Alphabeta MathWeak Laws and Series of Independent Random Variables
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Counting, Factorials and Binomial Coefficients
- Finite Probability Spaces and Random Variables
- Foundations of the Real Numbers for Analysis
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Independence Borel Cantelli and Zero One Laws
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Modes of Convergence Egorov and Lusin
- Modes of Convergence for Random Variables
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Probability Spaces Random Variables and Expectation
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Riemann Integral: Definition and Integrability
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Identical distribution and IID families
Definition
Partial sums, row sums and sample means
Theorem
Chebyshev weak law for uncorrelated arrays
Corollary
IID finite-variance weak law
Definition
Zero truncation at a positive level
Theorem
Khinchin weak law for integrable IID variables
Definition
Almost-sure convergence of a random series
Theorem
Kolmogorov maximal inequality
Theorem
Kolmogorov convergence criterion
Corollary
Kolmogorov two-series sufficiency
Definition
Symmetric real random variables
Lemma
Independent-copy symmetrization of random series
Lemma
Bounded centered convergent series have summable variances
Lemma
Necessity of the truncated mean and variance conditions
Theorem
Kolmogorov three-series theorem
Lemma
Kronecker summation lemma
Theorem
Strong law under summable normalized variances
Corollary
Strong law for independent nonidentical variables
Lemma
One-sided maximal inequality for symmetric independent sums
Lemma
Tail comparisons under independent-copy symmetrization
Theorem
Truncation weak law for independent arrays
Lemma
Vanishing tail control bounds truncated second moments
Lemma
Largest-summand bound for independent symmetric variables
Theorem
Exact tail criterion for a truncated-centered IID weak law
Lemma
Levy maximal bound from uniform tail bounds
Lemma
Cauchy sequences in probability have a measurable limit
Theorem
Convergence in probability and almost surely agree for independent series
Sources
- Section 2.2.1, p. 58, IID paragraph
- Section 2.2.2, p. 59, opening row-sum notation
- Theorems 2.2.1, 2.2.3 and 2.2.6, pp. 56–59
- Section 2.2.1, p. 58
- Theorem 3.2, pp. 55–56
- Section 2.2.3, p. 62
- Theorem 3.3, first proof, pp. 56–57
- Theorem 2.2.14, pp. 64–65
- Section 2.5, Example 2.5.2 p. 81 and series convention p. 84
- Section 3.4 opening, p. 61
- Theorem 2.5.5, p. 84
- Lemma 3.7, p. 62
- Theorem 2.5.6, pp. 84–85
- Theorem 3.10, pp. 65–66; L2 strengthening is proved locally by Fatou
- Theorem 3.11, p. 66
- Appendix A, Definition 4.16, p. 9
- Theorem 3.12 necessity, p. 67
- Appendix A, Example 4.17, p. 9
- Lemma 3.13 and complete proof, pp. 67–68
- Theorem 3.12 necessity and Lemma 3.13, pp. 67–68
- Theorem 2.5.8, p. 85
- Theorem 3.12, pp. 66–68
- Theorem 2.5.9 and full proof, pp. 85–86
- Theorems 2.5.6 and 2.5.9; proof of Theorem 2.5.11, pp. 84–87
- Theorems 2.5.6 and 2.5.9, pp. 84–86
- Section 2.2, Lemma 5.13 and proof, p. 8
- Appendix A, Lemma 4.18 and proof, pp. 9–10; symmetric-interval variant
- Theorem 2.2.11, pp. 62–63; second-moment sufficient form
- Section 2.1, Theorem 4.8 and proof, p. 4; variance form
- Theorem 2.2.12 proof with Lemma 2.2.13, pp. 63–64
- Roch Note 4, Appendix A, Lemma 4.19 and proof, p. 10
- Theorem 4.4, pp. 2–5 and Appendix A, pp. 9–11
- Theorem 2.2.12 and necessity remark, pp. 63–64
- Lemma 5.13, p. 8
- Lemma 3.8 and proof, pp. 62–63
- Varadhan, Chapter 3, Exercise 3.11, p. 65
- Theorem 3.9, implication (ii) to (iii), pp. 63–65