Moments and Projections of Semistable Probability Measures on p-adic Vector Spaces

In this paper, two topics on semistable probability measures on p-adic vector spaces are studied. One is the existence of absolute moments of operator-semistable probability measures and another is an answer to the question whether one can get semistability of a probability measure from that of all its projections. All results obtained here are extensions of known results for real vector spaces to p-adic vector spaces.     

MODp‐tests,almost independence and small probability spaces

In this paper, we consider approximations to probability distributions over Z . We present an approach to estimate the quality of approximations to probability distributions towards the construction of small probability spaces. These small spaces are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan, and Veličković [EGLNV], the methods which are used here are simple, and we get smaller sample spaces. Our investigations are motivated by recent work of Azar, Motwani, and Naor [AMN]. They considered the problem to construct in time respective space polynomial in n a good approximation to the joint probability distribution of the mutually independent random variables X₁, X₂,…,X_n. Each X_i has values in {0, 1} and satisfies X_i=0 with probability q and X_i=1 with probability 1−q where q∈[0, 1] is arbitrary. Our considerations improve on results in [EGLNV] and [AMN] for q=1/p and p a prime. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 293–313, 2000     

Optional decomposition of optional supermartingales and applications to filtering and finance

ABSTRACT

The classical Doob–Meyer decomposition and its uniform version the optional decomposition are stated on probability spaces with filtrations satisfying the usual conditions. However, the comprehensive needs of filtering theory and mathematical finance call for their generalizations to more abstract spaces without such technical restrictions. The main result of this paper states that there exists a uniform Doob–Meyer decomposition of optional supermartingales on unusual probability spaces. This paper also demonstrates how this decomposition works in the construction of optimal filters in the very general setting of the filtering problem for optional semimartingales. Finally, the application of these optimal filters of optional semimartingales to mathematical finance is presented.     

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.     

Multistage stochastic programming: Error analysis for the convex case

We consider convex stochastic multistage problems and present an approximation technique which allows to analyse the error with respect to time. The technique is based on barycentric approximation of conditional and marginal probability spaces and requiresstrict nonanticipativity for the constraint multifunction and thesaddle property for the value functions.Part of this work was carried out at the Institute of Operations Research of the University of Zurich.     

Isomorphisms of certain correlated probability spaces and of discrete memoryless correlated sources

Summary In an earlier paper [5], we defined a sufficient set of invariants for the isomorphy of discrete memoryless correlated sources with maximal correlation <1. Using the structure of isomorphisms of certain correlated probability spaces, we give here a sufficient set of invariants also for the case of maximal correlation equal to 1. We show, in particular, that two discrete memoryless stationary correlated sources with maximal correlation 1 may be isomorphic in a non-trivial way.     

The Geodesic Problem in Quasimetric Spaces

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x,y)≤σ(d(x,z)+d(z,y)) for some constant σ≥1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the d _α metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a “tree shaped” branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces. This work is supported by an NSF grant DMS-0710714.     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

On Γ-convergence of integral functionals defined on various weighted Sobolev spaces

We consider weighted Sobolev spaces correlated with a sequence of n-dimensional domains. We prove a theorem on the choice of a subsequence Γ-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 99–115, January, 2009.     

Factorization Theorems on Symmetric Spaces of Noncompact Type

We prove the analogs of the Khinchin factorization theorems for K-invariant probability measures on symmetric spaces X=G/K with G semisimple noncompact. We use the Kendall theory of delphic semigroups and some properties of the spherical Fourier transform and spherical functions on X.     

Random walks on almost connected locally compact groups: Boundary and convergence

We prove that, given an arbitrary spread out probability measure μ on an almost connected locally compact second countable groupG, there exists a homogeneous spaceG/H, called the μ-boundary, such that the space of bounded μ-harmonic functions can be identified withL ^∞ (G/H). The μ-boundary is an amenable contractive homogeneous space. We also establish that the canonical projection onto the μ-boundary of the right random walk of law μ always converges in probability and, whenG is amenable, it converges almost surely. The μ-boundary can be characterised as the largest homogeneous space among those homogeneous spaces in which the canonical projection of the random walk converges in probability.     

Duality theorems for marginal problems

Summary Given topological spaces X ₁, ..., X _n with product space X, probability measures _i on X _i together with a real function h on X define a marginal problem as well as a dual problem. Using an extended version of Choquet's theorem on capacities, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces X _i and the measurability resp. boundedness of the function h. Applications concern, among others, measures with given support, stochastic order and general marginal problems.     

A representation of the characteristic function of a stable probability measure on certain TV spaces

In this paper, a Lévy-Khintchine type representation of the characteristic function of a K-regular stable probability measure on real locally convex topological vector spaces, satisfying certain conditions, is presented.     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Global Rigidity: The Effect of Coning

Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al. in Characterizing generic global rigidity, arXiv:, 2007) running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity.     

Kolmogorov’s existence theorem for Markov processes inC* algebras

Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps betweenC* algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes. Dedicated to the memory of Professor K G Ramanathan     

Generic measures for hyperbolic flows on non-compact spaces

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense G _δ -subset consisting of ergodic measures fully supported on the non-wandering set. We also treat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces.     

Risk bounds for kernel density estimators

We use results from probability on Banach spaces and Poissonization techniques to develop sharp finite sample and asymptotic moment bounds for the L _p risk for kernel density estimators. Our results are shown to augment the previous work in this area. Bibliography: 19 titles.     

On the probability of independent sets in random graphs

Let k be the asymptotic value of the independence number of the random graph G(n, p). We prove that if the edge probability p(n) satisfies p(n) ? n^?2/5ln^6/5n then the probability that G(n, p) does not contain an independent set of size k ? c, for some absolute constant c > 0, is at most exp{?cn²/(k⁴p)}. We also show that the obtained exponent is tight up to logarithmic factors, and apply our result to obtain new bounds on the choice number of random graphs. We also discuss a general setting where our approach can be applied to provide an exponential bound on the probability of certain events in product probability spaces. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 1–14, 2003     

Fuzzy概率空间的Fuzzy格刻划

本文研究了Fuzzy概率空间中Fuzzy事件及概率的代数性质.利用Fuzzy概率空间中的概率为集函数这一特征和Fuzzy格的相关理论，得到了Fuzzy概率空间中的概率是从一个Fuzzy格到某个区间的Fuzzy格模同态，并将概率分解成Fuzzy格同态与Fuzzy格模同态的乘积。