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It is well known that every bivariate copula induces a positive measure on the Borel σ -algebra on [ 0 , 1 ] 2 , but there exist bivariate quasi-copulas that do not induce a signed measure on the same σ -algebra. In this paper we show that a signed measure induced by a bivariate quasi-copula can always be expressed as an infinite combination of measures induced by copulas. With this we are able to give the first characterization of measure-inducing quasi-copulas in the bivariate setting. [ABSTRACT FROM AUTHOR]

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation -\Delta u + {\rm e}^u({\rm e}^u-1) =\mu\quadin \Omega with the Dirichlet boundary condition. Approximating $\mu$ by a sequence $(\mu_n)_{n \in\mathbb N}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in\mathbb{N}$, we are interested in establishing the convergence of the sequence $(u_n)_{n\in\mathbb{N}}$ to a function $u^#$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^#$.

In this paper, we use stochastic integration in the framework of signed measures, together with the technique of time changes. Let Q be a bounded non-null signed measure on (Ω , F , P) , such that Q and P are equivalent. In the first part of the paper, we generalize the results of stochastic calculus in Beghdadi-Sakrani (Séminaire de probabilités XXXVI, Springer, 2003) to Q-local martingales and we give some examples. In the second part, we prove that the class of Q-semimartingales is invariant under time changes. We establish the famous formulas of time-changed local martingales as well as the representation of a Q-local martingale as a time-changed Brownian motion. [ABSTRACT FROM AUTHOR]

This work is divided in three parts. First, we introduce quasi-infinitely divisible (QID) random measures and formulate spectral representations. Second, we introduce QID stochastic integrals and present integrability conditions, continuity properties and spectral representations. Finally, we introduce QID processes, i.e. stochastic processes with QID finite dimensional distributions. For example, a process X is QID if there exist two ID processes Y and Z such that X + Y = d Z with Y independent of X. The class of QID processes is strictly larger than the class of ID processes. We provide spectral representations and Lévy–Khintchine formulations for potentially all QID processes. Many examples are presented. [ABSTRACT FROM AUTHOR]

A length-n random sequence X1, . . . , Xn in a space S is finitely exchangeable if its distribution is invariant under all n! permutations of coordinates. Given N >n, we study the extendibility problem: when is it the case that there is a length-N exchangeable random sequence Y1, . . . ,YN so that (Y1, . . . , Yn) has the same distribution as (X1, . . . , Xn)? In this paper, we give a necessary and sufficient condition so that, for given n and N, the extendibility problem admits a solution. This is done by employing functional-analytic and measure-theoretic arguments that take into account the symmetry. We also address the problem of infinite extendibility. Our results are valid when X1 has a regular distribution in a locally compact Hausdorff space S. We also revisit the problem of representation of the distribution of a finitely exchangeable sequence. [ABSTRACT FROM AUTHOR]

In this paper, we examined the connection between quantum systems’ indistinguishability and signed (or negative) probabilities. We do so by first introducing a measure-theoretic definition of signed probabilities inspired by research in quantum contextuality. We then argue that ontological indistinguishability leads to the no-signaling condition and negative probabilities.

Given a partition {I1, …, Ik} of {1, …, n}, let (X1, …, Xn) be random vector with each Xi taking values in an arbitrary measurable space (S,S) such that their joint law is invariant under finite permutations of the indexes within each class Ij. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class Ij. We provide a necessary condition for the existence of a nonnegative directing measure. This is related to the notions of infinite extendibility and reinforcement. In particular, given a finite exchangeable sequence of Bernoulli random variables, the directing measure can be chosen nonnegative if and only if two effectively computable matrices are positive semi-definite. [ABSTRACT FROM AUTHOR]

In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller’s characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the Lévy-Wiener theorem. This is a special case of an open problem which is proposed by Sato (2014), Chaumont and Yor (2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter λ in the DPCP class cannot be arbitrarily small. [ABSTRACT FROM AUTHOR]

We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (1964) and Roos (1999b) in combination with new smoothness inequalities, which could be of independent interest. [ABSTRACT FROM AUTHOR]

