Statistical maps: A categorical approach

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p _f , a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets. Supported by VEGA 1/2002/06.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

Description of random fields by means of one-point finite-conditional distributions

The aim of this note is to investigate the relationship between strictly positive random fields on a lattice ? ^ν and the conditional probability measures at one point given the values on a finite subset of the lattice ? ^ν . We exhibit necessary and sufficient conditions for a one-point finite-conditional system to correspond to a unique strictly positive probability measure. It is noteworthy that the construction of the aforementioned probability measure is done explicitly by some simple procedure. Finally, we introduce a condition on the one-point finite conditional system that is sufficient for ensuring the mixing of the underlying random field.     

Fuzzy measures and discreteness

The main result is to show that the space of nonmonotonic fuzzy measures on a measurable space (X,X) with total variation norm is separable if and only if the σ-algebra X is a finite set. Our result is related to fuzzy analysis, functional spaces and discrete mathematics.     

Attainable densities for random maps

We consider a random map T=T(Γ,ω), where Γ=(τ₁,τ₂,…,τK) is a collection of maps of an interval and ω=(p₁,p₂,…,pK) is a collection of the corresponding position dependent probabilities, that is, pk(x)?0 for k=1,2,…,K and . At each step, the random map T moves the point x to τk(x) with probability pk(x). For a fixed collection of maps Γ, T can have many different invariant probability density functions, depending on the choice of the (weighting) probabilities ω. Most of the results in this paper concern random maps where Γ is a family of piecewise linear semi-Markov maps. We investigate properties of the set of invariant probability density functions of T that are attainable by allowing the probabilities in ω to vary in a certain class of functions. We prove that the set of all attainable densities can be determined algorithmically. We also study the duality between random maps generated by transformations and random maps constructed from a collection of their inverse branches. Such representation may be of greater interest in view of new methods of computing entropy [W. S?omczyński, J. Kwapień, K. ?yczkowski, Entropy computing via integration over fractal measures, Chaos 10 (2000) 180-188].     

Some Characterizations of Ergodic Probability Measures

Let P be a Markov kernel defined on a measurable space (X, ??). A P-ergodic probability is an extreme point of the family of all P-invariant probability measures on ??. Several characterizations of P-ergodic probabilities are given. In particular, for the special case of a topological space X both P-ergodic Baire probabilities and P-ergodic Borel probabilities with additional regularity properties are characterized.     

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting on a Borel space S and on an underlying ??-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.     

Canonical and microcanonical Gibbs states

Summary The grand canonical Gibbs states for a system from classical statistical mechanics can be defined as the probability measures on an appropriate phase space which have certain specified conditional probabilities. These conditional probabilities are with respect to a family of -algebras associated with subsets of the space in which the system lies. If different families of -algebras are used then canonical and microcanonical Gibbs states are obtained. The relationship between these different Gibbs states is studied and, subject to various conditions, it is shown that each canonical and microcanonical Gibbs state can be written as a convex mixture of grand canonical Gibbs states.     

An existence theorem for probability measures invariant under a Markov kernel

LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allA ∈A∈A. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.     

Absolutely continuous invariant measures for random maps with position dependent probabilities

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τ_k is chosen from a finite collection of maps with constant probability p_k. In this note we allow the p_k's to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0,1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.     

Actuarial risk measures for financial derivative pricing

We present an axiomatic characterization of price measures that are superadditive and comonotonic additive for normally distributed random variables. The price representation derived involves a probability measure transform that is closely related to the Esscher transform, and we call it the Esscher-Girsanov transform. In a financial market in which the primary asset price is represented by a stochastic differential equation with respect to Brownian motion, the price mechanism based on the Esscher-Girsanov transform can generate approximate-arbitrage-free financial derivative prices.     

Convergence of stochastic empirical measures

Let P_n be a random probability measure on a metric space S. Let Pˆ_n be the empirical measure of k_n iid random variables, each distributed according to P_n. Our main theorem asserts that if {P_n} converges in distribution, as random probability measures on S, then so does {Pˆ_n}. Applications of the result to the study of bootstrap and other stochastic procedures are given.     

Polar Decomposition of Scale-Homogeneous Measures with Application to Lévy Measures of Strictly Stable Laws

A scaling on some space is a measurable action of the group of positive real numbers. A measure on a measurable space equipped with a scaling is said to be \(\alpha \)-homogeneous for some nonzero real number \(\alpha \) if the mass of any measurable set scaled by any factor \(t > 0\) is the multiple \(t^{-\alpha }\) of the set’s original mass. It is shown rather generally that given an \(\alpha \)-homogeneous measure on a measurable space there is a measurable bijection between the space and the Cartesian product of a subset of the space and the positive real numbers (that is, a “system of polar coordinates”) such that the push-forward of the \(\alpha \)-homogeneous measure by this bijection is the product of a probability measure on the first component (that is, on the “angular” component) and an \(\alpha \)-homogeneous measure on the positive half line (that is, on the “radial” component). This result is applied to the intensity measures of Poisson processes that arise in Lévy-Khinchin-Itô-like representations of infinitely divisible random elements. It is established that if a strictly stable random element in a convex cone admits a series representation as the sum of points of a Poisson process, then it necessarily has a LePage representation as the sum of i.i.d. random elements of the cone scaled by the successive points of an independent unit-intensity Poisson process on the positive half line each raised to the power \(-\frac{1}{\alpha }\).     

Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form

$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$

for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.

     

On uniform local dispersion on a family of G-orbits

Consider a topological space T which is the union of a family of G-orbits, where G is a locally euclidean group G acting on T. On every G-orbit consider a probability which is absolutely continuous with respect to the image measure of the normalized restriction of the Haar measure on some compact neighborhood of the identity in G. Assume that the densities of the probabilities on the orbits have a common upper bound. Let μ be a probability on T which is the integral over the measures on the orbits with respect to some probability μ′ on T. It is shown that this specific kind of integral representation of μ does not depend on the size of the compact neighborhood of the identity in G.     

An Elementary Approach to the Daniell-Kolmogorov Theorem and Some Related Results

We give a short elementary proof of the Daniell-Kolmogorov existence theorem for probability measures on product spaces, assuming nothing but the existence of Lebesgue measure on the unit interval. Related approaches are used to prove the existence of regular conditional distributions directly on Polish spaces, and to establish the existence of random measures and sets with given finite-dimensional distributions or hitting probabilities, respectively.     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

On the Parametrized Integral of a Multifunction: The Unbounded Case

Integration of set-valued maps (alias multifunctions) depending on a parameter is revisited. Results of Artstein, and of Saint-Pierre and Sajid are extended to the case of set-valued maps whose values may be unbounded. In the general case, this is achieved assuming that the transition probabilities involved in the integration procedure are absolutely continuous with respect to some fixed probability measure. However, when the integrating probability measure does not depend on the parameter this hypothesis is shown to be unnecessary. On the other hand, an alternative proof of a result of Saint-Pierre and Sajid is provided for convex compact-valued multifunctions. An application is given to the control of chattering systems. It is an extension of a result of Artstein to the case of set-valued maps with unbounded values. The proof of the main results is simple and essentially relies on measurable selections arguments.