Alphabeta MathRecurrence Transience and Hitting Times for Markov Chains
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✓ 19 results · all verified · 18 also independently AI-judged
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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
- 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
- Equivalent Forms of Completeness
- 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
- pi: the Equivalent Characterizations
- 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
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Real Gamma and Beta Functions
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Transition matrices and n-step probabilities
Lemma
Matrix Chapman–Kolmogorov equations
Definition
Accessibility, communication, and irreducibility
Lemma
Communication is an equivalence relation
Definition
Hitting, return, and visit times
Definition
Recurrent and transient states
Theorem
Renewal decomposition at successive returns
Theorem
Equivalent criteria for recurrence and transience
Definition
Green kernel of a transient chain
Lemma
Green-kernel resolvent identity
Theorem
Recurrence and transience are class properties
Corollary
Irreducible recurrence/transience dichotomy
Theorem
Hitting probability as minimal harmonic extension
Lemma
Geometric tail for hitting in a finite irreducible chain
Theorem
Bounded Dirichlet problem for hitting probabilities
Definition
Period of a state
Lemma
Period is constant on communicating classes
Definition
Aperiodic irreducible chain
Definition
Simple symmetric walk on the integer lattice
Corollary
One-dimensional simple symmetric walk is recurrent
Corollary
Two-dimensional simple symmetric walk is recurrent
Corollary
Higher-dimensional simple symmetric walks are transient
Definition
Nonnegative kernel action and finite drift
Theorem
First-step equations for nonnegative exit costs
Theorem
Superharmonic majorants bound exit costs
Corollary
Expected exit time solves the Poisson equation
Theorem