Distributional Transformations Without Orthogonality Relations

Distributional transformations characterized by equations relating expectations of test functions weighted by a given biasing function on the original distribution to expectations of the test function’s higher derivatives with respect to the transformed distribution play a great role in Stein’s method and were, in great generality, first considered by Goldstein and Reinert (J Theoret Probab 18(1):237–260, 2005. doi: 10.1007/s10959-004-2602-6). We prove two abstract existence and uniqueness results for such distributional transformations, generalizing their \(X-P\)-bias transformation. On the one hand, we show how one can abandon previously necessary orthogonality relations by subtracting an explicitly known polynomial depending on the test function from the test function itself. On the other hand, we prove that for a given nonnegative integer m, it is possible to obtain the expectation of the m-th derivative of the test function with respect to the transformed distribution in the defining equation, even though the biasing function may have \(k<m\) sign changes, if these two numbers have the same parity. We explain how these results can be used to guarantee the existence of two different generalizations of the zero-bias transformation by Goldstein and Reinert (Ann Appl Probab 7(4):935–952, 1997. doi: 10.1214/aoap/1043862419). Further applications include the derivation of Stein-type characterizations without needing to solve any Stein equation and the presentation of a general framework for estimating the distance from the distribution of a given real random variable X to that of a random variable Z, whose distribution is characterized by some mth-order linear differential operator. We also explain the fact that, in general, the biased distribution depends on the choice of the sign change points, if these are ambiguous. This new phenomenon does not appear in the framework from Goldstein and Reinert (2005).     

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

The Matsumoto–Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As was observed in Letac and Weso?owski (Ann Probab 28:1371–1383, 2000), the natural framework for the multivariate MY property is symmetric cones; however, they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.     

On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of <Emphasis Type="Italic">p</Emphasis>-Adic Numbers

We prove the following theorem. Let X be a discrete field, and \(\xi \) and \(\eta \) be independent identically distributed random variables with values in X and distribution \(\mu \). The random variables \(S=\xi +\eta \) and \(D=(\xi -\eta )^2\) are independent if and only if \(\mu \) is an idempotent distribution. A similar result is also proved in the case when \(\xi \) and \(\eta \) are independent identically distributed random variables with values in the field of p-adic numbers \({\mathbf {Q}}_p\), where \(p>2\), assuming that the distribution \(\mu \) has a continuous density.     

Distance-regular Cayley graphs with least eigenvalue $$$$

We classify the distance-regular Cayley graphs with least eigenvalue \(-2\) and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.     

Linearly embeddable designs

We address the problem of determining when a plane algebraic cubic curve is complete as an (n, 3)-arc in \(\mathrm {PG}(2,q)\). Theoretical results are given for absolutely irreducible singular cubic curves, while computer based results are given for \(q\le 81\).     

Classification of partial difference sets in Abelian groups of order $$4p^2$$

In this article we provide a complete classification of regular partial difference sets in Abelian groups of order \(4p^2\), p an odd prime. It turns out that the known examples are the only examples. These are, up to complements, the trivial examples, the PCP examples, and a sporadic example in \(\mathbb {Z}_2^2\times \mathbb {Z}_3^2\).     

Characterizing Gaussian Flows Arising from Itô’s Stochastic Differential Equations

In order to identify which of the strong solutions of Itô’s stochastic differential equations (SDEs) are Gaussian, we introduce a class of diffusions which ‘depend deterministically on the initial condition’ and then characterize the class. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flows generated by the SDEs, for an extended class of initial conditions, are the unique solutions of the class of stochastic partial differential equations introduced in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162 2008), ‘Probabilistic Representations of Solutions of the Forward Equations’.     

A Notion of Fine Continuity for BV Functions on Metric Spaces

In the setting of a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with respect to the codimension 1 Hausdorff measure.     

In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r ^α for r∈(0,1] and any point x from metric space.     

Fast algorithm for singly linearly constrained quadratic programs with box-like constraints

This paper focuses on a singly linearly constrained class of convex quadratic programs with box-like constraints. We propose a new fast algorithm based on parametric approach and secant approximation method to solve this class of quadratic problems. We design efficient implementations for our proposed algorithm and compare its performance with two state-of-the-art standard solvers called Gurobi and Mosek. Numerical results on a variety of test problems demonstrate that our algorithm is able to efficiently solve the large-scale problems with the dimension up to fifty million and it substantially outperforms Gurobi and Mosek in terms of the running time.