# Remembering Krishna Athreya

As I write this, he passed away about a week ago. While the end of a person’s life is a sad time, it also provides an opportunity to reflect on the past. The first memory that came to my mind was of sitting in Sid Resnick’s office in the Stanford Statistics department in 1974 (or 1975.) A group of us got together once a week to make our way through the new book on Branching Processes by Athreya and Peter Ney.  It was a welcome modernization of Harris’ 1963 book that started the subject. There were crisp proofs of the basic facts about Galton-Watson processes, Markov branching processes (with exponential lifetimes), the age-dependent case (general lifetimes), and multi-type branching processes. Over my career I have loaned this book to my graduate students many times to help them learn the subject. Remarkably I still have it but it is a little worse for wear.

#2 on my list of Athreya’s greatest hits is A new approach to the limit theory of recurrent Markov chains which appeared in the Transactions of the American Math Society, a paper that was written with Peter Ney in 1978. Again this is a contribution to an area founded by Ted Harris. While Markov chains on discrete state space are well understood, on a general state space numerous pathologies arise. Harris’ genius was to identify a class of these chains that have a tractable theory and cover a number of examples.

There is an elegant analytical theory described in the book by Revuz. However in 1978 several researchers, including Esa Nummelin who later developed a book based on this approach, had the same idea at the same time. I remember attending a session of talks at the 1978 meeting on Stochastic Processes and their Applications and hearing three talks on the topic. This was devasting for a Ph.D. student in the audience who was working on this for his thesis.

The idea is simple but brilliant: a Harris chain can be modified to have one state that is hit with positive probability starting from any state and having one such state is enough to carry out all the usual theory for the discrete case. Given this hint I am sure you can work out the details for yourself. I was so excited by the idea that I put it in the Markov chains chapter of my graduate text book.

Returning to a more traditional narrative: Krishna Athreya received his Ph.D. in 1967 from Stanford where he worked with Sam Karlin, a legendary probabilist with an impressive pedigree: son of Bochner, who was a grandson of Hilbert, and the mentor for 44 students including Tom Liggett and Charles Stone among many others. Athreya’s thesis topic was Multitype Continuous Time Markov Branching Processes and Some Classical Urn Schemes. Soon after he got his degree Athreya and Karin worked on Branching processes in random environments. Two papers were published in Annals of Mathematical Statistics in 1971 since the Annals of Probability which began in 1973 did not yet exist.

These two papers like many in Athreya’s top 20 most cited on MathSciNet contain a number of ideas that have not been fully explored. An example is the work with his Ph.D. student Jack Dai on random logistic maps. Last but not least, I would like to mention his 1994 paper on large deviations for branching processes which contains material that working probabilists should know. Athreya has left an impressive mathematical legacy that will enrich your life and research if you have the time to read it. It is sad that there will be no more work coming from him, but I hope others who read this will be inspired to continue his work. .