Prerequisites for LLN and CLT Failures Under Heavy Tails

Direct prerequisites (6)

1. Law of Large Numberslayer 0B, tier 1
2. Central Limit Theoremlayer 0B, tier 1
3. Characteristic Functionslayer 1, tier 1
4. Fat Tails and Heavy-Tailed Distributionslayer 2, tier 1
5. Common Probability Distributionslayer 0A, tier 1
6. Modes of Convergence of Random Variableslayer 0B, tier 1

Reachable through the chain (308)

These topics are not directly cited as prerequisites but are reached transitively by following the chain upward. Working through the direct prerequisites pulls these in.

• Random Variableslayer 0A, tier 1
• Kolmogorov Probability Axiomslayer 0A, tier 1
• Sets, Functions, and Relationslayer 0A, tier 1
• Basic Logic and Proof Techniqueslayer 0A, tier 2
• Expectation, Variance, Covariance, and Momentslayer 0A, tier 1
• Exponential Function Propertieslayer 0A, tier 1
• Integration and Change of Variableslayer 0A, tier 2
• Measure-Theoretic Probabilitylayer 0B, tier 1
• Cardinality and Countabilitylayer 0A, tier 2
• Zermelo-Fraenkel Set Theorylayer 0A, tier 2
• Joint, Marginal, and Conditional Distributionslayer 0A, tier 1
• Triangular Distributionlayer 0A, tier 2
• Borel-Cantelli Lemmaslayer 0B, tier 1
• Metric Spaces, Convergence, and Completenesslayer 0A, tier 1
• Moment Generating Functionslayer 0A, tier 2
• Convex Tinkeringlayer 2, tier 2
• Common Inequalitieslayer 0A, tier 1
• Non-Probability Samplinglayer 2, tier 1
• Double/Debiased Machine Learninglayer 3, tier 1
• Asymptotic Statistics: M-Estimators, Delta Method, LANlayer 0B, tier 1
• Maximum Likelihood Estimation: Theory, Information Identity, and Asymptotic Efficiencylayer 0B, tier 1
• Differentiation in Rⁿlayer 0A, tier 1
• Vectors, Matrices, and Linear Mapslayer 0A, tier 1
• Continuity in Rⁿlayer 0A, tier 1
• KL Divergencelayer 1, tier 1
• Information Theory Foundationslayer 0B, tier 2
• Distance Metrics Comparedlayer 1, tier 2
• Non-Euclidean and Hyperbolic Geometrylayer 1, tier 2
• Total Variation Distancelayer 1, tier 1
• Method of Momentslayer 0B, tier 2
• Radon-Nikodym and Conditional Expectationlayer 0B, tier 1
• Cramér-Rao Bound: Information Inequality, Achievability, and Sharper Variantslayer 0B, tier 1
• Fisher Information: Curvature, KL Geometry, and the Natural Gradientlayer 0B, tier 1
• Basu's Theoremlayer 0B, tier 3
• Sufficient Statistics and Exponential Familieslayer 0B, tier 2
• Positive Semidefinite Matriceslayer 0A, tier 1
• Eigenvalues and Eigenvectorslayer 0A, tier 1
• Matrix Operations and Propertieslayer 0A, tier 1
• Linear Independencelayer 0A, tier 1
• Inner Product Spaces and Orthogonalitylayer 0A, tier 1
• Matrix Normslayer 0A, tier 1
• Cramér-Wold Theoremlayer 1, tier 2
• Cross-Validation Theorylayer 2, tier 2
• Empirical Risk Minimizationlayer 2, tier 1
• Concentration Inequalitieslayer 1, tier 1
• Martingale Theorylayer 0B, tier 2
• Skewness, Kurtosis, and Higher Momentslayer 1, tier 1
• High-Dimensional Probability (Vershynin)layer 2, tier 1
• Loss Functions Cataloglayer 1, tier 1
