Probability: Standing Notation and Background

(i) Base. This setting extends Probability Spaces: Notation, Conditional Expectations: Notation, Convergence of Probability Measures and Random Elements: Standing Notation and Gaussian Random Vectors: Standing Notation: their conventions and notation are in force. Letters these settings use for different objects ($\mathcal{G}$, $T$, $Z$) take the meaning the statement using them gives. Background, relating the earlier and the current versions of the definitions the results below are stated with: The Earlier and the New Versions of the Definitions of Probability Spaces, Laws, Expectation, Independence and Modes of Convergence Agree, The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree, The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree.

(ii) Random elements and laws. Background: Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality, Properties of Distribution Functions, Expectations Computed from the Law, The Distance of Two Random Elements of a Separable Metric Space Is a Random Variable and The σ-Algebra Generated by Measurable Functions into Metric Spaces.

(iii) Independence. Background: Independence of σ-Algebras: π-Systems, Grouping and Pairs, Independent Random Elements: Preimages, Functions and Blocks, Independent Random Elements: Product Laws and Iterated Expectations, Expectations and Variances of Independent Random Variables and Borel-Cantelli Lemmas.

(iv) Moments. Background: Rules for Expectations, Variances and Covariances, Markov's and Chebyshev's Inequalities and Weak Law of Large Numbers.

(v) Densities and conditional expectation. Background: The Radon–Nikodym Theorem for a σ-Finite Measure, and Uniqueness of Densities, Existence and Uniqueness of Conditional Expectations, Rules for Conditional Expectations, Conditional Jensen Inequality, Lᵖ Contraction and Convergence Theorems for Conditional Expectations, Conditional Expectation as the Best Mean-Square Predictor and The Doob–Dynkin Factorization.

(vi) Modes of convergence. Background: Implications between Modes of Convergence of Random Elements, Convergence in Distribution to a Constant and Convergence in Probability, Convergence in Distribution of Real Random Variables through Distribution Functions, Smooth Test Functions Characterize Convergence in Distribution of Real Random Variables, Convergence in Distribution through Expectations of Bounded Continuous and of Bounded Lipschitz Functions and Slutsky's Theorem.

(vii) Weak convergence and tightness. Background: Lipschitz Test Functions Determine a Finite Borel Measure, and Uniqueness of Weak Limits, Portmanteau Theorem on a Metric Space, Continuous Mapping Theorem for Weak Convergence of Finite Borel Measures, Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight and Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence.

(viii) Gaussian vectors and the central limit theorem. Background: A Random Vector Is Gaussian with Mean m and Covariance Σ Exactly When Its Law Is N(m,Σ), Uncorrelated Jointly Gaussian Blocks are Independent, Independent Random Variables and Independent Standard Normal Coordinates on a Product Space, Lindeberg's Replacement Bound and The Central Limit Theorem for Independent Identically Distributed Square-Integrable Random Variables.