GENERAL I ARTICLE Norbert Wiener and Probability Theory Some Reflections G Kallianpur In this article I will be concerned only with what I think are Wiener's most important contributions to probability Gopinath Kallianpur, one theory. In the concluding section of this essay, I mention oflndia's distinguished some of my personal reminiscences about him. probabilists, is currently at the University of Brownian Motion North Carolina, Chapel Bill, USA. Earlier, he In his 1905 paper on the kinetic theory of heat, Einstein had was on the faculty of the University of Minnesota studied the irregular movement of particles suspended in a fluid. If the time interval is not too small, he made the assump­ tion that the movement of each particle is independent of the movement of all other particles and further concluded that the displacements (on the x-axis) of a particle over different time intervals are mutually independent random variables (to use the - modern terminology) and the displacement X t Xs over the time interval (s, t) is normal with mean 0 and variance = D (t-s) where D is the diffusion constant of interest to statistical physics. Five years before Einstein, Louis Bachelier, in his thesis on the theory of speculation, chose Brownian motion to model the fluc­ tuation of stock prices on the market. He derived (although not rigorously) several important properties of Brownian motion. It is under the influence of Bachelier's work that geometric Brow­ nian motion (a variant of Brownian motion) has become a basic model for a stock price process in the modern theory of finance. Wiener seems to have been unaware of Bachelier's work. In 1923, ten years before the foundations of probability were laid in the book by the great Russian probabilist Andrei Nikolayevich Kolmogorov, Norbert Wiener constructed a probability measure (now called the Wiener measure) on the space of continuous functions under which the entire paths of Brownian motion --------~-------- 32 RESONANCE I January 1999 GENERAL I ARTICLE could be regarded as random functions and the probabilities of certain sets of these paths could be calculated. It was the first concrete example of a stochastic process and the beginning of the modern theory. Prediction Theory The other area of probability theory which was virtually created by Wiener and Kolmogorov was the theory oflinear prediction of stationary time series. It is now an important part of stochastic processes with ramifications in other branches of mathematics such as harmonic analysis. Wiener characterized the 'time domain' properties of deter­ ministic and purely nondeterministic processes in terms of properties of the spectral density of the process, for example, the famous Paley-Wiener theorem. In his investigation of conti­ nuous time stationary processes, Wiener introduced the famous Wiener-Hopf equation to solve the prediction problem. Wiener at the Indian Statistical Institute My personal recollections of Professor Wiener relate to his two visits to the Indian Statistical Institute, Calcutta the first one a brief st~y of about two weeks in 1954 and a five month stay beginning in September 1955. During this period Wiener lectured about his work and also found time to discuss mathe­ matics with me. Almost every morning he would come to my office and start the day by asking his favorite question, 'what is new today?' He meant, of course, new in the world of mathematics. We would go to an empty classroom and Wiener would start writing on the blackboard while I sat at the desk listening or taking notes. Among the topics we talked about were mostly his own past work in ergodic theory, prediction theory and generalized harmonic analysis. We also worked together (but left unfinished) on non-linear prediction theory. At this stage in his life Wiener had health problems which, I believe, were serious enough to tire him out by lunch time. --------~-------- RESONANCE I January 1999 33 GENERAL I ARTICLE Norbert Wiener was a universal genius, fluent in many languages, with a deep interest in poetry and literature. He was one of the last representatives of a vanishing liberal tradition. I can only think of his great contemporaries, Bertrand Russell and John von Neumann, to compare with him in this respect. Wiener's stay at the Institute coincided with a period of intense excitement when the Institute was deeply involved with preparing India's Second Five Year Plan. The younger workers especially were eager to take part in seminars given by other prominent visitors like Professor Joan Robinson of Cambridge. I had the impression that these concerns did not touch Wiener. Perhaps the following episode (which I quote from an earlier article ofmine) would