• Logistic Regressionlayer 1, tier 1
• Convex Optimization Basicslayer 1, tier 1
• Dynamic Programminglayer 0A, tier 1
• Graph Algorithms Essentialslayer 0A, tier 2
• Greedy Algorithmslayer 0A, tier 2
• GraphSLAM and Factor Graphslayer 3, tier 2
• Inverse and Implicit Function Theoremlayer 0A, tier 2
• The Jacobian Matrixlayer 0A, tier 1
• Submodular Optimizationlayer 3, tier 3
• Taylor Expansionlayer 0A, tier 1
• The Hessian Matrixlayer 0A, tier 1
• Vector Calculus Chain Rulelayer 0A, tier 1
• Data Preprocessing and Feature Engineeringlayer 1, tier 1
• Linear Regressionlayer 1, tier 1
• The Elements of Statistical Learning (Hastie, Tibshirani, Friedman)layer 0B, tier 1
• Naive Bayeslayer 1, tier 2
• Robust Statistics and M-Estimatorslayer 3, tier 2
• Minimax and Saddle Pointslayer 2, tier 2
• Convex Dualitylayer 2, tier 1
• Subgradients and Subdifferentialslayer 1, tier 1
• Winsorizationlayer 1, tier 3
• Order Statisticslayer 1, tier 2
• Sequences and Series of Functionslayer 0A, tier 2
• Understanding Machine Learning (Shalev-Shwartz, Ben-David)layer 1, tier 1
• Bias-Variance Tradeofflayer 2, tier 2
• Elastic Netlayer 2, tier 2
• Ridge Regressionlayer 1, tier 1
• Shrinkage Estimation and the James-Stein Estimator: Inadmissibility, SURE, and Brown's Characterizationlayer 0B, tier 1
• Minimax Lower Bounds: Le Cam, Fano, Assouad, and the Reduction to Testinglayer 3, tier 1
• Empirical Processes and Chaininglayer 3, tier 2
• Rademacher Complexitylayer 3, tier 1
• VC Dimensionlayer 2, tier 1
• Counting and Combinatoricslayer 0A, tier 2
• Hypothesis Classes and Function Spaceslayer 2, tier 1
• PAC Learning Frameworklayer 1, tier 1
• Uniform Convergencelayer 2, tier 1
• Adaptive Learning Is Not IIDlayer 3, tier 2
• Bernstein Inequalitylayer 2, tier 1
• Bennett's Inequalitylayer 2, tier 1
• Chernoff Boundslayer 1, tier 1
• Hoeffding's Lemmalayer 1, tier 1
• Realizability Assumptionlayer 2, tier 1
• Loss Functionslayer 1, tier 2
• Slud's Inequalitylayer 2, tier 2
• Bias-Complexity Tradeofflayer 2, tier 2
• No-Free-Lunch Theoremlayer 2, tier 2
• Glivenko-Cantelli Theoremlayer 2, tier 2
• McDiarmid's Inequalitylayer 3, tier 1
• Sub-Gaussian Random Variableslayer 2, tier 1
• Epsilon-Nets and Covering Numberslayer 3, tier 1
• Contraction Inequalitylayer 3, tier 2
• Sub-Exponential Random Variableslayer 2, tier 1
• Chi-Squared Concentrationlayer 2, tier 1
• Symmetrization Inequalitylayer 3, tier 1
• Measure Concentration and Geometric Functional Analysislayer 3, tier 1
• Stochastic Processes for MLlayer 2, tier 2
• Gauss-Markov Theoremlayer 2, tier 1
• The Multivariate Normal Distributionlayer 0B, tier 1
• Maximum A Posteriori (MAP) Estimationlayer 0B, tier 1
• Bayesian Estimationlayer 0B, tier 2
• Lasso Regressionlayer 2, tier 1
• Generalized Additive Modelslayer 2, tier 2
• MARS (Multivariate Adaptive Regression Splines)layer 2, tier 3
• K-Nearest Neighborslayer 1, tier 2
• AIC and BIClayer 2, tier 1
• Decision Trees and Ensembleslayer 2, tier 2
