Topic

# Stationary distribution

About: Stationary distribution is a research topic. Over the lifetime, 4036 publications have been published within this topic receiving 74581 citations.

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TL;DR: The sampling theory of stationary processes in space is not completely analogous to that of stationary time series, due to the fact that the variate of a time series is influenced only by past values, while for a spatial process dependence extends in all directions as mentioned in this paper.

Abstract: The sampling theory of stationary processes in space is not completely analogous to that of stationary time series, due to the fact that the variate of a time series is influenced only by past values, while for a spatial process dependence extends in all directions. This point is elaborated in ?? 2-4. The estimation and test theory developed in ? 7 is applied in ? 8 to uniformity data for wheat and oranges. The final section is devoted to an examination of some particular two-dimensional processes.

1,575 citations

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TL;DR: In this paper, the authors presented a detailed analytical study of the spatial node distribution generated by random waypoint mobility and derived an exact equation of the asymptotically stationary distribution for movement on a line segment and an accurate approximation for a square area.

Abstract: The random waypoint model is a commonly used mobility model in the simulation of ad hoc networks It is known that the spatial distribution of network nodes moving according to this model is, in general, nonuniform However, a closed-form expression of this distribution and an in-depth investigation is still missing This fact impairs the accuracy of the current simulation methodology of ad hoc networks and makes it impossible to relate simulation-based performance results to corresponding analytical results To overcome these problems, we present a detailed analytical study of the spatial node distribution generated by random waypoint mobility More specifically, we consider a generalization of the model in which the pause time of the mobile nodes is chosen arbitrarily in each waypoint and a fraction of nodes may remain static for the entire simulation time We show that the structure of the resulting distribution is the weighted sum of three independent components: the static, pause, and mobility component This division enables us to understand how the model's parameters influence the distribution We derive an exact equation of the asymptotically stationary distribution for movement on a line segment and an accurate approximation for a square area The good quality of this approximation is validated through simulations using various settings of the mobility parameters In summary, this article gives a fundamental understanding of the behavior of the random waypoint model

1,122 citations

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TL;DR: A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...

Abstract: A randomised approximation scheme for the permanent of a 0–1s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0–1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class includes all dense matrices (those that contain sufficiently many 1’s) and almost all sparse matrices in some reasonable probabilistic model for 0–1 matrices of given density.For the approach sketched above to be computationally efficient, the Markov chain must be rapidly mixing: informally, it must converge in a short time to its stationary distribution. A major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...

878 citations

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TL;DR: A powerful and flexible MCMC algorithm for stochastic simulation that builds on a pseudo-marginal method, showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method.

Abstract: We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139--1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.

723 citations

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TL;DR: In this paper, two principles of a statistical mechanics of time-dependent phenomena are proposed and argued for, namely, the proper mathematical object to describe the physical situation is the stationary random process specified by the ensemble of time series ai(Xt)i=1−s and the distribution ρ(X).

Abstract: Two principles of a statistical mechanics of time‐dependent phenomena are proposed and argued for. The first states that the proper mathematical object to describe the physical situation is the stationary random process specified by the ensemble of time series ai(Xt)i=1···s and the distribution ρ(X). The set phase functions ai(X)i=1···s represent the set of grossly observable features of the system. Xt is the image of the phase X after time t. ρ(X) is a stationary distribution. The second principle is concerned with the very common case in which the phenomenological equations are of the first order in time and states that in this case the random process in question is a Markoff process. A Fokker‐Planck equation is derived for the process, and an entropy is defined and is shown always to increase. Phenomenological equations are derived as a first approximation to the Markoff process. These involve a certain matrix ξij which is shown to satisfy symmetry relations which are a generalization of Onsager's. The...

659 citations