We investigate the multi-dimensional super resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the \ell 1 -minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal \ell 1 -minimization exact recovery problem of super resolution that unleashes standard techniques (such as moment-sum-of-squares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the super resolution problem in dimension greater than 2 and for a large family of domains described by semi-algebraic sets. [ABSTRACT FROM PUBLISHER]

A random vector X = ( X 1 , … , X n ) with the X i taking values in an arbitrary measurable space ( S , S ) is exchangeable if its law is the same as that of ( X σ ( 1 ) , … , X σ ( n ) ) for any permutation σ . We give an alternative and shorter proof of the representation result (Jaynes [6] and Kerns and Székely [9] ) stating that the law of X is a mixture of product probability measures with respect to a signed mixing measure. The result is “finitistic” in nature meaning that it is a matter of linear algebra for finite S . The passing from finite S to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof (“canonical mixing measure”) by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on S ; but for S = R , one can choose a mixing measure on R n . [ABSTRACT FROM AUTHOR]

We discuss the extension of Jensen's inequality to the framework of quasiconvex functions. Moreover, it is proved that our results work for a class of signed measures larger than the class of probability measures. [ABSTRACT FROM AUTHOR]

We generalize geodesic finite elements to obtain spaces of higher approximation order. Our approach uses a Riemannian centre of mass with a signed measure. We prove well-definedness of this new centre of mass under suitable conditions. As a side product, we can define geodesic finite elements for non-simplex reference elements such as cubes and prisms. We prove smoothness of the interpolation functions and various invariance properties. Numerical tests show that the optimal convergence orders of the discretization error known from the linear theory are obtained also in the nonlinear setting. [ABSTRACT FROM AUTHOR]

In this paper, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, we obtain some properties of the first eigenvalue of a fractional differential equation. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence of positive solutions are established. [ABSTRACT FROM AUTHOR]

This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, etc.) using an ℓ 1 -regularization procedure. In particular, we provide an explicit quantitative localization of the spikes. [ABSTRACT FROM AUTHOR]

We consider a family of signed measures supported on the set of square-free numbers. We prove some local limit theorems for the prime divisor counting function ω( n) and establish new Erdős-Kac type results. [ABSTRACT FROM AUTHOR]

Abstract: We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized two-dimensional Navier–Stokes equations in vorticity form introduced by Kotelenez. [Copyright &y& Elsevier]

For the regression modely=Xβ+εwhere the errors follow the elliptically contoured distribution, we consider the least squares, restricted least squares, preliminary test, Stein-type shrinkage and positive-rule shrinkage estimators for the regression parameters,β. We compare the quadratic risks of the estimators to determine the relative dominance properties of the five estimators. [ABSTRACT FROM AUTHOR]

In this paper, the properties of the super quantum measures are studied. Firstly, the products of Dirac measures are discussed; Secondly, based on the properties of Dirac measures, the structures of super quantum measures are characterized; At last, we prove that any super quantum measure can determine a unique diagonally positive strongly symmetric signed measure. This result verifies the conjecture which was proposed by Gudder. [ABSTRACT FROM AUTHOR]

In an experiment of treatment selections, random samples are drawn from k populations with ordered means. The probability that a sample statistic from the population with the highest mean turns out to be ranked the highest is referred to as the probability of correct selection (PCS). An inequality was proved previously that shows the monotonicity of PCS with respect to change in variance of the samples. In this article, we first present a more general form of the probability inequality to be used to investigate PCS. An extension of the monotonicity of PCS to order statistics is considered. We show that the PCS of the smallest order statistic preserves the monotonicity. Additionally, a normal approximation method is used to further generalize the theory. The general order statistics will not enjoy the same properties, as we reveal the obstacles, and a numerical counter example. [ABSTRACT FROM AUTHOR]

Recently, in Dvurečenskij (, ), it was shown that if a pseudo effect algebra satisfies a kind of the Riesz decomposition property (RDP), then its state space is either empty or a nonempty simplex. This will allow us to prove a Yosida-Hewitt type and a Lebesgue type decomposition for measures on pseudo effect algebra with RDP. The simplex structure of the state space will entail not only the existence of such a decomposition but also its uniqueness. [ABSTRACT FROM AUTHOR]

