scholarly journals A structure theorem for stochastic processes indexed by the discrete hypercube

2021 ◽  
Vol 9 ◽  
Author(s):  
Pandelis Dodos ◽  
Konstantinos Tyros
Keyword(s):  
Stochastic Process ◽  
Positive Integer ◽  
Stochastic Processes ◽  
Probability Space ◽  
Structure Theorem ◽  
Structural Information ◽  
Cartesian Product ◽  
Stationarity Condition ◽  
Finite Set ◽  
As If

Abstract Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

scholarly journals Contextuality Scenarios Arising from Networks of Stochastic Processes

2016 ◽  
Vol 23 (03) ◽  
pp. 1650012
Author(s):  
Rodrigo Iglesias ◽  
Fernando Tohmé ◽  
Marcelo Auday
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Empirical Model ◽  
Joint Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Empirical Models ◽  
Classical Source ◽  
Input And Output ◽  
Long Run

An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class 𝒳 of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations are coherent, in the sense that the distribution on a face of a simplex coincides with the marginal of the distribution over the entire simplex. An empirical model is called contextual if its distributions cannot be obtained by marginalizing a joint distribution over 𝒳. Contextual empirical models arise naturally in quantum theory, giving rise to some of its counter -intuitive statistical consequences. In this paper, we present a different and classical source of contextual empirical models: the interaction among many stochastic processes. We attach an empirical model to the ensuing network in which each node represents an open stochastic process with input and output random variables. The statistical behaviour of the network in the long run makes the empirical model generically contextual and even strongly contextual.

scholarly journals A probability space of continuous-time discrete value stochastic process with Markov property

2016 ◽  
Vol 12 (3) ◽  
pp. 5975-5991
Author(s):  
Miloslawa Sokol
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Probability Space ◽  
Measure Theory ◽  
Markov Property ◽  
Finite Measure ◽  
Physical Models

Getting acquainted with the theory of stochastic processes we can read the following statement: "In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it". The classical results for limited stochastic and intensity matrices goes back to Kolmogorov at least late 40-s. But for some infinity matrices the sum of probabilities of all trajectories is less than 1. Some years ago I constructed physical models of simulation of any stochastic processes having a stochastic or an intensity matrices and I programmed it. But for computers I had to do some limitations - set of states at present time had to be limited, at next time - not necessarily. If during simulation a realisation accepted a state out of the set of limited states - the simulation was interrupted. I saw that I used non-quadratic, half-infinity stochastic and intensity matrices and that the set of trajectories was bigger than for quadratic ones. My programs worked good also for stochastic processes described in literature as without probability space. I asked myself: did the probability space for these experiments not exist or were only set of events incompleted? This paper shows that the second hipothesis is true.

On Two Hypotheses in Economic Analysis of Stochastic Processes

2018 ◽  
Vol 8 (8) ◽  
pp. 2391
Author(s):  
Dmitry Aleksandrovich ENDOVITSKY ◽  
Valery Vladimirovich DAVNIS ◽  
Viacheslav Vladimirovich KOROTKIKH
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Probability Space ◽  
Econometric Modeling ◽  
Model Errors ◽  
Expectations Hypothesis ◽  
Additional Information ◽  
Model Residuals ◽  
Econometric Approach ◽  
Probabilistic Nature

Purpose: Development of the apparatus of the stochastic processes econometric modeling. Discussion: The authors identify risk component in the dynamics of stochastic processesin the economy. Theoretical justification of the alternative and proportional expectations is usedto make probabilistic nature of the risk. Results: The authors suggest stochastic process decomposition based on econometric approach to allocate a probability space of risks, and to identify shocks realizations that lie beyond the boundary of this space. Proportional expectations hypothesis distinguished two types of the event influence on the stochastic process realization: continuous (risk) and discrete (shock). The authors suggest model errors and residualsas the main source of information for the identification of the probability space of risks. The technique of econometric modeling of the price and return processes on stock market under theconditions of the proposed hypotheses is considered in the empirical part of the study. F-testresults have not disproved the statement that the model residuals provide additional information about the simulated rate in the case of lack of relevant factors.

