to Weil's classic book, and to Hashimoto's ideal topologies. We make use of group norms, which cast light on the interplay between the group and measure structures. The Steinhaus–Weil interior-points theorem (‘on ’) plays a crucial role here; so too does its converse, the Simmons–Mospan theorem.
Bingham NH, Ostaszewski AJ, 2018, Additivity, subadditivity and linearity: automatic continuity and quantifier weakening, Indagationes Mathematicae, Vol: 29, Pages: 687-713, ISSN: 0019-3577
We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.
Bingham NH, Ostaszewski AJ, 2017, Category-measure duality: convexity, midpoint convexity and Berz sublinearity, Aequationes Mathematicae, Vol: 91, Pages: 801-836, ISSN: 0001-9054
Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over ℚ to prove that the graph of a measurable sublinear function that is ℚ+-homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set.
Kasahara Y, Bingham NH, 2017, Coefficient stripping in the matricial Nehari problem, Journal of Approximation Theory, Vol: 220, Pages: 1-11, ISSN: 0021-9045
This note deals with a matricial Schur function arising from a completely indeterminate Nehari problem. The Schur algorithm is characterized by a unilateral shift for a Nehari sequence.
Bingham NH, Gashi B, 2016, Voronoi means, moving averages, and power series, Journal of Mathematical Analysis and Applications, Vol: 449, Pages: 682-696, ISSN: 1096-0813
We introduce a non-regular generalisation of the Nörlund mean, and show its equivalence with a certain moving average. The Abelian and Tauberian theorems establish relations with convergent sequences and certain power series. A strong law of large numbers is also proved.
Bingham NH, 2016, Probability unfolding, 1965‒2015, Advances in Applied Probability, Vol: 48, Pages: 1-13, ISSN: 0001-8678
We give a personal (and we hope, not too idiosyncratic) view of how our subject of probability theory has developed during the last half-century, and the author in tandem with it.
Bingham NH, Ostaszewski AJ, 2015, Beurling moving averages and approximate homomorphisms, Indagationes Mathematicae, Vol: 27, Pages: 601-633, ISSN: 0019-3577
Bingham NH, Ostaszewski AJ, 2015, Cauchy's functional equation and extensions: Goldie's equation and inequality, the Golab-Schinzel equation and Beurling's equation, AEQUATIONES MATHEMATICAE, Vol: 89, Pages: 1293-1310, ISSN: 0001-9054
Bingham NH, 2015, Hardy, Littlewood and probability, BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Vol: 47, Pages: 191-202, ISSN: 0024-6093
Bingham NH, Gashi B, 2015, Logarithmic moving averages, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 421, Pages: 1790-1802, ISSN: 0022-247X
Bingham NH, 2015, ON SCALING AND REGULAR VARIATION, PUBLICATIONS DE L INSTITUT MATHEMATIQUE-BEOGRAD, Vol: 97, Pages: 161-174, ISSN: 0350-1302
Bingham NH, Missaoui B, 2014, ASPECTS OF PREDICTION, JOURNAL OF APPLIED PROBABILITY, Vol: 51, Pages: 189-201, ISSN: 0021-9002
Bingham NH, Ostaszewski AJ, 2014, Beurling slow and regular variation, Transactions of the London Mathematical Society, Vol: 1, Pages: 29-56, ISSN: 2052-4986
Bingham NH, 2014, Modelling and Prediction of Financial Time Series, COMMUNICATIONS IN STATISTICS-THEORY AND METHODS, Vol: 43, Pages: 1351-1361, ISSN: 0361-0926
Kasahara Y, Bingham NH, 2014, VERBLUNSKY COEFFICIENTS AND NEHARI SEQUENCES, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 366, Pages: 1363-1378, ISSN: 0002-9947
Bingham NH, 2014, THE WORLDWIDE INFLUENCE OF THE WORK OF B. V. GNEDENKO, THEORY OF PROBABILITY AND ITS APPLICATIONS, Vol: 58, Pages: 17-24, ISSN: 0040-585X
Bingham NH, Ostaszewski AJ, 2013, The Steinhaus theorem and regular variation: de Bruijn and after, INDAGATIONES MATHEMATICAE-NEW SERIES, Vol: 24, Pages: 679-692, ISSN: 0019-3577
Bingham NH, Inoue A, Kasahara Y, 2012, An explicit representation of Verblunsky coefficients, STATISTICS & PROBABILITY LETTERS, Vol: 82, Pages: 403-410, ISSN: 0167-7152
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