Discrete Random Variables - Introduction
Introduction to discrete random variables, exploring their definition, properties, and applications in probability theory and statistics.
The Key Equation Behind Probability - Entropy, Cross-Entropy, and KL Divergence
Explore probability theory fundamentals, from surprise to entropy, cross-entropy, and KL divergence. Learn key concepts applicable to neuroscience and machine learning.
Classical and Free Zero Bias for Infinite Divisibility
Explore classical and free zero bias in infinite divisibility with Larry Goldstein, delving into probability theory and statistical concepts.
The Atypical Growth in Random Interfaces - KPZ Equation and Unusual Growth Patterns
Explore random interface growth phenomena and the KPZ equation, combining probability, PDEs, and integrable systems to understand unusual growth behaviors.
Sample Amplification: Increasing Dataset Size Even When Learning is Impossible
Explore sample amplification techniques to generate larger datasets from limited samples, even when learning the original distribution is impossible.
Mean and Location Estimation with Optimal Constants - UW Madison
Explore advanced statistical techniques for mean and location estimation, focusing on finite sample theories and optimal constants in high-dimensional regimes.
Spectral Stability Under Real Random Absolutely Continuous Perturbations
Explore spectral stability in random matrix perturbations, focusing on eigenvector condition numbers and minimum eigenvalue gaps. Gain insights into numerical linear algebra algorithms.
Generalized Kyle-Back Insider Trading Model with Dynamic Information
Explore a generalized Kyle-Back insider trading model with dynamic information, focusing on Markovization, stochastic boundary value problems, and forward-backward SDEs.
Probabilistic Parking Functions - Exploring Randomness in Combinatorics
Explore probabilistic parking functions, their properties, and connections to combinatorics. Examine statistical transitions and discuss future research directions.
Lozenge Tilings via the Dynamic Loop Equation
Explore discrete Markov chains with particle interactions, focusing on lozenge tilings and their asymptotic behavior using holomorphic observables.
Anticoncentration in Ramsey Graphs and Proof of the Erdös-McKay Conjecture - Lecture
Explore Ramsey graphs and the proof of the Erdös-McKay Conjecture, focusing on edge-statistics and anticoncentration in these unique graph structures.
Integral Varadhan Formula for Nonlinear Heat Flow on Measured Finsler Manifolds
Explore integral Varadhan formula for nonlinear heat flow on measured Finsler manifolds, examining particle probability and distance relationships in geometric analysis.
Probability Distribution Formulas
Explore formulas for binomial, geometric, Poisson, uniform, and exponential probability distributions. Master key concepts to solve statistical problems effectively.
Counting Limit Theorems for Representations of Gromov-Hyperbolic Groups
Explore limit theorems for Gromov-hyperbolic group representations, examining statistical results analogous to random matrix products theory for spherical averages in the S-metric.
A Non-symmetric Kesten Criterion for Random Walks on Groups
Explore a non-symmetric version of Kesten's criterion for random walks on countable groups, relating to amenability and probability of return to identity.
Around the Joint Behavior of Independent Random Tensor Matrices - Lecture 3
Explore advanced topics in free probability theory, focusing on joint behavior of independent random tensor matrices and symmetry groups, with applications in matrix analysis.
Around the Joint Behavior of Independent Random Tensor Matrices - Lecture 2
Explore advanced topics in free probability theory, focusing on joint behavior of independent random tensor matrices and symmetry groups, with applications in matrix analysis.
Functional Relations for Higher Order Free Cumulants and Moments
Explore functional relations between generating series of cumulants and moments in higher order free probability. Gain insights into matrix models and topological recursion.
Around the Joint Behavior of Independent Random Tensor Matrices - Lecture 1
Explore joint behavior of independent random tensor matrices with smaller symmetry groups. Learn about tensor structures, generalizations of free probability, and new analytical techniques.
Singularity of Stationary Measures in Markov Chains
Explore stationary measures of compound Markov chains, focusing on examples using boundary processes of random walks on the modular group PSL(2,Z) and related classical constructions.
Constrained Stochastic Processes: From Brownian Motion to Run-and-Tumble Particles
Explore constrained stochastic processes, including Brownian motion, Lévy flights, and run-and-tumble particles. Learn effective dynamics for generating constrained trajectories in various scenarios.