This volume contains the proceedings of the AMS Special Session on Unimodularity in Randomly Generated Graphs, held from October 8-9, 2016, in Denver, Colorado. Unimodularity, a term initially used ...in locally compact topological groups, is one of the main examples in which the generalization from groups to graphs is successful. The "randomly generated graphs", which include percolation graphs, random Erdős-Rényi graphs, and graphings of equivalence relations, are much easier to describe if they result as random objects in the context of unimodularity, with respect to either a vertex-transient "host"-graph or a probability measure. This volume tries to give an impression of the various fields in which the notion currently finds strong development and application: percolation theory, point processes, ergodic theory, and dynamical systems.
Our ability to acknowledge and recognize our own identity — our “self” — is a characteristic doubtless unique to humans. Where does this feeling come from? How does the combination of ...neurophysiological processes coupled with our interaction with the outside world construct this coherent identity? We know that our social interactions contribute via the eyes, ears, etc. However, our self is not only influenced by our senses. It is also influenced by the actions we perform and those we see others perform. Our brain anticipates the effects of our own actions and simulates the actions of others. In this way, we become able to understand ourselves and to understand the actions and emotions of others. This book describes the new field of “Motor Cognition”. Though motor actions have long been studied by neuroscientists and physiologists, it is only recently that scientists have considered the role of actions in building the self. How consciousness of action is part of self-consciousness, how one's own actions determine the sense of being an agent, how actions performed by others impact on ourselves for understanding others, differentiating ourselves from them and learning from them: these questions are raised and discussed throughout the book, drawing on experimental, clinical, and theoretical bases. The advent of new neuroscience techniques, such as neuroimaging and direct electrical brain stimulation, together with a renewal of behavioral methods in cognitive psychology, provide new insights into this area. Mental imagery of action, self-recognition, consciousness of actions, imitation can be objectively studied using these new tools. The results of these investigations shed light on clinical disorders in neurology, psychiatry, and in neuro-development.
The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a ...continuous version of graphical models indexed by graphs with an embedded time structure-- so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.
The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation ...decay--a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behavior, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications. After a short historical overview, this book describes the current state of knowledge about selfsimilar processes and their applications. Concepts, definitions and basic properties are emphasized, giving the reader a road map of the realm of selfsimilarity that allows for further exploration. Such topics as noncentral limit theory, long-range dependence, and operator selfsimilarity are covered alongside statistical estimation, simulation, sample path properties, and stochastic differential equations driven by selfsimilar processes. Numerous references point the reader to current applications.
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8.
Tempering stable processes ROSINSKI, Jan
Stochastic processes and their applications,
06/2007, Volume:
117, Issue:
6
Journal Article
Peer reviewed
Open access
A tempered stable Lévy process combines both the
α
-stable and Gaussian trends. In a short time frame it is close to an
α
-stable process while in a long time frame it approximates a Brownian motion. ...In this paper we consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove short and long time behavior of tempered stable Lévy processes and investigate their absolute continuity with respect to the underlying
α
-stable processes. We find probabilistic representations of tempered stable processes which specifically show how such processes are obtained by cutting (tempering) jumps of stable processes. These representations exhibit
α
-stable and Gaussian tendencies in tempered stable processes and thus give probabilistic intuition for their study. Such representations can also be used for simulation. We also develop the corresponding representations for Ornstein–Uhlenbeck-type processes.
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Gaussian predictive process models for large spatial data sets Banerjee, Sudipto; Gelfand, Alan E.; Finley, Andrew O. ...
Journal of the Royal Statistical Society. Series B, Statistical methodology,
September 2008, Volume:
70, Issue:
4
Journal Article
Peer reviewed
Open access
With scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. Over the last decade, hierarchical ...models implemented through Markov chain Monte Carlo methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics. However, fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data are collected at frequent time points and spatiotemporal process models are used. With regard to this challenge, our contribution is to work with what we call predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower dimensional subspace, thereby reducing the computational burden. Hence, we achieve the flexibility to accommodate non-stationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large data sets. We discuss attractive theoretical properties of these predictive processes. We also provide a computational template encompassing these diverse settings. Finally, we illustrate the approach with simulated and real data sets.
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