Notes on probability spaces, random variables, expectation, conditional expectation, independence, convergence, laws of large numbers, and central limit theorems.
Probability Theory
This section collects foundational notes in probability theory, with emphasis on concepts that recur in asymptotic statistics, stochastic modelling, machine learning theory, and decision-making under uncertainty.
A consolidated set of notes on the foundations of probability theory, including probability spaces, random variables, distributions, expectation, conditional probability, convergence, laws of large numbers, and central limit theorems.
A focused note on almost sure convergence, convergence in probability, convergence in distribution, convergence in mean, and the logical relationships among these modes of convergence.
Notes on sample spaces, sigma-algebras, probability measures, measurable functions, induced distributions, and distribution functions.
Notes on expectation as integration, variance, covariance, moment inequalities, and basic expectation identities.
A conceptual note on conditional expectation, sigma-algebras, measurability, tower property, and the Hilbert-space projection interpretation.
A note comparing weak and strong laws of large numbers, with examples showing why averaging stabilizes random quantities.
A note on classical central limit theorems, asymptotic normality, normalization, and their role in statistical inference.