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.