Haar's Lemma (1918) deals with the algebraic characterization of the inclusion of polyhedral sets. This Lemma has been involved many times in automatic control of linear dynamical systems via positive invariance of polyhedrons. More recently, it has been used to characterize stochastic comparison w.r.t. linear/integral ordering of Markov (reward) chains. In this paper we develop a state space oriented approach to the control of Discrete Event Systems (DES) based on the remark that most of control...

In this note we focus attention on identifying optimal policies and on elimination suboptimal policies minimizing optimality criteria in discrete-time Markov decision processes with finite state space and compact action set. We present unified approach to value iteration algorithms that enables to generate lower and upper bounds on optimal values, as well as on the current policy. Using the modified value iterations it is possible to eliminate suboptimal actions and to identify an optimal policy...

We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...

We present three new identities in law for quadratic functionals of conditioned bivariate Gaussian processes. In particular, our results provide a two-parameter generalization of a celebrated identity in law, involving the path variance of a Brownian bridge, due to Watson (1961). The proof is based on ideas from a recent note by J.-R. Pycke (2005) and on the stochastic Fubini theorem for general Gaussian measures proved in Deheuvels et al. (2004).

Let $X=X\left(t\right),t\in {ℝ}^N$ be a Gaussian random field in ${ℝ}^d$ with stationary increments. For any Borel set $E\subset {ℝ}^N$, we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

En este trabajo se describe la implementación de un algoritmo para el cálculo de polinomios zonales, así como dos aplicaciones explícitas de éstos en el ámbito del análisis multivariante. Concretamente, esta implementación permite obtener resultados de sumación aproximados para funciones hipergeométricas de argumento matricial que, a su vez, pueden utilizarse en la génesis de distribuciones multivariantes discretas con frecuencias simétricas. De igual forma, se pone en práctica un conocido resultado...

In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product. We consider then the Stochastic Collocation method, and use the previous estimates to introduce...

Having Polish spaces $𝕏$, $𝕐$ and $\mathbb{Z}$ we shall discuss the existence of an $𝕏\times 𝕐$-valued random vector $(\xi ,\eta )$ such that its conditional distributions $K_x=ℒ(\eta \mid \xi =x)$ satisfy $e(x,K_x)=c\left(x\right)$ or $e(x,K_x)\in C\left(x\right)$ for some maps $e:𝕏\times {ℳ}_1\left(𝕐\right)\to \mathbb{Z}$, $c:𝕏\to \mathbb{Z}$ or multifunction $C:𝕏\to 2^{\mathbb{Z}}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $K:𝕏\to {ℳ}_1\left(𝕐\right)$ defined implicitly by $e(x,K_x)=c\left(x\right)$ or $e(x,K_x)\in C\left(x\right)$ respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the $(e,c)$- or $(e,C)$-problem and illustrate...