Articles

Probability

Eight Ways to Derive the Characteristic Function of the Normal Distribution

chf of the normal.pdf

A Simple Probabilistic Proof of a Wallis-type Formula for the Gamma Function

Wallis via Gamma.pdf

The Strong Law of Large Numbers without Recycling

SLLN with no recycling.pdf

A Relatively Simple and Elementary Proof of the Baum—Katz Theorem

Baum-Katz.pdf

A Rigorous Treatment of a Fundamental Result in Self-similar Traffic Modeling

self-similar traffic modeling.pdf

An Elementary Exposition on the Proof of the Hartman—Wintner Law of Iterated Logarithm

LIL.pdf

Exchangeability and Moment Problems

exch_moments.pdf

Real Analysis

An Intuitive Proof of the Hahn—Kolmogorov Theorem

hahn-kolmogorov.pdf

A Proof of Hölder's Inequality using the Layer Cake Representation

Hölder using layer cake.pdf

Proving Integral Convergence Theorems by Viewing Integrals as Areas

integral_area.pdf

Comparison of the Powers of Convergence Theorems for Integrals

Convergence Theorems.pdf

Set Theory

Borel's Proof of the Heine—Borel Theorem

Borel's proof of Heine-Borel.pdf

Perturbing All Points in R^d into a General Position

perturb-general.pdf

A Note on the Axiom of Choice

AC.pdf

Topology

Characterization of Continuous Maps in terms of Preservation of Connected Sets

Continuity and connectedness.pdf

An Application of Complete Metric Space Theory: the Picard—Lindelöf Theorem

picard-lindelof.pdf

Filters in General Topology

filters.pdf

Countable Products of Finite Spaces

countable-products-of-finite-spaces.pdf

An Application of Ascoli's Theorem: the Cauchy—Peano Existence Theorem

cauchy-peano.pdf

Algebra

A Short and Self-contained Proof of the Insolvability of the Quintic by Radicals

quintic v3.pdf

An Overkill Proof of Wilson's Theorem using a Sylow Theorem

wilson by sylow.pdf

A Linear-algebraic Overkill Proof that Finite Fields are not Algebraically Closed

finite fields not closed.pdf

LaTeX

I gave a 3-hour introductory lecture on LaTeX for KAIST students in Korean on May 16th, 2021. Click here for the video recording of the lecture. Please download intro_to_latex_210516 before attending the lecture. You can find answers to the exercises in after_lecture.