Parameterized Metatheory for Continuous Markovian Logic

This work presents three increasingly expressive Dynamic Logics in which the programs are described in a language based on CCS. Our goal is to build dynamic logics that are suitable for the description and verification of properties of communicating concurrent systems, in a similar way as PDL is used for the sequential case. In order to accomplish that, CCS’s operators and constructions are added to a basic modal logic. Doing this, the semantics of CCS’s parallel operator allows us to build dynamic logics that support communicating and concurrent programs. We build a simple Kripke semantics for these logics, provide complete axiomatizations for them and show that they have the finite model property. This contrasts with other dynamic logics with parallel operators presented in the literature, such as Peleg’s Concurrent PDL with Channels, where either the parallel programs cannot communicate, or at least one of the properties mentioned above (simple Kripke semantics, complete axiomatization and finite model property) is missing.

Interest in the study of dichotomy for various classes of probability measures arose after the well-known Kakutani theorem [1] concerning the equivalence and singularity of product measures had appeared. The development and various generalizations of this result can be found in Feldman [2], Hajek [3], Fernique [4], Kanter [5], and Okazaki [6]. In the last paper, Okazaki proved a dichotomy theorem for H-ergodic probability measures on a locally convex Hausdorff space equipped with a cylindrical c-algebra C(E, {x~ }), where {x*},~z C E*; this theorem generalizes all earlier dichotomy results, since in every case one can find an appropriate set H with respect to which the measures are ergodic. The proof of the Okasaki theorem uses topological properties of the original space and the separability of the c-algebra C(E, {x~ }) on which the measure p is defined. In this paper we prove a dichotomy theorem for ergodic measures on linear measure spaces in the most general situation under rather weak requirements. We do not assume that the linear space is equipped with a topological structure, and the measurable space structure need not agree with the linear operations. Let (E, ~) be an arbitrary measure space such that E is a linear set and the a-algebra ~ is invariant under translations by elements z E E. Let p be a probability measure on ~, and let H C E.

Current Temporal Logics are all oriented towards the description of totally ordered sequences. This limits their usefulness for reasoning about systems whose computations cannot easily be mapped into totally ordered sequences. Here, we propose a temporal logic geared towards describing partially ordered sets and apply it to dynamic distributed systems. Even though the logic we define does not have the finite model property, we establish that it has a one exponential decision procedure and a complete axiomatization.

Suppose we observe an ergodic Markov chain on an arbitrary state space. The usual nonparametric estimator of a linear functional of the stationary distribution is the empirical estimator. If the stationary distribution obeys finitely many known linear constraints, we can improve the empirical estimator by empirical likelihood weights. Since the observations are dependent, an optimal choice of weights is determined by weighting averages over disjoint blocks of observations with slowly increasing length. We show that the improved empirical estimator is efficient. We also introduce two additively corrected empirical estimators that are asymptotically equivalent to the weighted empirical estimator, hence also efficient.

Potential theory for finitely additive Markov chains

For a Markov chain with an arbitrary nonempty state space, with stationary finitely additive transition probabilities and with initial distribution concentrated on a recurrent state, it is shown that the probability of every tail set is either zero or one. This generalizes and in particular gives an alternative proof of the result due to Blackwell and Freedman [1] in case the state space is countable and all transition probabilities are countably additive.

In this paper stationary Markov decision problems are considered with arbitrary state space and compact space of strategies. Conditions are given for the existence of an average optimal strategy. This is done by using the fact that a continuous function on a compact space attains its minimum. To prove the continuity of the average costs as function on the space of strategies some perturbation results for quasi-compact linear operators are used. In a first set of conditions the boundedness of the one-period cost functions and the quasi-compactness of the Markov processes are assumed. In more general conditions the boundedness of the cost functions is replaced by the boundedness, on a subset A of the state space, of the recurrence time and costs until A and the quasi-compactness of the Markov processes are replaced by the quasi-compactness of the embedded Markov processes on A.

Partial derivative with respect to the measure and its application to general controlled mean-field systems

Uniform consistency and weak convergence is proved of estimators of the transition probabilities of an arbitrary finite state space Markov renewal process, based on n independent and identically distributed “right censored” realizations of the process. The approach uses the theory of stochastic integrals and counting processes. It is shown how the results may be extended to the non-identically distributed case and to general censorship under suitable conditions.

In this paper, we study the fluctuations of observables of metric measure spaces which are random discrete approximations Xn of a fixed arbitrary (complete, separable) metric measure space X=(X,d,μ). These observables Φ(Xn) are polynomials in the sense of Greven–Pfaffelhuber–Winter, and we show that for a generic model space X, they yield asymptotically normal random variables. However, if X is a compact homogeneous space, then the fluctuations of the observables are much smaller, and after an adequate rescaling, they converge towards probability distributions which are not Gaussian. Conversely, we prove that if all the fluctuations of the observables Φ(Xn) are smaller than in the generic case, then the measure metric space X is compact homogeneous. The proofs of these results rely on the Gromov reconstruction principle, and on an adaptation of the method of cumulants and mod-Gaussian convergence developed by Féray–Méliot–Nikeghbali. As an application of our results, we construct a statistical test of the hypothesis of symmetry of a compact Riemannian manifold.

Diffusion polynomial frames on metric measure spaces

In the language of random counting measures, many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology, are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.

Allowing for incomplete information, this paper characterizes the social choice functions that can be approximated by the equilibrium outcomes of a mechanism: incentive compatibility is necessary and almost sufficient for virtual Bayesian implementability. In conjunction with a second condition, Bayesian incentive consistency, incentive compatibility is also sufficient. This new condition is weak-under standard topological and informational assumptions, it is satisfied by every social choice function. The type sets of the agents are taken to be arbitrary (possibly infinite) measurable spaces. An example shows that there are virtually (in fact, exactly) Bayesian implementable social choice functions that are not virtually implementable in iteratively undominated strategies.

For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures, including as special cases Choquet and Sugeno integrals. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced.

Let ${\rm X}=({\rm x}_1,{\rm x}_2,\ldots,{\rm x}_n)$ be a sample consisting of n independent observations in an arbitrary measurable space ${\cal X}$ such that the first $\theta$ observations have a distribution F while the remaining $n-\theta$ ones follow $G\neq F$, the distributions F and G being unknown and quantities n and $\theta$ large. In [A. A. Borovkov and Yu. Yu. Linke, Math. Methods Statist., 14 (2005), pp. 404–430] there were constructed estimators $\theta^*$ of the change-point $\theta$ that have proper error (i.e., such that ${\bf P}_\theta\{|\theta^*-\theta|>k\}$ tends to zero as k grows to infinity), under the assumption that we know a function h for which the mean values of $h({\rm x}_j)$ under the distributions F and G are different from each other. Sequential procedures were also presented in that paper. In the present paper, we obtain similar results under a weakened form of the above assumption or even in its absence. One such weaker version assumes that we have functions $h_1,h_2,\ldots,h_l$ on $\cX$ such that for at least one of them the mean values of $h_j({\rm x}_i)$ are different under F and G. Another version does not assume the existence of known to us functions $h_j$, but allows the possibility of estimating the unknown distributions F and G from the initial and terminal segments of the sample ${\rm X}$. Sequential procedures are also dealt with.