Doob's martingale convergence theorems | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Doob%27s_martingale_convergence_theorems


00:00:17 1 Statement of the theorems
00:00:48 1.1 Doob's first martingale convergence theorem
00:01:43 1.2 Doob's second martingale convergence theorem
00:04:40 1.3 Corollary: convergence theorem for continuous martingales
00:05:19 2 Discrete-time results
00:06:06 3 Convergence of conditional expectations: Lévy's zero–one law
00:09:46 4 Doob's upcrossing inequality
00:12:27 5 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8387727260148604
Voice name: en-GB-Wavenet-B


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob.