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  • Probability
    • Asymptotic (limsup) Growth ⇒ Global Bounds
    • Characteristic Functions
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    • martingales
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    • useful
    • MTH868
      • MTH 868 — Lecture 04
      • Lecture 5 — Quotients and Tangent Spaces
      • Lecture 6
      • MTH 868 — Lecture 07
      • MTH 868 – Lecture 1 Expanded Notes
      • Topology — Lecture 2
      • MTH 868 – Lecture 03 (Jan 16, 2026)
    • STT881
      • 1 — Cardinality, Cantor’s Theorem, Series, Measure Theory
      • 2 — Algebras, Measures, Length Measure, Continuity, σ Algebras
      • 3 — π systems, λ systems, π–λ Theorem, Outer Measure, Lebesgue Measure
      • 4 — Outer Measure, Measurable Sets, Cantor Set, Non Measurable Sets, σ Finite Measures, Integration
      • 5 — Cantor Set, Borel vs Lebesgue σ Algebras, and Integration of Positive Functions
      • 6 — Monotone Convergence, σ Finiteness, and Integrating General Measurable Functions
      • 7 — Integral Properties, Jensen, Hölder, and the $L^p$ Triangle Inequality
      • 8 — Hölder’s Inequality Refinements, $L^p$ Norms, Triangle Inequality, Bounded Convergence, Convergence in Measure
      • 9 — Fatou’s Lemma, MCT, DCT
      • 10 — MCT for Series, Fatou Applications, Change of Variables, Uniform Integrability
      • 11 — Uniform Integrability (UI)
      • 12 — Uniform Integrability, Product Measures, and Fubini
      • 13 — Kolmogorov Extension Theorem
      • 14 — Fubini’s Theorem and Applications
      • 15 Lebesgue Decomposition, Radon Nikodym, and Probability Measures
      • Lecture 16 — Gaussian Tail Bounds and Independence
      • Lecture 17 — Weak Law of Large Numbers & Bernstein Polynomials
      • Lecture 18 — Weak Law of Large Numbers & Triangular Arrays
      • Lecture 19 — Truncation Methods and Infinite Mean Laws of Large Numbers
      • Lecture 20 — Borel–Cantelli, Subsequence Convergence, DCT Applications
      • Lecture 21 — Borel–Cantelli II, SLLN via 4th Moment, and Extensions
      • Lecture 22 — Extended Borel–Cantelli II and Applications
      • Lecture 23 — Longest Runs & The Strong Law
      • Lecture 24 — Formal SLLN Proof via Truncation and Block Arguments
      • Lecture 25 — SLLN Completion, Weighted Triangular Arrays, Random Series
      • Lecture 26 — Kolmogorov’s 0–1 Law and Maximal Inequalities
      • Lecture 27 — A.s. Convergence via Variance Summability & Kolmogorov’s 3–Series Theorem
      • Lecture 28 — Kronecker’s Lemma and Applications
      • Lecture 29 — Feller’s Theorem and Completion of the M–Z Strong Law
      • Lecture 30 — Feller’s Theorem, Complex Inequalities, and Weak Convergence
      • Lecture 31 — Convergence in Distribution
      • Lecture 32 — Scheffé’s Theorem, Total Variation, and Portmanteau Properties
      • Lecture 33 — Uniformly Continuous Functions, Helly Selection, and Tightness
      • Lecture 34 — CLT Without Characteristic Functions
      • Lecture 35 — Lindeberg–Feller CLT and Characteristic Functions
      • Lecture 36 — Characteristic Functions, Their Properties, and Inversion
      • Lecture 37 — Continuity Theorem
      • Lecture 38 — Lindeberg–Feller CLT via Characteristic Functions and Truncation
      • Lecture 39 — Poisson Convergence & Domains of Attraction
      • Lecture 40 — Poisson Approximation in Total Variation
      • Lecture 41 — Poisson Convergence Under Dependence, Derangements, Empty Boxes, and Compound Poisson Limits
    • STT882