Abstract: We show that there exist bivariate proper quasi-copulas that do not induce a doubly stochastic signed measure on . We construct these quasi-copulas from the so-called proper quasi-transformation square matrices. [Copyright &y& Elsevier]

Given $\mathcal{X}$, some measurable subset of Euclidean space, one sometimes wants to construct a finite set of points, $\mathcal{P}\subset\mathcal {X}$, called a design, with a small energy or discrepancy. Here it is shown that these two measures of design quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^{2}(=\mathcal{X}\times\mathcal {X})\to\mathbb{R}$. The error of approximating the integral $\int_{\mathcal{X}}f(\boldsymbol{x})\,\mathrm{d}\mu(\boldsymbol{x})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms of the energy or discrepancy of $\mathcal{P}$. The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the measure μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. If $\mathcal{X}$ is the orbit of a compact, possibly non-Abelian group, $\mathcal{G}$, acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action, then it is shown that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$. This allows us to calculate explicit representations of equilibrium measures. [ABSTRACT FROM AUTHOR]

Abstract: Using Schauder''s fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincaré inequalities’, Indiana Univ. Math. J., 49 (2000) 143–175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem: under the usual conditions for initial value problem. We also extend our existence and uniqueness theorems in the case where v is just a signed measure with and [Copyright &y& Elsevier]

Abstract: We study the relationship between multivariate quasi-copulas and measures that they may or may not induce on . We first study the mass distribution of the pointwise best possible lower bound for the set of n-quasi-copulas for . As a consequence, we show that not every n-quasi-copula induces a signed measure on . [Copyright &y& Elsevier]

Abstract: In this paper the conditions under which a broad class of Stein-type estimators dominates the best invariant unbiased estimator of the mean of an elliptically contoured population have been established. The superiority conditions are derived for both known and unknown scale structures. Also an example is given when the general scale matrix is assumed to be known in linear regression. [Copyright &y& Elsevier]

Let ( X, d) be a compact metric space and let $$ \mathcal{M} $$( X) denote the space of all finite signed Borel measures on X. Define I: $$ \mathcal{M} $$( X) → ℝ by I(μ) = ∫ X∫ X d( x, y) dμ(x)dμ( y), and set M( X) = sup I(μ), where μ ranges over the collection of measures in $$ \mathcal{M} $$( X) of total mass 1. The space ( X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in $$ \mathcal{M} $$( X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M( X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L1-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13]. [ABSTRACT FROM AUTHOR]

Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X. Define I : M)X) → ℝ by I (μ) = ∫X∫X d(x, y) dμ(x) dμ(y), and set M(X) = sup I (μ); where μ ranges over the collection of signed measures in M(X) of total mass 1. The metric space (X, d) is quasihypermetric if for all n ϵ ℕ, all α1,…, …n ϵ ℝ satisfying ∑i=1n αi and all x1,…,xn ϵ X, the inequality ∑i,j=1n αiαjd(xi, xj) ≤ 0 holds. Without the quasihypermetric property M(X) is infinite, while with the property a natural semi-inner product structure becomes available on M0(X), the subspace of M(X) of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of (X, d), the semiinner product space structure of M0(X) and the Banach space C(X) of continuous real-valued functions on X; conditions equivalent to the quasihypermetric property; the topological properties of M0(X) with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-* topology and the measure-norm topology on M0(X); and the functional-analytic properties of M0(X) as a semi-inner product space, including the question of its completeness. A later paper [P. Nickolas and R. Wolf, Distance geometry in quasihypermetric spaces. II, Math. Nachr., accepted] will apply the work of this paper to a detailed analysis of the constant M(X). [ABSTRACT FROM AUTHOR]

Abstract: We obtain sharp estimates in signed binomial approximation of binomial mixtures with respect to the total variation distance. We provide closed form expressions for the leading terms, and show that the corresponding leading coefficients depend on the zeros of appropriate Krawtchouk polynomials. The special case of Pólya–Eggenberger distributions is discussed in detail. Our approach is based on a differential calculus for linear operators represented by stochastic processes, which allows us to give unified proofs. [Copyright &y& Elsevier]