be a fitting conclusion to my reminiscences. Towards the end of his stay, the Soviet leaders Khrushchev and Bulganin visited Calcutta during their state visit to India. Their visi t to the Institute had been arranged by Professor Mahalanobis. The honored guests were expected to arrive at the lSI in the late afternoon. By noon time, every window and vantage point in the Institute grounds was jammed with lSI workers and their families and the streets from Baranagore all the way to the center of the city were packed with crowds bringing all traffic ~o a standstill. When I met Wiener outside the guest house (then located in the Research and Training School building), I could see at once the excitement in his face. 'We are not interested in seeing Khrushchev, are we? Let's go to a blackboard'. Without waiting for my answer he led the way until, with some difficulty, we were able to find an empty seminar room. Wiener then jubilantly announced that he had a mathematical model for Indian economic'development. I was astonished but pleased at his interest in such practical matters. Then he began to explain his ideas: 'Let T be a measure preserving transformation ... '. The rest of the conversation has not remained in my memory, I regret to say. Wiener himself never alluded to it again. (Inci­ dentally, Khrushchev and Bulganin never 'made it through the Calcutta crowds and the visit to the lSI had to be cancelled.) After we both left India, he in 1955 and I a little later. I had the --------~-------- 14 RESONANCE I January 1999 GENERAL I ARTICLE opportunity to meet Wiener on only two other occasions. The last meeting was a brief one at the International Congress of Mathematicians in Stockholm in 1962. Earlier, in 1960, I saw Wiener when he visited Indiana University in Bloomington. He openly expressed his pleasure at meeting three of his collaborators - Professors Eberhard Hopf, Pesi Masani and (I assume as a matter of courtesy) myself. This meeting with Wiener was, from my own professional point of view, the most fruitful in that it gave direction to my future research. Wiener explained to me that he considered his work on homogeneous Ghaos expansions to be the beginning of a non­ Address for Correspondence linear theory of prediction. Strangely, he seemed not to be G Kallianpur familiar with K Ito's work in this area. In any case those Department of Statistics profound remarks by Wiener directed me to I to's theory of University of North Carolina Chapel Hill, NC 27514 USA stochastic integration and its application to non-linear prediction e-mail: [email protected] and filtering theory. Memories of Einstein l' Vl Albert Einstein is probably the best known physicist of this century. His revolutionary ideas ~ of space, time and gravitation made him a legend in his own lifetime. The great physicist is also .I~ " remembered for his strong dislike for formality of all sorts. He hated wearing even simple business clothing and was most happy in his sailboat, smoking his pipe and wearing an old sweater. There are many interesting stories about Einstein which illustrate some of his characteristics. The incident described below occurred sometime between 1930 and 1931, when Einstein visited the California Institute of Technology (CIT). R A Millikan, the American experimentalist, who measured the charge on the electron by his famous oil drop experiments had established an association called the 'The Athenaeum' for raising funds for the CIT. Its associates were wealthy business men and some distinguished guests who had a general interest in the advancement of American science. The launching of 'The Athenaeum' began with a grand dinner and for this purpose Millikan succeeded in inducing a celebrity no less than Einstein to come from" Princeton to lend fame to the occasion. Einstein, who hated formality came to attend the dinner in the conventional evening attire! It chanced that the immediate neighbour of Einstein was a very successful businessman from New York city, Leon L Watters who had a doctorate in chemistry from Columbia University. As the dinner progressed and speeches were being given by the associates of the Athenaeum, Watters, sitting next to Einstein realizing that the great physicist might have taken refuge from the pompous gathering to think on the unified field theory, wrote a few lines on a piece of paper and passed it on to Einstein. Thou lost amidst host of unknown faces with all the world a jangle-and-a-jar I meditate on interstellar space and smoke a mild cigar. Einstein liked this small poem tremendously and after this incident the two became very warm friends. Atul Khanna, GND University, Amritsar, India --------~-------- RESONANCE I January 1999 35 .