      • 1 Random Vectors, Multinomial, and Multivariate Normal
      • 2 Characteristic Functions, Independence, and Weak Convergence in $\mathbb{R}^d$
      • 3 — Multivariate CLT in $\mathbb{R}^d$
      • 4 — Stopping Times
      • 5 — Filtrations
      • 6 — Wald’s Equations and Hitting Times
      • 7 — H–S 0–1 Law, Applications, and Return Time Results
      • 8 Hewitt–Savage 0–1 Law, Symmetric Difference Basics, Conditioning
      • 9 — Conditional Expectation
      • 10 Regular Conditional Probability / Distribution
      • 11 — Doob’s Upcrossing Lemma and Martingale Convergence
      • 12 — Doob’s Upcrossing Lemma and Martingale Convergence
      • 13 — Upcrossing Lemma, Doob’s Inequality, and MGCT
      • 14 — Doob’s Decomposition and Uniqueness
      • 15 — Extended Borel–Cantelli II, Polya’s Urn, Martingales
      • 16 — Radon–Nikodym, Lebesgue Decomposition, and Kakutani
      • 17 — Optional Stopping, Doob Inequalities, Maximal Inequalities
      • 18 — Doob’s $L^p$ Inequality and Branching Processes
      • 19 — L² Martingales, Doob Decomposition, L² Inequality, and Kronecker Lemma
      • 20 — L² Martingales, BCII+, Polya Scheme, Extensions
      • 21 — Doob’s $L^p$ Inequality, BDG Inequality, and Uniform Integrability
      • 22 — Convergence of Conditional Expectations
      • 23 — Conditioning, UI vs DCT, Reverse Martingales
      • 24 — Backwards Martingales, Exchangeability, de Finetti
      • 25 — Optional Stopping (OST), Backwards Martingales, Simple Random Walk
      • 26 — Optional Stopping for Random Walks, Expected Hitting Times
      • 27 — Central Limit Theorem for Martingales
      • 28 — Brownian Motion (March 28)
      • 29 — Brownian Motion: Non Differentiability, Kolmogorov Continuity, and Lévy Modulus
      • 30 — Lévy Modulus of Continuity and Haar Basis Construction of Brownian Motion
      • 31 Brownian Motion Construction Using Hilbert Spaces
      • 32 — Markov Property of Brownian Motion
      • 33 – Brownian motion: Markov property, right–continuous filtration, time inversion, tail events
      • 34 — Strong Markov Property, Hitting Times, Stopping Times
      • 35 — Strong Markov Property and Reflection Principle
      • 36 — Scaling of Hitting Times, Stability, and Zero Set Structure
      • 37 — Martingales for Brownian Motion
      • 38 — Two Sided Hitting, Gambler’s Ruin for BM, and Exponential Martingales
      • 39 — Embedding Random Variables into Brownian Motion (Skorokhod Embedding) and CLT via Brownian Motion
    • STT996
      • High Dimensional Probability — Lecture 01
      • High Dimensional Probability — Lecture 2
      • STT 996 – High Dimensional Probability
      • STT 996 – High Dimensional Probability
      • STT 996 – High Dimensional Probability
      • STT 996 – High Dimensional Probability
      • STT 996 – High Dimensional Probability
      • STT 996 — High Dimensional Probability
      • STT 996 — High Dimensional Probability
      • Chapter 1
    • STT997
      • Gaussian Processes
      • STT 997 — Homework 1
      • Homework 05
      • STT 997 — Lecture 01
      • STT 997 — Lecture 02
      • STT 997 — Lecture 03
      • Lecture 06 — Efficient Leave One Out Prediction, Precision Matrices, and Simulation
    • convergence
      • Almost Sure Convergence (a.s.)
      • Convergence in Distribution
      • $L^p$ Convergence
      • Convergence in Probability
      • Weak Laws of Large Numbers
    • distributions
    • high dimensional
      • 2.6 Subgaussian Distributions
      • 2.8 Bernstein’s Inequality
      • 2.7.1 A More General View: Orlicz Spaces.
