- Characterization of Continuous Maps in Terms of Preservation of Connected SetsWe 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”
- Simple Proofs of the Kolmogorov Extension Theorem and Prokhorov’s TheoremWe 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”
- Comparison of the Powers of Convergence Theorems for IntegralsWe 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.
- A Relatively Simple and Elementary Proof of the Baum-Katz TheoremWe 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.
- A Rigorous Treatment of a Fundamental Result in Self-Similar Traffic ModelingThis 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 without RecyclingThe 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”
- A Simpler Probabilistic Proof of a Wallis-type Formula for the Gamma FunctionWe simplify our probabilistic proof of a Wallis-type formula for the Gamma function by using Gamma distributions instead of normal distributions.
- An Intuitive Proof of the Hahn-Kolmogorov TheoremThe 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.
- An Overkill Proof of Wilson’s Theorem Using a Sylow TheoremWe prove Wilson’s theorem, which is an elementary result in number theory, using one of the Sylow theorems.
- A Linear-Algebraic Overkill Proof that Finite Fields are not Algebraically ClosedWe prove the simple fact that no finite field is algebraically closed using rational and Jordan canonical forms.
- Perturbing All Points in R^d into a General PositionWe prove that there is a way to move each point in less than a prescribed amount, so that no two points are moved to the same point, and the resulting set of points are in a general position (i.e. no three points are in a line, no four points are in a plane, andContinue reading “Perturbing All Points in R^d into a General Position”
- Eight Ways to Derive the Characteristic Function of the Normal DistributionWe present eight proofs of where is a standard normal random variable. This fact is more or less equivalent to saying that the Gaussian is its own Fourier transform. We start with proofs that are rather elementary, and move on to more sophisticated proofs.
- A Proof of Hölder’s Inequality Using the Layer Cake RepresentationWe prove Hölder’s inequality using the so-called layer cake representation and the tensor power trick. The heart of the proof is the following one line:
- Borel’s Proof of the Heine-Borel TheoremWe present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.