Alphabeta MathWeak Convergence Tightness and Representation
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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
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- 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
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Probability
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
- Countability and Uncountability
- Equivalent Forms of Completeness
- 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
- 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 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
Borel probability law on a polish space
Definition
Weak convergence of borel probability measures
Definition
Convergence in distribution of random elements
Theorem
Portmanteau theorem
Lemma
Real cdf and bounded continuous definitions agree
Corollary
Weak limits are unique
Theorem
Continuous mapping theorem
Theorem
Converging together lemma
Definition
Tight family of probability measures
Definition
Relative sequential compactness for weak convergence
Theorem
Every borel probability on a polish space is tight
Lemma
Countable uniformly dense tests on a compact metric space
Lemma
Probability laws on a compact metric space have weakly convergent subsequences
Theorem
Prokhorov tightness theorem on polish spaces
Corollary
Weakly convergent sequences are tight
Corollary
Tightness extracts a weakly convergent subsequence
Definition
Levy prokhorov metric
Lemma
Levy prokhorov distance is a metric
Lemma
Countable boundary null partitions of a separable metric space
Theorem
Levy prokhorov metric metrizes weak convergence
Lemma
Interval realization from refining small diameter partitions
Theorem
Skorokhod representation on polish spaces
Remark
Skorokhod representation does not couple the original variables
Lemma
Countable compactly supported tests determine euclidean weak convergence
Theorem
Empirical measures of iid euclidean samples converge weakly
Lemma
The standard normal density has total mass one
Definition
Standard normal and normal laws
Sources
- van Gaans, §2, p. 3
- van Gaans, Definition 3.1, pp. 6–7
- van Gaans, Definition 3.1 applied to laws
- van Gaans, Theorem 3.2, pp. 7–9
- Durrett, Theorem 3.2.9, pp. 119–120
- van Gaans, Theorem 4.1, definiteness argument, pp. 9–10
- van Gaans, Theorem 3.2; local closed-preimage argument
- van Gaans, Theorem 3.2 and closed-neighborhood argument
- van Gaans, Definition 5.1, p. 14
- van Gaans, Theorem 5.2, p. 14 (closure of the family)
- van Gaans, Theorem 2.6, pp. 5–6
- van Gaans, Proposition 5.3, pp. 15–16; explicit countable-test replacement for its Alaoglu step
- van Gaans, Proposition 5.3 and §6; diagonal replacement for compact-case functional argument
- van Gaans, Theorem 5.2, Proposition 5.3, Lemma 5.4, pp. 14–18
- Durrett, §§2.4–2.5, pp. 76–87
- van Gaans, §4, pp. 9–10 (open-enlargement convention; equivalence must be proved locally)
- van Gaans, Theorem 4.1, pp. 9–10
- van Gaans, Lemma 4.3, pp. 10–11; refining-partition consequence
- van Gaans, Theorem 4.2 and Lemma 4.3, pp. 10–12
- Advanced Probability, Theorem 5.29, pp. 65–67; representative-limit repair of the nonclosed-atom intersection step
- Advanced Probability, Theorem 5.29, pp. 65–67, repaired as in the preceding lemma
- Advanced Probability, Theorem 5.29
- van Gaans, compact-test approximation in Proposition 5.3 and tightness transfer; explicit Euclidean construction
- Durrett, strong law Theorem 2.5.10 plus countable-test argument
- Durrett, Probability: Theory and Examples, Example 1.6.11, p.34; local normalization from the earlier published Gaussian integral