      • Chapter 3 Random Vectors in High Dimensions
    • prelim
      • 2018.Q1 — Gaussian Identities, Stein’s Lemma, and Moment Recursions”
      • 2018.Q2 — Gaussian Maxima, Extreme Value Limits, and Gumbel Convergence
      • 2018 Q3 – Bernoulli Binary Expansion, Almost Sure Convergence, and Characteristic Functions
      • 2018.Q4 — Integrals of Brownian Motion and Measurability
      • 2018.Q5 — Random Walks, Stopping Times, and Invariance Principle
      • 2018.Q6 — Martingale / Supermartingale Construction and Convergence
      • 2021.Q1: Conditional Expectation via Symmetry and Radon–Nikodym
      • 2021.Q2: Time Reversal and the Maximum of Brownian Motion
      • 2022 Q1 – Exponential Markov Bounds, MGFs, and Rademacher Tails
      • 2022 Q2 — Least Squares Projection, Orthogonality, and Conditional Expectation
      • 2022 Q4 — Upper Bounds for Brownian Motion Maxima
      • Prob F22 – Question 5 (Martingale Normalization via Quadratic Variation)
      • 2022.Q6: Random Walk Exit Times and Exponential Martingales
      • 2023 Q2 – Equivalences and Counterexamples in Modes of Convergence
      • 2023 Q3 – Heavy Tailed Symmetric Variables, Truncation, and Limit Laws
      • 2023 Q4 Brownian representation of a 3 point random variable
      • 2023 Q5 Optional stopping for a simple symmetric random walk
      • 2023 Q6 Kolmogorov type inequality via stopped square martingale
      • 2024.Q1 – Variance Asymptotics, Normalized Convergence, and Almost Sure Limits
      • 2024.Q2 – Heavy Tailed Sums, Truncation, and Almost Sure Convergence
      • 2024.Q3 – Stable Limits, Characteristic Functions, and Densities
      • 2024.Q4 – Lindeberg Condition, CLT, and Variance Asymptotics
      • 2024.Q5 – Wald’s Identity, Stopping Times, and Independence
      • 2025 Q1 – Characteristic Functions of Compound Poisson Sums
      • 2025 Q2 – Symmetric Variables, Conditional Expectation, and Quadratic Variation
      • 2025 Q3 – Characteristic Functions, Independence, and Poisson Limits in ℝ³
      • 2025 Q4 – Truncation, Tail Control, and a Strong Law with Variable Normalization
      • 2025 Q5 (Downcrossings): exam optimal solution skeleton
      • Review Sheet
    • tu
      • Point set Topology
  • Research
    • ai homework
      • discovery ready
      • INTAKE
    • concepts
      • Angular Opposition Interpolation (AOI)
      • Bayesian View of Activation Patterns in ReLU Networks
      • Conditional Covariance in ReLU Networks via Activation Fields
      • Covariance Tensors
      • Crop Lifecycle Temporal Modeling Notes
      • Ecological Currency and Productive Activity Backed Value
      • Matrix Model
      • Gradient Ownership
      • How I Survived My Probability Prelim
      • Intrinsic Geometry of Mathematics and Latent Structure under Embedding
      • Quantum Skip Gram Proof Solver (Lean/mathlib)
      • wind flow sim
    • intake
    • prob proof
      • A Learned Prior for Tactic Selection in Smooth Manifold Proofs
      • A Learned Prior for Tactic Selection in Smooth Manifold Proofs
  • Fiction
    • 24system
      • Training & Advancement
      • Armor
      • Atrocitas Sine Auctore
      • authors note
      • Character Creation (24 System)
      • combat tactics
      • Basic Combat Actions (RHLF)
      • 24 System: Success & Failure Levels
      • Size
      • 24 System Combat (Current Notes)
      • Damage Types
      • Defense Actions (24 System)
      • Equipment
      • Fear Track (24 System)
      • Initiative
      • Light Levels
      • Movement (Playtest Rules)
      • Narrative Based Perception
      • 24 System Combat Reference
      • 24 System Player Playtest Packet
      • 24 System Project State
      • Push Your Luck
      • quote
      • The RHLF (“Ralph”) Action System
      • Skills
      • Spellcasting Axioms
      • Status Effects
      • Weapons & Weapon Properties
      • Wounds, Trauma, and Disability (24 System)
      • Alhazred
        • Armor of Alhazred
        • Economics of Alhazred
        • Equipment of Alhazred
        • Rituals of Alhazred
      • monsters
        • The Echo Born Other
        • Monster Rules
        • Natural Creatures: Canines
        • Natural Creatures: Rats
        • Star Bound Enforcer
        • Abyssal Tentacle Horror
      • test plan
        • 01 Core Resolution Test
    • Oort Factor
    • president rock
      • Invitation to Collaborate
      • President Rock
      • episodes
        • Episode: The Silence of the Rock
        • Episode: The Birth of PIT
        • Episode: The Loophole
        • Episode: The Loophole
        • Episode: The Bloodiest Day
        • Episode: The Day the Rock Moved
    • space voxel
      • Procedural Modeling & Mechanical Articulation – Concept Document (v0)
      • Unified Build UI Philosophy
      • Missile Design Philosophy
      • Galaxy Physics Backbone Architecture
      • Nozzle Generator – Parameter Schema (v0.1)
      • Project State – SpaceVoxel Alpha Slice
      • Hierarchical Resource Graph & Subgraph Model
      • Manager AI – Soft Thresholding & Impact Preview Specification
      • Tier 1 Logistics: Autonomous Transport and Player Engagement
    • whisper
      • Character Decisions Snapshot
      • Historical Lead Up to the Present Day
      • Whisper — Core Plot & Philosophy
      • Whisper — Revised High Level Plotline (with Moral Externalization)
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