Borel Σ-field Research Articles

Kernel based method for the k-sample problem with functional data

In this paper, we deal with the problem of testing for the equality of k probability distributions defined on where is a metric space and is the corresponding Borel σ-field. We introduce a test statistic based on reproducing kernel Hilbert space embeddings and derive its asymptotic distribution under the null hypothesis. Simulations show that the introduced procedure outperforms a known method.

An Approach of Randomness of a Sample Based on Its Weak Ergodic Limit

For a Polish Sample Space with a Borel σ-field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

Two-Person Zero-Sum Stochastic Games with Semicontinuous Payoff

Consider a two-person, zero-sum stochastic game with Borel state space S, finite action sets A,B, and Borel measurable law of motion q. Suppose the payoff is a bounded function f of the infinite history of states and actions that is measurable for the product of the Borel σ-field for S and the σ-fields of all subsets for A and B, and is lower semicontinuous for the product of the discrete topologies on the coordinate spaces. Then the game has a value and player II has a subgame perfect optimal strategy.

A Skorohod representation theorem for uniform distance

Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X n such that X n ~ μ n for all n ≥ 0 and ||X n − X 0|| → 0 in probability, where ||·|| is the sup-norm. Such conditions do not require μ 0 separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.

Tightness of probability measures on function spaces

Let C E = C ( [ 0 1 ] , E ) be the Banach space, with the supremum norm, of all continuous functions f from the unit interval [ 0 1 ] into the Banach space E. If E = R we put C R = C . Function spaces under consideration are equipped with their Borel σ-field. This paper deals with the tightness property of some classes of probability measures (p.m) on the function space C E . We will be concerned mainly with the specific cases E = R , E = C and more generally E a separable Banach space. We give sufficient conditions for tightness by extending and strengthening the conditions developed by Prohorov in connection with limit theorems of stochastic processes. In the general case of a separable Banach space E, the property of tightness will be settled under conditions of different nature from those of Prohorov. Finally weak convergence of p.m on C E will be established under the condition of weak convergence of their finite dimensional distributions. This extends a similar result valid in the space C.

Bassalygo-Dobrushin uniqueness for continuous spin systems on irregular graphs

The theory of Gibbs random fields indexed by countable sets (e.g., Gibbs states of lattice models) is now well elaborated, see [1]. At the same time, only a few papers deal with the field having index sets presented by irregular graphs. Among them there is a paper by L. A. Bassalygo and R. L. Dobrushin, [2]. It presents a technique of proving uniqueness if the single-site state spaces (single-spin spaces) are finite. Our aim is to develop a similar technique, which covers the case where the single-site state spaces are general metric spaces. In this case, with slight abuse of terminology we say that the spins are continuous, which is reflected in the title above. The basic assumption, however, is that the interaction between the spins is bounded. As possible applications of our theory we mention the theory of Euclidean Gibbs states of the following quantum models. In each of them, quantum particles are located at sites (one particle per site), which form an irregular structure. The single-particle Hamiltonians have discrete spectra and the interparticle interaction is pair-wise and bounded. As a quantum particle, one can take: (a) a free particle moving in a compact Riemannian manifold (e.g. quantum rotator), see [3,4]; (b) a free particle moving in a compact subset of R; (c) a quantum anharmonic oscillator, see e.g. [5]. Let (L,E) be a graph with (infinite) countable sets of vertices, L, and edges, E. We also suppose that the graph is simple, i.e., it has no loops, isolated vertices, and multiple edges. Two vertices `, ` are called adjacent if there exists an edge 〈`, `〉. The number n` of the vertices adjacent to ` is called degree. For each ` ∈ L, let X` be a complete separable metric space (Polish space), B(X`) be the corresponding Borel σ-field, and χ` be a finite Borel measure on (X`,B(X`)). For an edge 〈`, `〉, let a bounded symmetric continuous function (potential) V``′ : X` × X`′ → R be given. Under certain conditions, these objects define a Gibbs random field on the product space X = ∏ `∈LX`. If all V``′ equal zero, there exists only one Gibbs field. Thus, one can expect the same uniqueness if the potentials are sufficiently small, which certainly depends on the underlying graph. If the latter is regular (each vertex has the same degree), the proof of the uniqueness by small potentials is quite standard. The case where n`’s are different but globally bounded (there exists n ∈ N, such that n` 6 n for all ` ∈ L) can be handled similarly. The situation changes substantially if sup`∈L n` = +∞. This can be seen from the example considered in Section 4 below, where the graph is so dense that the ferromagnetic Ising model defined on this graph has multiple Gibbs states for arbitrary non-zero interactions. For sparse graphs of a certain kind, which in

Skorohod representation on a given probability space

Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the X n are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (n k ). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).

