Simple Proofs of the Kolmogorov Extension Theorem and Prokhorov’s Theorem

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”

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

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.

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

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 without Recycling

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”

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

We 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 Theorem

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.

Eight Ways to Derive the Characteristic Function of the Normal Distribution

We 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.