Alphabeta MathMarkov Kernels and Markov Chains — Examples
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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
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
Combinatorics
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
Probability
- Conditional Distributions and Regular Conditional Probability
- Conditional Expectation
- Discrete Time Martingales
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Markov Kernels and Markov Chains
- Probability Spaces Random Variables and Expectation
- Stopping Times and Optional Stopping
- Weak Convergence Tightness and Representation
- Weak Laws and Series of Independent Random Variables
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- 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
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- 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
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
IID sequences as Markov chains with state-independent kernel
Example
A deterministic dynamical system as a Markov kernel
Example
Simple random-walk transition kernel
Example
Absorbing gambler's-ruin chain
Example
Gaussian AR(1) chain
Example
Random-mapping representation for a finite transition matrix
Counterexample
Identical one-time marginals do not determine a Markov chain
Counterexample
The Markov property can fail for a larger filtration
Counterexample