A computation of the Littlewood exponent of stochastic processes

1988 ◽  
Vol 103 (2) ◽  
pp. 367-370 ◽  
Author(s):  
Ron C. Blei ◽  
J.-P. Kahane
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Total Variation ◽  
Probability Space ◽  
Complex Measure ◽  
Bona Fide ◽  
Image Position

A stochastic process X = {X(t): t ∈ [0, 1]} on a probability space (Ω, , ℙ) is said to have finite expectation if the function defined on the measureable rectangles in Ω × [0, 1] byfor A ∈ and (s, t) ⊂ [0, 1] gives rise to a complex measure in each of its two coordinates (see [1], definition 1·1). Equivalently, X has finite expectation ifis finite. The function defined by (1), effectively a generalization of the Doléans measure (see e.g. [4] pp. 33–35), is extendible to a bona fide complex measure on Ω × [0, 1] if and only if its ‘total variation’

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

scholarly journals A Double-indexed Functional Hill Process and Applications

2016 ◽  
Vol 8 (4) ◽  
pp. 144
Author(s):  
Modou Ngom ◽  
Gane Samb Lo
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Order Statistics ◽  
Finite Dimension ◽  
Domain Of Attraction ◽  
Extreme Value ◽  
Extreme Value Index

<div>Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (\textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes</div><div> </div><div>\begin{equation}<br />T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log<br />X_{n-j,n}\right)^{s} ,  \label{fme}<br />\end{equation}</div><div> </div><div>indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}%^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies</div><div> </div><div>\begin{equation*}<br />1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .<br />\end{equation*}</div><div> </div><div>We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.</div>

scholarly journals Applications of Generating Functions to Stochastic Processes and to the Complexity of the Knapsack Problem

2021 ◽  
Author(s):  
Jorma Jormakka ◽  
Sourangshu Ghosh
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Polynomial Time ◽  
Knapsack Problem ◽  
Generating Functions ◽  
General Theorem ◽  
Main Idea ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Linear Transform

The paper describes a method of solving some stochastic processes using generating functions. A general theorem of generating functions of a particular type is derived. A generating function of this type is applied to a stochastic process yielding polynomial time algorithms for certain partitions. The method is generalized to a stochastic process describing a rather general linear transform. Finally, the main idea of the method is used in deriving a theoretical polynomial time algorithm to the knapsack problem.

scholarly journals Markov Stochastic Processes in Biology and Mathematics -- the Same, and yet Different

2018 ◽  
Vol 14 (1) ◽  
pp. 7540-7559
Author(s):  
MI lOS lAWA SOKO
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Stochastic Matrix ◽  
Random Number Generator ◽  
Point Of View ◽  
State Transitions ◽  
Similar Principle ◽  
Probabilistic Space ◽  
And Mathematics

Virtually every biological model utilising a random number generator is a Markov stochastic process. Numerical simulations of such processes are performed using stochastic or intensity matrices or kernels. Biologists, however, define stochastic processes in a slightly different way to how mathematicians typically do. A discrete-time discrete-value stochastic process may be defined by a function p : X0 × X → {f : Î¥ → [0, 1]}, where X is a set of states, X0 is a bounded subset of X, Î¥ is a subset of integers (here associated with discrete time), where the function p satisfies 0 < p(x, y)(t) < 1 and  EY p(x, y)(t) = 1. This definition generalizes a stochastic matrix. Although X0 is bounded, X may include every possible state and is often infinite. By interrupting the process whenever the state transitions into the X −X0 set, Markov stochastic processes defined this way may have non-quadratic stochastic matrices. Similar principle applies to intensity matrices, stochastic and intensity kernels resulting from considering many biological models as Markov stochastic processes. Class of such processes has important properties when considered from a point of view of theoretical mathematics. In particular, every process from this class may be simulated (hence they all exist in a physical sense) and has a well-defined probabilistic space associated with it.

On the modeling of linear system input stochastic processes with given accuracy and reliability

2018 ◽  
Vol 24 (2) ◽  
pp. 129-137
Author(s):  
Iryna Rozora ◽  
Mariia Lyzhechko
Keyword(s):  
Banach Space ◽  
Stochastic Process ◽  
Linear System ◽  
Stochastic Processes ◽  
Impulse Response Function ◽  
Output Process ◽  
Gaussian Stochastic Process ◽  
System Input ◽  
Time Invariant ◽  
Gaussian Stochastic Processes

AbstractThe paper is devoted to the model construction for input stochastic processes of a time-invariant linear system with a real-valued square-integrable impulse response function. The processes are considered as Gaussian stochastic processes with discrete spectrum. The response on the system is supposed to be an output process. We obtain the conditions under which the constructed model approximates a Gaussian stochastic process with given accuracy and reliability in the Banach space{C([0,1])}, taking into account the response of the system. For this purpose, the methods and properties of square-Gaussian processes are used.

Type A Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Algebraic Number ◽  
Type A ◽  
Algebraic Number Field ◽  
Basis Vector ◽  
Roots Of Unity ◽  
Finite Set ◽  
Multiple Dirichlet Series

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

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