• Kolmogorov Complexity and MDLlayer 2, tier 2
• Class Imbalance and Resamplinglayer 1, tier 2
• Confusion Matrices and Classification Metricslayer 1, tier 1
• Multi-Class and Multi-Label Classificationlayer 1, tier 2
• Signal Detection Theorylayer 2, tier 2
• Hypothesis Testing for MLlayer 2, tier 2
• Benford's Lawlayer 1, tier 2
• Confusion Matrix: MCC, Kappa, and Cost-Sensitive Evaluationlayer 1, tier 1
• Differential Privacylayer 3, tier 2
• Federated Learninglayer 3, tier 2
• Optimizer Theory: SGD, Adam, and Muonlayer 3, tier 1
• Adam Optimizerlayer 2, tier 1
• Gradient Descent Variantslayer 1, tier 1
• Stochastic Gradient Descent Convergencelayer 2, tier 1
• Coordinate Descentlayer 2, tier 2
• Mirror Descent and Frank-Wolfelayer 3, tier 2
• Online Convex Optimizationlayer 3, tier 2
• No-Regret Learninglayer 3, tier 2
• Projected Gradient Descentlayer 2, tier 2
• Proximal Gradient Methodslayer 2, tier 1
• Quasi-Newton Methodslayer 2, tier 1
• Newton's Methodlayer 1, tier 1
• Line Search Methodslayer 2, tier 2
• Secant Methodlayer 1, tier 2
• Automatic Differentiationlayer 1, tier 1
• Matrix Calculuslayer 1, tier 1
• Information Geometrylayer 3, tier 3
• Whitening and Decorrelationlayer 2, tier 2
• Principal Component Analysislayer 1, tier 1
• Singular Value Decompositionlayer 0A, tier 1
• Gram Matrices and Kernel Matriceslayer 1, tier 1
• Matrix Multiplication Algorithmslayer 1, tier 2
• The Kernel Tricklayer 2, tier 1
• Support Vector Machineslayer 2, tier 1
• Perceptronlayer 1, tier 2
• Bayesian Linear Regressionlayer 2, tier 1
• Conjugate Priorslayer 0B, tier 1
• High-Dimensional Covariance Estimationlayer 3, tier 2
• Matrix Concentrationlayer 3, tier 1
• NMF (Nonnegative Matrix Factorization)layer 2, tier 3
• Tensors and Tensor Operationslayer 0A, tier 1
• Pandas and NumPy Fundamentalslayer 4, tier 3
• Floating-Point Arithmeticlayer 0A, tier 1
• Preconditioned Optimizers: Shampoo, K-FAC, and Natural Gradientlayer 3, tier 2
• Conjugate Gradient Methodslayer 2, tier 2
• Numerical Linear Algebralayer 1, tier 2
• Riemannian Optimization and Manifold Constraintslayer 3, tier 2
• Equivariant Deep Learninglayer 4, tier 2
• Convolutional Neural Networkslayer 3, tier 2
• Feedforward Networks and Backpropagationlayer 2, tier 1
• Activation Functionslayer 1, tier 1
• Deep Learning (Goodfellow, Bengio, Courville)layer 0B, tier 1
• Gradient Boostinglayer 2, tier 1
• AdaBoostlayer 2, tier 2
• Cubist and Model Treeslayer 2, tier 3
• Fast Fourier Transformlayer 1, tier 2
• Complex Numbers for Fourierlayer 0A, tier 2
• Signals and Systems for MLlayer 1, tier 2
• Skip Connections and ResNetslayer 2, tier 1
• SVM for RF Classificationlayer 4, tier 3
• Graph Neural Networkslayer 3, tier 2
• Clustering for Gene Expressionlayer 4, tier 3
• K-Means Clusteringlayer 1, tier 1
• Self-Organizing Mapslayer 2, tier 3
• t-SNE and UMAPlayer 2, tier 2
• Spectral Clusteringlayer 2, tier 2
• PageRank Algorithmlayer 2, tier 2
• Attention for Protein Structure: AlphaFold and Successorslayer 4, tier 3
