Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.

Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Abstract: Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.

Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice

Abstract: Preface. Acknowledgments. INTRODUCTION. Random Vectors. Linear Operators. Infinitely Divisible Distributions and Triangular Arrays. MULTIVARIATE REGULAR VARIATION. Regular Variations for Linear Operators. Regular Variation for Real-Valued Functions. Regular Variation for Borel Measures. MULTIVARIATE LIMIT THEOREMS. The Limit Distributions. Central Limit Theorems. Related Limit Theorems. APPLICATIONS. Applications to Statistics. Self-Similar Stochastic Processes. References. Index.

Stochastic limit theory : an introduction for econometricians

Abstract: This is a survey of the recent developments in the rapidly expanding field of asymptotic distribution theory, with a special emphasis on the problems of time dependence and heterogeneity. The book is designed to be useful on two levels. First as a textbook and reference work, giving definitions of the relevant mathematical concepts, statements, and proofs of the important results from the probability literature, and numerous examples; and second, as an account of recent work in the field of particular interest to econometricians, including a number of important new results. It is virtually self-contained, with all but the most basic technical prerequisites being explained in their context; mathematical topics include measure theory, integration, metric spaces, and topology, with applications to random variables, and an extended treatment of conditional probability. Other subjects treated include: stochastic processes, mixing processes, martingales, mixingales, and near-epoch dependence; the weak and strong laws of large numbers; weak convergence; and central limit theorems for nonstationary and dependent processes. The functional central limit theorem and its ramifications are covered in detail, including an account of the theoretical underpinnings (the weak convergence of measures on metric spaces), Brownian motion, the multivariate invariance principle, and convergence to stochastic integrals. This material is of special relevance to the theory of cointegration.

Stochastic monotonicity and stationary distributions for dynamic economies

Abstract: The existence and stability of invariant distributions for stochastically monotone processes is studied. The Knaster-Tarski fixed point theorem is applied to establish existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering. Global convergence of a monotone Markov process to its unique invariant distribution is established under an easily verified assumption. Topkis' theory of supermodular functions is applied to stochastic dynamic optimization, providing conditions under which optimal stationary decisions are monotone functions of the state and induce a monotone Markov process. Applications of these results to investment theory, stochastic growth, and industry equilibrium dynamics are given.

Gaussian Random Processes

Abstract: There are several types of random processes that have found wide application because of their realistic physical modeling yet relative mathematical simplicity. In this and the next two chapters we describe these important random processes. They are the Gaussian random process, the subject of this chapter; the Poisson random process, described in Chapter 21; and the Markov chain, described in Chapter 22. Concentrating now on the Gaussian random process, we will see that it has many important properties. These properties have been inherited from those of the N-dimensional Gaussian PDF, which was discussed in Section 14.3.