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.
We prove the simple fact that no finite field is algebraically closed using rational and Jordan canonical forms.
We 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”