The "famous four" at the Indian Statistical Institute

During 1956-1963 a group of four mathematicians, Veeravalli S Varadarajan, Ramaswamy Ranga Rao, K R Parthasarathy and S R Srinivasa Varadhan, who studied at the Indian Statistical Institute in Calcutta became known as the "famous four". In the article 'Some mathematical reminiscences', Methods and Applications of Analysis 9 (3) (2002), v-xviii, Varadarajan discusses how this group came about.

The "famous four" at the Indian Statistical Institute.

Let me begin my story with an account of what it was like to be an aspiring graduate student interested in studying mathematics in the 1950's in India. You must keep in mind that our country had become independent in 1947, just a few years back. Although in ancient times India had been prosperous and very creative in mathematics, she had entered a long period of darkness and was only slowly coming out of it in the 1950's. Although there were isolated flashes of achievements like those of Ramanujan, Raman, Saha, and a few others, there were very few places with a tradition of doing good science; moreover, there was a feeling among most people, unstated and implicit but nevertheless very real, that under those conditions, it was only by going to some place in Europe or the US that one would be able to do good work. However there were two places which were very attractive for young people to get into. One was the Tata Institute of Fundamental Research in Bombay, and the other was the Indian Statistical Institute in Calcutta. I joined the Statistical Institute in 1956 as a research fellow. A little later R Ranga Rao and K R Parthasarathy also joined the Institute. Varadhan came to the Institute in 1959.

In the mid 1950's and early 1960's, the Statistical Institute was a place with an extraordinary ambiance for doing creative work in many areas. It had been founded in 1931 by P C Mahalanobis with the goal of developing statistical methods and techniques in an environment that was deeply concerned with actual applications. In the 1950's the need for planning the economic development of India forced the Institute to the forefront of statistical and planning activities. Mahalanobis, the director of the Institute, and C R Rao, the head of the research and training school of the Institute, had the vision to realise that it was essential to create and maintain a high level of scientific atmosphere at the Institute to foster the discovery and development of new ideas. The main focus of the Institute was on statistics and its many applications, mainly to economics and planning, but there was activity in many other areas such as psychometry, biometry, and genetics. There was a constant stream of visiting scientists, such as Oskar Lange, Leo Boron, Charles Bettelheim, Norbert Wiener, John Galbraith, R A Fisher, Jerzy Neyman, A N Kolmogorov, N N Bogolyubov, R Vaidyanathaswamy, J B S Haldane, Nathan Keyfitz, and scores of others. I think, for young students like us, this exposure to so many different areas of interest was very beneficial. It allowed us to put what we wanted to do in its proper place in the big scheme of things and gave us an overview that would prove to be crucial for our development and growth as scientists.

For those of us who wanted to work in Statistics, sample surveys, genetics, or other areas closer to applications, there were plenty of people around who were doing world class work in those subjects and so guidance was available. Many of our friends, like J Sethuraman, Vasant Korde, G P Patil, went in that direction. But our small group was interested in probability theory and mathematics and there was almost no one available to guide us. S D Chatterji, who joined the Institute a year before me, left to work in the US a few months after I came in. G Kallianpur, who was a central figure in the probability group at the Institute, at that time, had left the Institute to go to the US permanently. It is a tribute to C R Rao and Mahalanobis that we were not pressured in any way to do things we did not like to, and so we were left pretty much to ourselves. But this isolation did not bother us very much. We were a very confident bunch of youngsters and felt we did not need anyone to tell us what we should do! Because of our deep interest in probability theory we started studying measure theory and topology. The books of Halmos on measure theory, finite dimensional vector spaces and Hilbert space, the book of Kelly on set topology, and the great classics of Gnedenko and Kolmogorov (limit theorems for sums of independent variables), Cramer (mathematical methods of Statistics), Doob and Feller constituted the background for everything we did. At some point R A Gangolli joined our group but only for a few months; he left for MIT for further work.

Of course merely reading books and discussing them in seminars, while a very necessary activity, is hardly sufficient to take the next step, namely to do something new. The impulse for this came suddenly. In 1956 C R Rao had gone to Moscow for a conference on probability and information theory where he had spent some time with Kolmogorov. On his return he had told me that in Kolmogorov's view information theory was going to be a very basic field in the years to come; he had also given me a reprint that Kolomogorov had given him of a paper of Kolomogorov and Prokhorov that was presented by them in a conference in Berlin in 1954. This paper treated weak convergence of probability measures in complete separable metric spaces (polish spaces). I became quite excited and began studying the topological aspects of the space of probability measures on topological spaces, especially metric spaces and polish spaces. When Ranga Rao and Parthasarathy joined me later, I communicated to them my excitement. This intervention of C R Rao gave a real boost to our solitary efforts and started us on a path from which we never looked back. In the meantime, R R Bahadur had joined the Institute faculty. With his interest in the more mathematical side of things and his great interest in us, the atmosphere for our group improved considerably. Ranga Rao worked on uniform approximations of measures and obtained deep and far reaching generalisations to higher dimensional spaces of classical uniform approximation theorems in probability. Parthasarathy started looking into measure and information theoretic aspects of dynamical systems. We had, in some miraculous way, formed a nucleus of a modest school and were looking at real research problems of some current interest. At that time there was no other place in India except the Tata Institute of Fundamental Research where young people were doing real mathematics.

