Sequence Of Probability Measures Research Articles

Multiple partition structures and harmonic functions on branching graphs

We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman's partition structures, and are motivated by a problem of population genetics. They are related to harmonic functions and coherent systems of probability measures on a certain branching graph. The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter. If the value of the Jack parameter is equal to one the branching graph under considerations reflects the branching rule for the irreducible representations of the wreath product of a finite group with the symmetric group. If the value of the Jack parameter is zero then the coherent systems of probability measures are precisely the multiple partition structures. We establish a bijective correspondence between the set of harmonic functions on the graph and probability measures on the generalized Thoma set. The correspondence is determined by a canonical integral representation of harmonic functions. As a consequence we obtain a representation theorem for multiple partition structures.We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. If the finite group contains the identity element only then it coincides with the well-known Ewens probability measure on the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.

On Wiener's lemma for locally compact abelian groups

Inspired by an extension of Wiener's lemma on the relation of measures μ on the unit circle and their Fourier coefficients μˆ(kn) along subsequences (kn) of the natural numbers by Cuny, Eisner and Farkas [1], we study the validity of the lemma when the Fourier coefficients are weighted by a sequence of probability measures. By using convergence with respect to a filter derived from these measure sequences, we obtain similar results, now also allowing the consideration of locally compact abelian groups other than T and R. As an application, we present an extension of a result of Goldstein [6] on the action of semigroups on Hilbert spaces.

Parallel sequential Monte Carlo for stochastic gradient-free nonconvex optimization

We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth-order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples and the dimension of the search space. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad “flat” regions which are hard to minimize using gradient-based techniques.

Peacocks nearby: Approximating sequences of measures

A peacock is a family of probability measures with finite mean that increases in convex order. It is a classical result, in the discrete time case due to Strassen, that any peacock is the family of one-dimensional marginals of a martingale. We study the problem whether a given sequence of probability measures can be approximated by a peacock. In our main results, the approximation quality is measured by the infinity Wasserstein distance. Existence of a peacock within a prescribed distance is reduced to a countable collection of rather explicit conditions. This result has a financial application (developed in a separate paper), as it allows to check European call option quotes for consistency. The distance bound on the peacock then takes the role of a bound on the bid–ask spread of the underlying. We also solve the approximation problem for the stop-loss distance, the Lévy distance, and the Prokhorov distance.

Error analysis of the moving least-squares method with non-identical sampling

ABSTRACTWe derive the convergence rate of the moving least-squares learning algorithm for regression under the assumption that the samples are drawn from a non-identical sequence of probability measures. The error analysis is carried out by analysing the drift error and using the probability inequalities for the non-identical sampling. When the sequence of marginal distributions converges exponentially to marginal distribution in the dual of a Hölder space, we obtain the satisfactory capacity dependent error bounds of the algorithm that can be arbitrarily close to the rate .

Asymptotic arbitrage in fractional mixed markets

We consider a family of mixed processes given as the sum of a fractional Brownian motion with Hurst parameter $H\in(3/4,1)$ and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyze the underlying markets with methods from large financial markets. More precisely, we show the existence of a strong asymptotic arbitrage (defined as in Kabanov and Kramkov [Finance Stoch. 2(2), 143--172 (1998)]) when the scaling factor converges to zero. We apply a result of Kabanov and Kramkov [Finance Stoch. 2(2), 143--172 (1998)] that characterizes the notion of strong asymptotic arbitrage in terms of the entire asymptotic separation of two sequences of probability measures. The main part of the paper consists of proving the entire separation and is based on a dichotomy result for sequences of Gaussian measures and the concept of relative entropy.

An iterated Azéma–Yor type embedding for finitely many marginals

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $\mu_{1},\ldots,\mu_{n}$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalization of the Azema and Yor [In Seminaire de Probabilites, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (1979) 90–115 Springer] solution. In particular, we recover the stopping boundaries obtained by Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578] and Madan and Yor [Bernoulli 8 (2002) 509–536]. Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$. In our analysis we compute the law of the maximum at each of the $n$ stopping times. This is used in Henry-Labordere et al. [Ann. Appl. Probab. 26 (2016) 1–44] to show that the construction maximizes the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

On normalized multiplicative cascades under strong disorder

Multiplicative cascades, under weak or strong disorder, refer to sequences of positive random measures n; ;n = 1;2;::: , parameterized by a positive disorder parameter , and defined on the Borel -field B of @T = f0;1;:::b 1g 1 for the product topology. The normalized cascade is defined by the corresponding sequence of random probability measures prob n; := Z 1 n; n; ;n = 1;2:::; normalized to a probability by the partition function Zn; . In this note, a recent result of Madaule [27, 2011] is used to explicitly construct a family of tree indexed probability measures prob1; for strong disorder parameters > c, almost surely defined on a common probability space. Moreover, viewingfprobn; : > cg 1 n=1 as a sequence of probability measure valued stochastic process leads to finite dimensional weak convergence in distribution to a probability measure valued processfprob1; : > cg. The limit process is

Von Neumann and Birkhoff ergodic theorems for negatively curved groups

We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

Weak*-Convergence to Minimum Energy Measure and Dispersed-Dot Halftoning

In this paper, we investigate a weak*-convergence property of certain sequences of probability measures to an energy-minimizing measure and then derive a simple heuristic algorithm to realize optimal point distributions constrained by a given density function as a nearly minimum energy state. With the derived results, we develop a new dispersed-dot halftoning technique, which achieves uniform distributions of dots while approximating the original image in the weak*-sense.

