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Toss a fair coin ten times and getting six or more heads is ordinary. Toss it a thousand times and getting six hundred or more is almost impossible. Probability theory has long said that averages settle down. Srinivasa Varadhan's work answers the next question: how fast does the chance of a big deviation shrink, and what does the system do when one happens anyway? This post tells his story and shows the idea with coin tosses.
Who is S. R. S. Varadhan?
According to his MacTutor biography, Sathamangalam Ranga Iyengar Srinivasa Varadhan was born in Madras, now Chennai, on 2 January 1940. His father was a science teacher who became a school principal in Ponneri, near Madras.
He credits a high school teacher who invited good students home at weekends to work on extra problems. "That gave me the idea that mathematics is something that you can enjoy doing like playing chess or solving puzzles," he said in an interview quoted by MacTutor.
At Presidency College, Madras, he chose statistics over mathematics because he expected to work in industry, earning a BA with honours in 1959 and a master's degree in 1960. He then went to the Indian Statistical Institute in Kolkata.
What changed at the Indian Statistical Institute?
He was first asked to study statistical quality control and found it unsatisfying. Then he met three young researchers there, Varadarajan, Parthasarathy and Ranga Rao, who worked on probability as pure mathematics. In his words, quoted by MacTutor: "Let us work on a problem." They picked a problem about probability distributions on groups, solved it, and learned the tools they needed along the way.
His formal adviser was C. R. Rao, and his 1963 thesis was on convolution properties of distributions on topological groups. The great Soviet mathematician Andrei Kolmogorov, visiting the institute, examined it and wrote: "This is not the work of a student, but of a mature master." That autumn Varadhan went to the Courant Institute at New York University as a postdoctoral fellow. He stayed, became a full professor in 1972, and twice served as the institute's director.
What is a large deviation?
Toss a fair coin n times. The law of large numbers says the fraction of heads gets close to one half. A large deviation is the fraction landing far away, say at 60% or more. Here are the exact probabilities, from the binomial distribution:
10 tosses: about 0.377, so more than one time in three
100 tosses: about 0.028, roughly one time in 35
1,000 tosses: about 1.4 × 10⁻¹⁰
The probability does not just shrink; it shrinks exponentially in n. In his own survey of the field, Large deviations (2008), Varadhan works through exactly this coin example. For k heads out of n with k ≈ nx, the logarithm of the probability is approximately −nH(x), where
H(x) = log 2 + x log x + (1 − x) log(1 − x)
is a relative entropy, called the rate function. For x = 0.6, H(0.6) = 0.693 − 0.306 − 0.367 ≈ 0.020. So each extra toss multiplies the chance of a 60% result by roughly e⁻⁰·⁰²⁰. Check with 1,000 tosses: −ln(1.4 × 10⁻¹⁰) ÷ 1,000 ≈ 0.023, already close to 0.020, and the gap closes as n grows.
What did Varadhan add?
Results like this for sums of independent random quantities go back to Harald Cramér in 1938, as Varadhan's survey notes. Varadhan's contribution was to make large deviations a general theory. His 1966 paper, Asymptotic probabilities and differential equations, set out a general principle, and from 1975 a series of papers with Monroe Donsker, beginning with Asymptotic evaluation of certain Markov process expectations for large time, extended it to random processes that evolve in time.
The Abel Prize for 2007 went to Varadhan "for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations." The prize committee's citation, quoted by MacTutor, lists uses in quantum field theory, statistical physics, population dynamics, finance and traffic engineering, and in simulating rare events on computers.
Earlier, he shared the American Mathematical Society's 1996 Steele Prize with his long-time collaborator Daniel Stroock for their work on diffusion processes. He was elected a Fellow of the Royal Society in 1998 and received a Padma award from India in 2008.
What can a student take from Varadhan's way of working?
Learn tools by attacking a problem. He and his friends did not finish a syllabus first. They picked a problem and learned what it needed.
Treat mathematics as a game worth playing. His school teacher's weekend problems made the subject enjoyable rather than frightening.
Work with people. His major results came from long partnerships, with Stroock and with Donsker.
Ask the second question. Everyone knew averages settle down. He asked how unlikely the exceptions are, and what they look like.
Large deviations are graduate-level mathematics, but the coin example rests on Class 11 and 12 probability. If the binomial distribution or logarithms feel rusty, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revise them one at a time. Read our post on common probability mistakes, and the story of C. R. Rao, Varadhan's adviser at the Indian Statistical Institute.
Try it now: compute the chance of 8 or more heads in 10 tosses, then compare it with 80 or more in 100.
