Filtration: a sequence of embedding sigma-algebra

Martingale: w.r.t. A filtration is a stochastic process where (1) finite expectations (2) each r.v. is measurable w.r.t. respective sigma-algebra in the filtration (3) the expectation of next r.v. conditioned on current sigma-algebra is the current r.v.

Conditional expectation of a r.v. on a sigma-algebra is another r.v. which: (1) is measurable w.r.t. That sigma-algebra (2) for any element (event) from that sigma-algebra, on which those 2 r.v. have the same conditional expectation.

That is the reason why the conditional expectation is that r.v. itself if it is measurable. We call this “conditional const”.

Levy’s upward and downward theorem: a.s. And L1 convergence of martingales

Useful preliminary links:

Sigma-algebra: https://en.wikipedia.org/wiki/%CE%A3-algebra

Measure: https://en.wikipedia.org/wiki/Measure_(mathematics)

Probability space: https://en.wikipedia.org/wiki/Probability_space

Borel set: https://en.wikipedia.org/wiki/Borel_set

I heard this story from Dr. Roman Vershynin, my professor in Brownian motion class. When we went over the reflection principle and Lévy’s theorem, he was very excited to praise how amazing Lévy is, “such a great mathematician”, several times. Two of Lévy’s works, Théorie de l’addition des variables aléatoires in 1937 and Processus stochastiques et mouvement brownien in 1948, were just like world war 2 never even happened! How could that even be possible for him to do such great work during the most fierce war in human history! Earlier this year (2022), when Dr. Vershynin taught high-dimensional probability to Ukrainian students in Kyiv, he also told them the story about Lévy. It turned out that Lévy’s story encouraged and inspired students a lot! They were fully engaged in this online class during local evening time, taught from the other side of the earth, meanwhile under the threat of military strikes from Russian troops and constant power shortage.

Our department has a group of great Jewish and Ukrainian professors. I heard some very impressive stories from them, very unique and interesting.