• Attention Mechanism Theorylayer 4, tier 2
• Softmax and Numerical Stabilitylayer 1, tier 1
• Linear Layer: Shapes, Bias, and Memorylayer 2, tier 1
• Word Embeddingslayer 2, tier 2
• Information Retrieval Foundationslayer 2, tier 1
• Transformer Architecturelayer 4, tier 2
• Attention Mechanisms Historylayer 3, tier 2
• Recurrent Neural Networkslayer 3, tier 2
• Macroeconomic Time-Series Forecastinglayer 4, tier 3
• Time Series Forecasting Basicslayer 2, tier 2
• Time Series Foundationslayer 2, tier 2
• Byte-Level Language Modelslayer 4, tier 3
• Tokenization and Information Theorylayer 4, tier 3
• Distributional Semanticslayer 2, tier 2
• NLP for Economic Text Analysislayer 4, tier 3
• Natural Language Processing Foundationslayer 2, tier 2
• RNNs for Signal Sequenceslayer 4, tier 3
• Token Prediction and Language Modelinglayer 3, tier 2
• Hyperbolic Embeddings for Graphslayer 2, tier 2
• Training Dynamics and Loss Landscapeslayer 4, tier 2
• Stability and Optimization Dynamicslayer 2, tier 2
• Evaluation Metrics and Propertieslayer 2, tier 2
• Neyman-Pearson and Hypothesis Testing Theorylayer 2, tier 2
• Reproducibility and Experimental Rigorlayer 2, tier 2
• Git and GitLab for ML Researchlayer 4, tier 3
• Python for ML Researchlayer 4, tier 3
• Weights and Biases for Experiment Trackinglayer 4, tier 3
• Survival Analysislayer 3, tier 2
• Feature Importance and Interpretabilitylayer 2, tier 2
• Exploratory Data Analysislayer 1, tier 2
• ML Project Lifecyclelayer 1, tier 2
• Hardware for ML Practitionerslayer 1, tier 2
• Train-Test Split and Data Leakagelayer 1, tier 1
• Mechanistic Interpretability: Features, Circuits, and Causal Faithfulnesslayer 4, tier 1
• Kolmogorov-Arnold Networks (KANs)layer 4, tier 2
• Universal Approximation Theoremlayer 2, tier 1
• Residual Stream and Transformer Internalslayer 4, tier 2
• Forgetting Transformer (FoX)layer 4, tier 2
• Attention Variants and Efficiencylayer 4, tier 2
• Sparse Attention and Long Contextlayer 4, tier 2
• Gemini and Google Modelslayer 5, tier 2
• Gradient Flow and Vanishing Gradientslayer 2, tier 1
• RLHF and Alignmentlayer 4, tier 2
• Policy Gradient Theoremlayer 3, tier 1
• Markov Decision Processeslayer 2, tier 1
• Bayesian State Estimationlayer 2, tier 2
• Gaussian Processes in Astronomylayer 4, tier 3
• Gaussian Processes for Machine Learninglayer 4, tier 3
• Kernels and Reproducing Kernel Hilbert Spaceslayer 3, tier 2
• Dimensionality Reduction Theorylayer 2, tier 2
• Functional Analysis Corelayer 0B, tier 2
• Hanson-Wright Inequalitylayer 3, tier 2
• Regularization Theorylayer 2, tier 2
• Overfitting and Underfittinglayer 2, tier 1
• XGBoostlayer 2, tier 2
• Gaussian Process Regressionlayer 3, tier 2
• Kernel Methods for Moleculeslayer 4, tier 3
• Kalman Filterlayer 2, tier 1
• No-U-Turn Sampler and Neal's Funnellayer 3, tier 2
• Hamiltonian Monte Carlolayer 3, tier 2
• Metropolis-Hastings Algorithmlayer 2, tier 1
• Markov Chain Monte Carlolayer 2, tier 1
• Markov Chains and Steady Statelayer 1, tier 2
