Time-like graphical models /

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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Tadić, Tvrtko (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Providence, RI : American Mathematical Society, [2019]
Colección:Memoirs of the American Mathematical Society ; no. 1262.
Temas:
Distribution (Probability theory)
Distribution (Théorie des probabilités)
distribution (statistics-related concept)
Distribución (Teoría de probabilidades)
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title page
  • Introduction
  • Construction and properties
  • Natural Brownian motion and the stochastic heat equation
  • Processes on general and random time-like graphs
  • Open questions and appendix
  • Part 1 . Construction and properties
  • Chapter 1. Geometry of time-like graphs
  • 1.1. Basic definitions
  • 1.2. TLG* family
  • 1.3. Consistent representation of a TLG*-tower, spines and (re)construction
  • 1.4. Interval TLG*'s
  • 1.5. Topology on TLG's
  • 1.6. TLG* as a topological lattice
  • 1.7. Cell collapse transformation and the stingy algorithm
  • 1.8. TLG's with infinitely many vertices
  • Chapter 2. Processes indexed by time-like graphs
  • 2.1. Spine-Markovian property
  • 2.2. Consistent distributions on paths
  • 2.3. Construction from a consistent family
  • 2.4. Processes on TLG's with infinite number of vertices
  • Chapter 3. Markov properties of processes indexed by TLG's
  • 3.1. Cell-Markov properties
  • 3.2. Graph-Markovian and time-Markovian property
  • 3.3. Processes on TLG's for Markov family \cM
  • 3.4. Homogeneous Markov family \cM_{\cP}
  • 3.5. Three simple examples
  • Chapter 4. Filtrations, martingales and stopping times
  • 4.1. Expanding the filtrations
  • 4.2. Markov martingales
  • 4.3. Optional sampling theorem for martingales indexed by directed sets
  • 4.4. TLG
  • valued stopping times
  • 4.5. A simple coupling and branching process
  • Part 2 . Natural Brownian motion and the stochastic heat equation
  • Chapter 5. Maximums of Gaussian processes
  • 5.1. Sequence of Brownian bridges
  • 5.2. Sequence of normal variables
  • 5.3. Some concentration and convergence results
  • Chapter 6. Random walk and stochastic heat equation reviewed
  • 6.1. Modification of the Local Limit Theorem
  • 6.2. Approximations of the classical heat equation solution
  • 6.3. Euler method for the stochastic heat equation
  • 6.4. Convergence of interpolation of the Euler method
  • 6.5. Euler method with initial value condition and no external noise
  • Chapter 7. Limit of the natural Brownian motion on a rhombus grid
  • 7.1. Natural Brownian motion on a rhombus grid
  • 7.2. Network of Brownian bridges
  • 7.3. The main result
  • Part 3 . Processes on general and random time-like graphs
  • Chapter 8. Non-simple TLG's
  • 8.1. New definitions
  • 8.2. Embedding TLG's into simple TLG's
  • 8.3. TLG** family
  • Chapter 9. Processes on non-simple TLG's
  • 9.1. Processes on TLG**
  • 9.2. Properties of constructed processes
  • 9.3. Properties for Markov family \cM
  • 9.4. Processes on time-like trees
  • Chapter 10. Galton-Watson time-like trees and the Branching Markov processes
  • 10.1. TLG's with an infinite number of vertices
  • 10.2. Galton -Watson time-like tree
  • 10.3. Processes on TLG**'s with infinite number of vertices
  • 10.4. Natural \cP-Markov process
  • 10.5. Branching \cP-Markov process
  • Open questions and appendix
  • Chapter 11. Open questions
  • 11.1. Construction of process on all TLG's

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