Alphabeta MathMartingale Inequalities and Convergence — 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
- 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
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- 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 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
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- 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 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
2 · Logical flowchart
3 · Items
Example
Doob maximal bounds for a centered random walk
Example
A nonnegative martingale converges almost surely
Example
A dyadic martingale converges to the original L1 variable
Example
A reverse martingale and the tail sigma-algebra
Example
An Lp-bounded martingale with an Lp terminal value
Counterexample
An L1-bounded martingale need not converge in L1
Counterexample
Almost-sure martingale convergence need not preserve expectation
Counterexample
Doob Lp maximal inequality excludes p equals one
Example
Azuma bound for simple random walk
Sources
- van der Vaart, Martingales, Diffusions and Financial Mathematics, §2.9, pp. 22–24
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Exercise 2.22, p. 15
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Corollary 2.24, p. 16
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Example 2.28 and Exercise 2.31, pp. 17–18
- van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorems 2.23–2.25, pp. 15–16
- van der Vaart, Martingales, Diffusions and Financial Mathematics, martingale-convergence warnings in §§2.5 and 2.8
- Roch, Notes 20: Azuma's Inequality, Theorem 20.8, pp. 3–4