Alphabeta MathMartingale Inequalities and Convergence
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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
Functional Analysis
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- 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
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- 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 Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Probability
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Conditional Expectation
- Discrete Time Martingales
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Modes of Convergence for Random Variables
- Probability Spaces Random Variables and Expectation
- Weak Convergence Tightness and Representation
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
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- 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
- Mixed Partials, Taylor Formulae, and Extrema
- 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 Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- 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
Topology
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Upcrossing number of an interval
Lemma
Doob upcrossing inequality
Theorem
Doob submartingale convergence theorem
Theorem
Doob L1 maximal inequality
Theorem
Doob Lp maximal inequality
Theorem
Lp-bounded martingale convergence
Theorem
Uniformly integrable martingale convergence
Theorem
Closed martingale characterization
Definition
Reverse filtration and reverse martingale
Theorem
Reverse martingale convergence
Theorem
Levy upward convergence of conditional expectations
Theorem
Levy downward convergence of conditional expectations
Corollary
Kolmogorov zero-one law from martingale convergence
Lemma
Conditional Hoeffding bound for bounded martingale differences
Theorem
Azuma-Hoeffding inequality
Corollary
Symmetric bounded-increment Azuma bound
Definition
Square-integrable martingale-difference array and variance clock
Theorem
Martingale central limit theorem
Sources
- van der Vaart, Martingales, Diffusions and Financial Mathematics, §2.4, pp. 13–14
- Durrett, Probability: Theory and Examples, 5th ed., Theorem 4.2.10 (upcrossing inequality) with proof
- Durrett, Probability: Theory and Examples, 5th ed., Theorem 4.2.11 (martingale convergence theorem) with proof
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Lemma 2.43, p. 22
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 2.44 and proof, pp. 22–23
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 2.25 and §2.9, pp. 16, 23–24
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 2.23 and proof, pp. 15–16
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 2.23 and discussion, pp. 15–16
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Definition 2.27, p. 17
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 2.30 and Exercise 2.31, pp. 17–18
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Corollary 2.24 and proof, p. 16
- van der Vaart, Martingales, Diffusions and Financial Mathematics, §2.6, pp. 17–18
- Roch, Notes 20: Azuma's Inequality, Lemma 20.6 and Theorem 20.8 proof, pp. 2–3
- Roch, Notes 20: Azuma's Inequality, Theorem 20.8 and proof, pp. 3–4
- Roch, Notes 20: Azuma's Inequality, Theorem 20.8, pp. 3–4
- Roch, Notes 19: Martingale CLT, Theorem 19.15 setup, pp. 4–5
- Roch, Notes 19: Martingale CLT, Theorem 19.15 and proof, pp. 4–8