• Monte Carlo Methodslayer 2, tier 1
• Gibbs Samplinglayer 2, tier 1
• Griddy Gibbs Samplinglayer 2, tier 3
• Variance Reduction Techniqueslayer 2, tier 2
• Importance Samplinglayer 2, tier 1
• Number Theory and Machine Learninglayer 4, tier 3
• Peano Axiomslayer 0A, tier 2
• Rejection Samplinglayer 1, tier 2
• Squeezed Rejection Samplinglayer 2, tier 3
• Burn-in and Convergence Diagnosticslayer 2, tier 2
• Coupling Arguments and Mixing Timelayer 3, tier 3
• MCMC for Markov Random Fieldslayer 3, tier 3
• Perfect Samplinglayer 3, tier 3
• Slice Samplinglayer 2, tier 3
• Multi-Armed Bandits Theorylayer 2, tier 2
• Bayesian Optimization for Hyperparameterslayer 3, tier 2
• Online Learning and Banditslayer 3, tier 2
• Test-Time Training and Adaptive Inferencelayer 5, tier 2
• Continuous Thought Machineslayer 5, tier 3
• Neural ODEs and Continuous-Depth Networkslayer 4, tier 3
• Classical ODEs: Existence, Stability, and Numerical Methodslayer 1, tier 1
• Equilibrium and Implicit-Layer Modelslayer 4, tier 2
• Implicit Differentiationlayer 2, tier 2
• Lyapunov-Based Machine Learning for Chaoslayer 4, tier 3
• Nonlinear Dynamics and Chaos Fundamentalslayer 4, tier 3
• Physics-Informed Neural Networkslayer 4, tier 2
• Divergence, Curl, and Line Integralslayer 0A, tier 2
• PDE Fundamentals for Machine Learninglayer 1, tier 2
• Stochastic Differential Equationslayer 3, tier 2
• Ito's Lemmalayer 3, tier 2
• Stochastic Calculus for MLlayer 3, tier 3
• Symbolic Regression and Equation Discoverylayer 4, tier 3
• Sparse Recovery and Compressed Sensinglayer 4, tier 3
• Q-Learninglayer 2, tier 1
• Value Iteration and Policy Iterationlayer 2, tier 1
• Bellman Equationslayer 2, tier 1
• Stochastic Approximation Theorylayer 2, tier 2
• Temporal Difference Learninglayer 2, tier 2
• Actor-Critic Methodslayer 3, tier 2
• Reward Systems and Reinforcement Learning Neurosciencelayer 4, tier 3
• Fine-Tuning and Adaptationlayer 3, tier 1
• Sparse Autoencoders for Interpretability: TopK, JumpReLU, Matryoshka, and Scalinglayer 4, tier 1
• Autoencoderslayer 2, tier 2
• Boltzmann Machines and Hopfield Networkslayer 2, tier 3
• EM Algorithm Variantslayer 3, tier 2
• The EM Algorithmlayer 2, tier 1
• Truth Directions and Linear Probeslayer 4, tier 2
• Model Evaluation Best Practiceslayer 1, tier 1
• Proper Scoring Ruleslayer 2, tier 2
• ROC Curve and AUClayer 2, tier 2
• Statistical Significance and Multiple Comparisonslayer 2, tier 2
• Causal Inference Basicslayer 3, tier 3
• Weighted Conformal Prediction Under Covariate Shiftlayer 3, tier 1
• Split Conformal Predictionlayer 2, tier 1
• Tabular Foundation Models as Bayesian Inference Engineslayer 3, tier 1
• Prompt Engineering and In-Context Learninglayer 5, tier 2
• Kelly Criterionlayer 2, tier 2
• Expected Utility Theorylayer 2, tier 2
• Bounded Rationalitylayer 2, tier 1
• Decision Theory Foundationslayer 2, tier 2
• Causal Inference and the Ladder of Causationlayer 3, tier 1
• Leverage Points in Complex Systemslayer 3, tier 2