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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]

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]

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 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]

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]

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]

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]

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]

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]

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 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]