Alphabeta MathDiscrete Time Martingales
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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
- Pointwise Extended-Real Function Products
- 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
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
Definition
Filtration and filtered probability space
Definition
Adapted and integrable stochastic process
Definition
Natural filtration of a process
Definition
Martingale submartingale and supermartingale
Lemma
Multistep martingale characterization
Lemma
Conditional expectation process is a martingale
Definition
Martingale difference sequence
Theorem
Martingales and martingale differences correspond
Lemma
Martingale differences are orthogonal in l2
Theorem
Convex functions of martingales are submartingales
Corollary
Absolute value and powers of a martingale are submartingales
Definition
Predictable discrete time process
Definition
Discrete martingale transform
Theorem
Bounded predictable transforms preserve martingales
Corollary
Nonnegative predictable transforms preserve submartingale gains
Definition
Compensator and doob decomposition
Theorem
Doob decomposition of an integrable adapted process
Corollary
Submartingale doob decomposition has increasing compensator
Definition
Predictable quadratic variation in discrete time
Theorem
Square minus predictable quadratic variation is a martingale
Corollary