Alphabeta MathStrong Laws of Large Numbers
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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
- Measure-Preserving Systems and Mixing Criteria
- 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
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
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
- 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
Strong law of large numbers for a sequence
Corollary
Kolmogorov strong law for independent uniformly bounded variances
Corollary
Iid finite variance strong law
Lemma
Tail sum integrability equivalence
Lemma
Iid linear truncation occurs only finitely often
Lemma
Summability of truncated normalized variances
Lemma
Cesaro limit of truncated means
Theorem
Kolmogorov iid l1 strong law
Theorem
Integrability is necessary for an iid finite mean strong law
Theorem
Etemadi strong law for pairwise independent iid variables
Corollary
Iid strong law implies the weak law
Lemma
The maximal ergodic inequality on a probability space
Theorem
Birkhoff's theorem for an ergodic probability system
Corollary
Birkhoff strong law for iid coordinate shifts
Remark
Strong law does not assert a rate
Theorem
Finite variance logarithmic rate for iid sums
Sources
- Durrett, §§2.4–2.5, pp. 76–87
- Roch, Note 5, Theorems 5.8–5.9, printed pp. 5–6 (mutual independence specialization only)
- Durrett, Theorem 2.3.8 and §2.4
- Durrett, Theorem 2.4.1, Lemmas 2.4.2–2.4.4 and complete proof, pp. 76–78
- Durrett, Probability: Theory and Examples, 5th ed., Lemma 6.2.2, printed p.335; complete proof read
- Durrett, Probability: Theory and Examples, 5th ed., Theorem 6.2.1 and its complete proof, printed pp.335–337; ergodic specialization proved without conditional expectation
- Durrett, Examples 6.1.4–6.1.5 pp.332–333; Theorem 6.2.1, Lemma 6.2.2 and Example 6.2.3, pp.335–337
- Durrett, Theorem 2.5.11, p. 87; Roch, Theorem 5.9, pp. 5–6