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 and decompress the following file before attending the lecture. Click below for the answers to the exercises.
We prove that if X is locally connected and Y is locally compact Hausdorff, then f: X -> Y is continuous if and only if f has a closed graph and f(C) is connected for any connected subset C of X. This result is not new. We show by examples that the conditions on XContinue reading “Characterization of Continuous Maps in Terms of Preservation of Connected Sets”
We provide simple proofs of the Kolmogorov extension theorem and Prokhorov’s theorem. The proof of the Kolmogorov extension theorem is based on the simple observation that and the product measurable space are Borel isomorphic. To show Prokhorov’s theorem, we observe that we can assume that the underlying space is . Then the proof of Prokhorov’sContinue reading “Simple Proofs of the Kolmogorov Extension Theorem and Prokhorov’s Theorem”
We show that many integral convergence theorems are equivalent to the dominated convergence theorem, and that they are strictly weaker than the monotone convergence theorem. The precise sense of these assertions are given.
We provide a relatively simple proof of the generalization of the Hsu-Robbins-Erdős theorem by Baum and Katz (1965). Our proof avoids using some advanced inequalities, and is essentially based on the idea used to prove the strong law of large numbers under finite fourth moment condition.
This note provides a rigorous treatment of the setting and the proof of the main theorem of Taqqu, Willinger, and Sherman (1997) on self-similar traffic modeling.
The strong law of large numbers heavily depends on the fact that each of appears repeatedly in the sequence . We prove a version of the strong law of large number in which no random variable gets “recycled.” More precisely, we prove the Hsu-Robbins-Erdős theorem, which states that for i.i.d. , we have if andContinue reading “The Strong Law of Large Numbers without Recycling”
We simplify our probabilistic proof of a Wallis-type formula for the Gamma function by using Gamma distributions instead of normal distributions.
The Hahn-Kolmogorov theorem (which is sometimes called the Carathéodory extension theorem) is used to construct various kind of measures. We provide an intuitive proof of this theorem which avoids using Carathéodory’s (rather) unintuitive criterion for measurability.
We prove Wilson’s theorem, which is an elementary result in number theory, using one of the Sylow theorems.