High Risk Central Limit

High Risk Central Limit

While teaching probability this semester, I had naturally to think again at the central limit theorem and took this as an opportunity to warm up a bit something I abandoned in 2022 after learning that what I was following has been understood already by mathematicians like Levy or Gnedenko. Still, my approach to the high risk central limit theorem is fresh and looks at a handy class of random variables which are in the domain of attraction of the Cauchy distribution, a so called stable law. I looked at the class \mathcal{C} of random variables for which the mean exists in a Cauchy sense and for which the scaled truncated variance has a limit as an average. This is the property of the Cauchy distribution with PDF (\pi (1+x^2))^{-1}.

I wrote this presentation in Switzerland. I had to go there as my father Marcus Knill has passed. A short obituary has been written by Matthias Ackeret in the Schaffhauser Nachrichten, the local newspaper in Schaffhausen. The initial end end scenes in the movie below were recorded from my parents house. The rheinfalls are very close. At the end, I took my camera to a spin in the direction of Ellikon at the rhein, passing over the Rheinau abbey which was rounded around 800. The area has a lots of history. The camera also passed close to the celtic settlement between Altenburg and Rheinau. The Romans were here like at the Oppidum in Altenburg-Rheinau. Juliuas Cesar had been fighting there in the first century AD. A fun area for history. Keltic walls, Roman settlements, relicts from the Napoleon wars. The bridge in Feuerthalen (which I had to pass daily 4 times by bike in high school ) had been burned down by Napoleon in 1899. In the forest, I liked to run (and also ran again last week) there is a lake called “Russian Lake” because Russian armies watering his horses there during the Swiss campaign. The Napoleonic wars also still have visible traces in the area (a cannon ball is still stuck in the church). Back to the drone run over Rheinau: one crosses the river several times. A bit after passing the abbey, one can spot sees the Rheinau psychiatric university clinic. One sees from above the “closed section”, a building in which the court yards are crossed by wired tunnels. By the way, this whole area is also great for hiking, biking and running. I would already as a 12 year old bike through this area (rheinau was popular). There was once a school excursion (Schulreise) by foot to Eglisau which is further than what the drone has reached there. While in Switzerland, I managed to get to almost 10 km. Also there, a historical marker is crossed: the ruine Radegg, which was built around 1200.