We show that a finite collection of exchangeable random variables on an arbitrary measurable space is a signed mixture of i.i.d. random variables. Two applications of this idea are examined, one concerning Bayesian consistency, in which it is established that a sequence of posterior distributions continues to converge to the true value of a parameter θ under much wider assumptions than are ordinarily supposed, the next pertaining to Statistical Physics where it is demonstrated that the quantum statistics of Fermi-Dirac may be derived from the statistics of classical (i.e. independent) particles by means of a signed mixture of multinomial distributions. [ABSTRACT FROM AUTHOR]

Abstract: We generalize the identities of J. Pečarić and A.M. Fink for the Chebyshev functional. The identities are shown to be very useful in establishing various bounds for the Chebyshev functional. [Copyright &y& Elsevier]

Let $$X = (X_t)_{t \geq 0}$$ be a continuous semimartingale and let $$b: \mathbb{R}_+ \rightarrow \mathbb{R}$$ be a continuous function of bounded variation. Setting $$C = \{(t, x) \in \mathbb{R} + \times \mathbb{R} | x < b(t)\}$$ and $$D = \{(t,x) \in \mathbb{R}_+ \times \mathbb{R} | x > b(t)\}$$ suppose that a continuous function $$F: \mathbb{R}_+ \times \mathbb{R} \rightarrow \mathbb{R}$$ is given such that F is C1,2 on $$\bar{C}$$ and F is $$C^{1,2}$$ on $$\bar{D}$$. Then the following change-of-variable formula holds: $$\eqalign{ F(t,X_t) = F(0,X_0)+\int_0^{t} {1 \over 2} (F_t(s, X_s+) + F_t(s,X_s-)) ds\cr + \int_0^t {1 \over 2} (F_x(s,X_s+) + F_x(s,X_s-))dX_s\cr + {1 \over 2} \int_0^t F_{xx} (s,X_s)I (X_s \neq b(s)) d \langle X, X \rangle_s\cr + {1 \over 2} \int_0^t (F_x(s,X_s+)-F_x(s,X_s-)) I(X_s = b(s)) d\ell_{s}^{b} (X),\cr} $$ where $$\ell_{s}^{b}(X)$$ is the local time of X at the curve b given by $$\ell_{s}^{b}(X) = \mathbb{P} - \lim_{\varepsilon \downarrow 0} {1 \over 2 \varepsilon} \int_0^s I(b(r)- \varepsilon < X_r < b(r) + \varepsilon) d \langle X, X \rangle_{r} $$ and $$d\ell_{s}^{b}(X)$$ refers to the integration with respect to $$s \mapsto \ell_{s}^{b}(X)$$. A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping. [ABSTRACT FROM AUTHOR]

In an experiment of treatment selections, random samples are drawn from k populations with different means. The probability that, among the samples, the sample mean from the population with the highest underlying expected value turns out to be the highest is called the probability of correct selection (PCS). In this article, a probability inequality motivated by the PCS problem, is obtained with respect to change in variances. As an application of the inequality in normal mean selection, it is shown that in a multiple treatment experiment setting, by increasing sample sizes, and thus obtaining a higher precision in the estimation of means, in general (but not always), one will have a higher PCS. However, there are special case exceptions. [ABSTRACT FROM AUTHOR]

In this paper we study signed measures. Our main results are as follows: the Fubini theorem is not true in the general case; the Jordan parts of a transition measure are not necessarily transition measures; the operation of taking the Jordan parts does not necessarily commute with multiplying by the initial measure; the product of σ-bounded measures need not be a σ-bounded measure. [ABSTRACT FROM AUTHOR]

In this paper, we examined the connection between quantum systems' indistinguishability and signed (or negative) probabilities. We do so by first introducing a measure-theoretic definition of signed probabilities inspired by research in quantum contextuality. We then argue that ontological indistinguishability leads to the no-signaling condition and negative probabilities. [ABSTRACT FROM AUTHOR]