Some of my work on convergence of measures on metric spaces had much overlap with work of Prokhorov in Moscow and Le Cam at Berkeley. At about the same time Patrick Billingsley at Princeton was also doing things similar to what I had been doing. Prokhorov was the examiner for my thesis. This fact requires an explanation. Although I worked at the Institute, the Institute had not yet been given the power to grant the doctoral degree and we had to submit our theses to the Calcutta University. Indian Universities, operating on an ancient system that was based on total distrust of the candidate's supervisor, insisted on having a foreign examiner for any Ph.D. thesis, and Calcutta was no exception. But surprisingly they accepted my suggestion that in my case Prokhorov should be the foreign examiner.

This was the situation when Varadhan joined the Institute as a research fellow in 1959. He had had a spectacular career in the Madras University. His total of marks for the masters degree was the highest in the history of the University. In spite of this he was very modest, willing to do anything and look at any question that people asked him. Everyone who came into contact with him was immediately impressed by his quickness and his power of analysis of problems. Haruki Morimoto, who was a statistician from Osaka, Japan, and who was a frequent visitor to the Institute, told me years later that a conversation with Varadhan at that time changed his entire perspective on a whole series of problems and led to many papers of which he is very fond of. Unfortunately I did not have much chance to interact with Varadhan in this period. I had received an offer of a postdoctoral fellowship from Princeton University and was about to leave. In the weeks and months before my departure, Ranga Rao, Parthasarathy, and myself had been running a seminar on topological groups (essentially studying Pontryagin's famous book). In view of our background and also of our being a part of the prevailing culture of the Institute, it is not surprising that that the idea came to us that it would be nice to prove limit theorems for sums of independent variables with values in general locally compact abelian groups and even Hilbert space. This problem is of course closely related to the determination of the structure of infinitely divisible distributions on those groups.

After I left for Princeton Varadhan joined this group. Their very first problem was to examine whether a typical probability measure on a metric group was decomposable as a convolution of two other non trivial probability measures. Parthasarathy has told me how one day Varadhan came in and showed them the argument that produced many indecomposable measures; building on this they then succeeded in showing that on a complete separable metric group the set of indecomposable measures was a dense G_delta in the space of probability measures on the group equipped with the weak topology, a result that surprised Bochner (to whom they had communicated this work to ask for his opinion). They then went on to work out completely the theory of infinitely divisible distributions and the associated limit theorems in all locally compact abelian groups. But the extension to Hilbert space proved quite difficult. This is due to the fact that infinite dimensional Hilbert space is not locally compact, and so to prove the tightness of a family of probability measures, i.e., to find compact sets whose complements have small probability uniformly in the family, is always technically difficult because compact sets are harder to locate in spaces which are not locally compact. Varadhan succeeded in extending the locally compact theory to the case of independent random variables with values in a Hilbert space. This was a substantial achievement, not only because of the infinite dimensionality of the context, but also because the techniques he used were quite original and were based on a deep use of Levy's concentration function but adapted to the context of Hilbert space. This work, done about 1962, was more or less his thesis for the Ph. D degree.

Our group was changing once again. Ranga Rao had gone to Urbana, Illinois, Parthasarathy had gone to Moscow and I myself had returned from the US. By this time the Institute itself had begun to grant the Ph. D degree. However the system of going to foreign examiners was still retained. For Varadhan's thesis the foreign examiner was Kolmogorov. His report was in Russian. Since I was one of the few people in our department who was familiar with the Russian language, the report eventually found its way to me.

I have since tried to trace a copy of that report but without success. But I still remember two sentences which stood out in that report. Kolmogorov wrote that

this thesis is not that of a student but that of a mature master ... the thesis deserves the second degree in the Soviet Union.

I should remind you that the first degree in the (then) Soviet Union is called candidate's degree and is roughly the same as a Ph.D in the US, while the second degree, Doctorate of Sciences, is given only for distinguished work, usually several years after the candidate's degree. For instance, Prokhorov's famous 1956 paper was essentially his thesis for his second degree.

Last Updated October 2026