Hiroshi SUGITA

Email sugita(
Keywords Probabilistic number theory, limit theorem, infinite-dimensional stochastic analysis, Monte-Carlo method
Office Science Building B-412 (Toyonaka Campus)

I specialized in Probability theory. In particular, I am interested in infinite dimensional stochastic analysis, Monte-Carlo method, and probabilistic number theory. Here I write about the Monte-Carlo method. One of the advanced features of the modern probability theory is that it can deal with "infinite number of random variables". It was E. Borel who first formulated "infinite number of coin tosses" on the Lebesgue probability space, i.e., a probability space consisting of [0,1)-interval and the Lebesgue measure. It is a remarkable fact that all of useful objects in probability theory can be constructed upon these "infinite number of coin tosses". This fact is essential in the Monte-Carlo method. Indeed, in the Monte-Carlo method, we first construct our target random variable S as a function of coin tosses. Then we compute a sample of S by plugging a sample sequence of coin tosses --- , which is computed by a pseudo-random generator, --- into the function. Now, a serious problem arises: How do we realize a pseudo-random generator? Can we find a perfect pseudo-random generator? People have believed it to be impossible for a long time. But in 1980s, a new notion of "computationally secure pseudo-random generator" let people believe that an imperfect pseudo-random generator has some possibility to be useful for practical purposes. A few years ago, I constructed and implemented a perfect pseudo-random generator for Monte-Carlo integration, i.e., one of Monte-Carlo methods which computes the mean values of random variables by utilizing the law of large numbers.