Alphabeta MathMarkov Kernels and Markov Chains
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
Measure Theory
- Lebesgue-Stieltjes Measures and Distribution Functions
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
Probability
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
Topology
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Countability Axioms and Cardinal Functions
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Time-homogeneous Markov chain with transition kernel
Lemma
Bounded-function form of the Markov property
Definition
Conditional independence given a sigma-algebra
Lemma
Conditional-independence equivalences and preservation
Lemma
Conditional-independence splice lemma
Definition
Initial distribution of a Markov chain
Definition
Iterated transition kernels
Theorem
Chapman-Kolmogorov equations
Theorem
Finite-dimensional laws of a Markov chain
Theorem
Ionescu-Tulcea construction of a Markov chain
Corollary
Canonical Markov chain on path space
Theorem
A Markov-chain law is determined by its initial law and kernel
Definition
Shift operator and future-coordinate sigma-algebra
Theorem
Markov property for bounded future path functionals
Theorem
The Markov property is past-future conditional independence
Theorem
Discrete strong Markov property
Corollary
The post-hitting chain restarts from the hit state
Definition
Killed and absorbed transition kernels
Lemma
Killed and absorbed kernels are probability kernels
Definition
Discrete generator of a countable-state transition matrix
Theorem
Countable-state martingale-problem characterization
Corollary
Bounded harmonic functions yield Markov-chain martingales
Sources
- Durrett, Probability: Theory and Examples, Sections 5.1-5.2
- Durrett, Probability: Theory and Examples, Section 5.1
- Aldous-Chewi probability notes, Lecture 9
- Varadhan, Probability Theory, Chapter 4
- Levin, Peres, Wilmer, Markov Chains and Mixing Times
- Shalizi, Building Infinite Processes from Finite-Dimensional Distributions, Theorem 33
- Durrett, Probability: Theory and Examples, Section 5.2
- Roch, Markov Chains: Martingale Methods, Note 24, Section 1
- Roch, Markov Chains: Martingale Methods, Theorem 24.2