Spectral gap of positive operators and applications

Let E be a Polish space equipped with a probability measure μ on its Borel σ-field B , and π a non-quasi-nilpotent positive operator on L p ( E , B , μ ) with 1 < p < ∞ . Using two notions, tail norm condition (TNC for short) and uniformly positive improving property ( UPI / μ for short) for the resolvent of π, we prove a characterization for the existence of spectral gap of π, i.e., the spectral radius r sp ( π ) of π being an isolated point in the spectrum σ ( π ) of π. This characterization is a generalization of M. Hino's result for exponential convergence of π n , where the assumption of existence of the ground state, i.e., of a nonnegative eigenfunction of π for eigenvalue r sp ( π ) , in M. Hino's result, is removed. Indeed, under TNC only, we prove the existence of ground state of π. Furthermore, under the TNC, we also establish the finiteness of dimension of eigenspace of π for eigenvalue r sp ( π ) and a interesting finite triangularization of π, which generalizes L. Gross' famous result by removing his assumption of symmetry and weakening his assumption of hyperboundedness. Finally, we give several applications of the characterization for spectral gap to Schrödinger operators, some invariance principles of Markov processes, and Girsanov semigroups respectively. In particular, we present a sharp condition to guarantee the existence of spectral gap for Girsanov semigroups.

Estimating the Correlation Dimension from a Chaotic System with Dynamic Noise

where F (x) = t(F (x), x1, . . . , xd−1) for x = (x1, x2, . . . , xd). Assuming Y t ∈ Ω for closed Ω ⊆ R and also assuming the ergodicity of {Y t}, we formulate the system as (Ω,F , μ,F ), where F is the completion of the Borel σ-field with respect to μ, and μ is an invariant measure, i.e. μ(F−1A) = μ(A) for A ∈ F . See Carlsson (2002) for a description of the sufficient conditions for the existence of the unique invariant measure. This paper is concerned with the correlation dimension which is defined by Grassberger and Procaccia (1983a, b) as follows. Putting

Measurability for fuzzy valued functions

In this paper, we characterize the Borel σ-field of the space of fuzzy sets endowed with the Skorokhod metric. As a result, the measurability for a fuzzy valued function is discussed.

Construction of markov process with markov resolvent onL P (E, m)

LetE be a Hausdorff topological space and let β(E) be its Borel σ-field. Letm be a σ-finite measure on (E, β(E)). A necessary and sufficient condition for a Markov resolvent onL P (E,m) to be associated with anm-tightm-special standard process (with state spaceE) is given. Furthermore some new examples which do not belong to the framework of Dirichlet space are also given.

The closeness of the range of a probability on a certain system of random events -- an elementary proof

The closeness of the range of a probability on a certain system of random events -- an elementary proof

Extension of stationary stochastic processes

LetX be an arbitrary Hausdorff space, and consider a stationary stochastic process inX with time interval [0, 1], i.e. a tight probability onX[0, 1], equipped with the Borel σ-field of the product space. We prove the existence of a stationary extension of this process to ℝ0+. Furthermore, we show that the extended process may be chosen to have continuous paths if the original process has this property. Under stronger topological assumptions, we derive the corresponding results whenX[0, 1] is equipped with the product of the Borel σ-fields.

On topological support of Dirichlet prior

Ferguson (1973) states that the topological support of a Dirichlet prior on the Borel σ-field (with respect to the weak convergence topology) of the set of all probability measures on the lines is the set of all probability measures with their topological supports contained in that of the parameter measure of the Dirichlet prior. We prove that statement with the line replaced by an arbitrary separable metric space and cite an application of our result to decision theory.

Fourier-Laplace transforms decaying exponentially

Let ƒ(x) be a C-valued function which is integrable with respect to a positive measure m on the Borel σ-field of R = (– ∞, ∞). We shall give necessary and sufficient conditions under which the following Fourier-Laplace transform f(λ) of ƒ(x) decays exponentially as λ → ∞:where θ is a constant with –π/2 ≤ θ ≤ π/2.

An exact rate of convergence in the functional central limit theorem for special martingale difference arrays

Since the topology of weak convergence of probability distributions on the Borel σ-field of the space C= C([0, 1]) is metrizable, it is natural to describe the speed of convergence in weak functional limit theorems by means of an appropriate metric. Using the metric proposed by Prokhorov it is shown that under suitable conditions the rate of convergence in the functional central limit theorem for C-valued partial sum processes based on martingale difference arrays is the same as in the special case of row-wise independent random variables where this rate is known to be an optimal one.

Messbare Mengen von Massen und Inhalten

Let X be a completely regular space. The customary σ-field is the coarsest σ-field on the space of Bairemeasures on X which makes μ→μ(A) measurable for any Baire set A. We compare the customary σ-field with the Baire and Borel σ-field induced by the weak* topology which lies on the dual space C(X)′. In (2.3) it is shown that the customary σ-field is just the Baire σ-field. In part 3 necessary and sufficient conditions are given under which the set of τ-smooth measures is measurable with respect to the Borel σ-field which lies on the positive cone of the space of finitely additive, regular measures C(X)′. Finally, a decomposition theorem for generalized kernels is proved. The τ-smooth part of a generalized kernel is a kernel again if certain conditions are fulfilled.

Weak convergence on non-separable metric spaces

AbstractFor certain problems in the weak convergence theory of probability measures on non-separable metric spaces it is convenient to consider measures defined on a σ-field B0 smaller than the Borel σ-field. A suitable theory has been developed by Dudley and Wichura. In this paper it is shown that some of the key results in that theory can be deduced directly from the better known weak convergence theory for Borel measures. This is achieved by a remetrization of the underlying space to make it separable. The σ-field B0 contains the new Borel σ-field and weak convergence in Dudley's sense becomes equivalent to convergence of restrictions of the measures to the latter σ-field.1980 Mathematics Subject Classification (Amer. Math. Soc.): primary 60 B 10, 60 B 05.