Random Kähler metrics

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kähler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kähler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kähler metrics. Several examples are considered.

Almost Everywhere Convergence of Convolution Measures

AbstractLet (X, ℬ, m, τ) be a dynamical system with (X, ℬ, m) a probability space and τ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in L1(X) of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures {vi} defined on ℤ. We then exhibit cases of such averages where convergence fails.

A note on weak convergence

We show that in a Polish space if {Pn} is a sequence of probability measures then the existence of \(\displaystyle \lim_n \int f dP_n\) for every bounded continuous function guarantees the existence of a probability P such that Pn converges weakly to P.

Coefficient regularized regression with non-iid sampling

In this paper, we study a more general kernel regression learning with coefficient regularization. A non-iid setting is considered, where the sequence of probability measures for sampling is not identical but the sequence of marginal distributions for sampling converges exponentially fast in the dual of a Holder space; the sampling z i , i ≥ 1 are weakly dependent, which satisfy a strongly mixing condition. Satisfactory capacity independently error bounds and learning rates are derived by the techniques of integral operator for this learning algorithm.

Stability Analysis of Two-Stage Stochastic Mathematical Programs with Complementarity Constraints via NLP Regularization

This paper presents numerical approximation schemes for a two-stage stochastic programming problem, where the second stage problem has a general nonlinear complementarity constraint. First the complementarity constraint is approximated by a parameterized system of inequalities with a well-known regularization approach by Scholtes [SIAM J. Optim., 11 (2001), pp. 918–936] in deterministic mathematical programs with equilibrium constraints; the distribution of the random variables of the regularized two-stage stochastic program is then approximated by a sequence of probability measures. By treating the approximation problems as a perturbation of the original (true) problem, we carry out a detailed stability analysis of the approximated problems, including continuity and local Lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points. A particular focus is given to the case where the probability distribution is approximated by the empirical probability measure which is also known as sample average approximation.

Generalization of Portmanteau Theorem with Respect to the Pseudoweak Convergence of Random Closed Sets

The main result of this paper is a theorem of portmanteau type for pseudoweak convergent sequences of capacity functionals for a sequence of random closed sets. For that purpose the classical Lebesgue integral had been substituted with a more general one, known as general pseudo-integral, and there is introduced the pseudoweak convergence of capacity functionals. A connection between weak convergence of a sequence of probability measures induced by the sequence of random closed sets and convergence of pseudo-integral with respect to the corresponding sequence of capacity functionals is given.

Uniform measures on inverse limit spaces

Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let where X is a compact metric space and f is continuous, onto and piecewise one-to-one and . Then starting with a measure μ1 on the Borel sets , we recursively construct a sequence of probability measures on satisfying for each and . This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y. If μ1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recursively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evaluation and in dynamical systems in providing an ambient measure (when Lebesgue measure is not available) with which to define a Sinai–Ruelle–Bowen (SRB) measure or a metric attractor for the shift map on the inverse limit space.

Stability of sequential Monte Carlo samplers via the Foster–Lyapunov condition

Sequential Monte Carlo (SMC) samplers [Del Moral, P., Doucet, A., Jasra, A., 2006. Sequential Monte Carlo samplers. J. Roy. Statist. Soc. B 68, 411–436] are designed to simulate from a sequence of probability measures on a common measurable space (E,E). One way to measure the accuracy of the resulting Monte Carlo estimates is the asymptotic variance in the central limit theorem (CLT). We investigate the conditions, for algorithms used in practice, which are sufficient to ensure that the resulting expression is upper bounded, of which, the typical conditions (e.g. [Chopin, N., 2004. Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32, 2385–2411]) are quite restrictive. We use the Foster–Lyapunov condition and contractions in the f-norm of the Markov kernels [Douc, R., Moulines, E., Rosenthal, J.S., 2004. Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab. 14, 1643–1665] to establish quantitative bounds on the asymptotic variance.

Weak Convergence and Vector-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi

In this paper we present a theorem that establishes a relation between continuous, norm-bounded functions from a metric space into a separable Hilbert space and weak convergence of sequences of probability measures on the metric space. After establishing this result, it’s application to the stability theory of Feynman’s operational calculi will be illustrated. We will see that the existing time-dependent stability theory of the operational calculi will be significantly improved when the operator-valued functions take their values in $\mathcal{L}(\mathcal{H})$ , $\mathcal{H}$ a separable Hilbert space.

A note on the weak convergence of probability measures in the D[0,1] space

Let τ be a regular metric as defined below for the D = D [ 0 , 1 ] space. Even when ( D , τ ) is not a separable and complete metric space we show (i) that the usual conditions on a sequence of probability measures in ( D , τ ) ensures its weak convergence and (ii) that Prohorov's theorem in ( D , τ ) can be derived as a consequence of our results.