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E-grāmata: Time-Like Graphical Models

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Time-Like Graphical ModelsPalielināt
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"We study 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 anembedded time structure - so called time-like graphs. We extend the notion of time-like graphs and find properties of processes indexed by them. In particular, we solve the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. We provide a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, our treatment of time-like graphical models reveals connections to Markovrandom fields, martingales indexed by directed sets and branching Markov processes"--

In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure-time-like graphs-says Tadic, and he extends that notion and finds properties of processes indexed by them. In particular, he solves the conjecture of uniqueness of the distribution for the process indexed by graphs with an infinite number of vertices. He also provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. Annotation ©2020 Ringgold, Inc., Portland, OR (protoview.com)
Introduction 1(4)
Construction and properties
3(1)
Natural Brownian motion and the stochastic heat equation
3(1)
Processes on general and random time-like graphs
3(1)
Open questions and appendix
4(1)
Part 1 Construction and properties
5(72)
Chapter 1 Geometry of time-like graphs
7(24)
1.1 Basic definitions
7(2)
1.2 TLG* family
9(4)
1.3 Consistent representation of a TLG*-tower, spines and (re)construction
13(3)
1.4 Interval TLG*'s
16(2)
1.5 Topology on TLG's
18(2)
1.6 TLG* as a topological lattice
20(2)
1.7 Cell collapse transformation and the stingy algorithm
22(6)
1.8 TLG's with infinitely many vertices
28(3)
Chapter 2 Processes indexed by time-like graphs
31(18)
2.1 Spine-Markovian property
32(2)
2.2 Consistent distributions on paths
34(1)
2.3 Construction from a consistent family
35(9)
2.4 Processes on TLG's with infinite number of vertices
44(5)
Chapter 3 Markov properties of processes indexed by TLG's
49(14)
3.1 Cell-Markov properties
49(2)
3.2 Graph-Markovian and time-Markovian property
51(1)
3.3 Processes on TLG's for Markov family M
51(6)
3.4 Homogeneous Markov family M-p
57(4)
3.5 Three simple examples
61(2)
Chapter 4 Filtrations, martingales and stopping times
63(14)
4.1 Expanding the filtrations
63(4)
4.2 Markov martingales
67(4)
4.3 Optional sampling theorem for martingales indexed by directed sets
71(2)
4.4 TLG - valued stopping times
73(2)
4.5 A simple coupling and branching process
75(2)
Part 2 Natural Brownian motion and the stochastic heat equation
77(42)
Chapter 5 Maximums of Gaussian processes
81(6)
5.1 Sequence of Brownian bridges
81(2)
5.2 Sequence of normal variables
83(1)
5.3 Some concentration and convergence results
84(3)
Chapter 6 Random walk and stochastic heat equation reviewed
87(24)
6.1 Modification of the Local Limit Theorem
87(3)
6.2 Approximations of the classical heat equation solution
90(7)
6.3 Euler method for the stochastic heat equation
97(8)
6.4 Convergence of interpolation of the Euler method
105(3)
6.5 Euler method with initial value condition and no external noise
108(3)
Chapter 7 Limit of the natural Brownian motion on a rhombus grid
111(8)
7.1 Natural Brownian motion on a rhombus grid
111(4)
7.2 Network of Brownian bridges
115(1)
7.3 The main result
115(4)
Part 3 Processes on general and random time-like graphs
119(36)
Chapter 8 Non-simple TLG's
121(8)
8.1 New definitions
121(1)
8.2 Embedding TLG's into simple TLG's
122(2)
8.3 TLG** family
124(5)
Chapter 9 Processes on non-simple TLG's
129(12)
9.1 Processes on TLG**
129(4)
9.2 Properties of constructed processes
133(3)
9.3 Properties for Markov family M
136(1)
9.4 Processes on time-like trees
137(4)
Chapter 10 Galton-Watson time-like trees and the Branching Markov processes
141(10)
10.1 TLG's with an infinite number of vertices
141(1)
10.2 Galton - Watson time-like tree
142(1)
10.3 Processes on TLG's with infinite number of vertices
143(1)
10.4 Natural P-Markov process
144(1)
10.5 Branching P-Markov process
144(5)
Open questions and appendix
149(2)
Chapter 11 Open questions
151(4)
11.1 Construction of process on all TLG's
151(1)
11.2 Reconstruction of TLG's based on the process
152(2)
11.3 Strong Markov property, parametrization, evolution over time,...
154(1)
Appendix A Independence and processes
155(10)
A.1 Conditional independence and expectations
155(1)
A.2 Construction of a conditional sequence
155(1)
A.3 Markov and Brownian bridges
156(3)
A.4 Markov random fields
159(1)
A.5 White noise
160(1)
A.6 The stochastic heat equation
161(2)
A.7 Crump - Mode - Jagers trees
163(1)
A.8 Branching Markov processes and branching Brownian motion
164(1)
Acknowledgments 165(2)
Bibliography 167(2)
Index 169
Tvrtko Tadic, University of Washington, Seattle, Washington, and University of Zagreb, Croatia.
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