Category Archives: Probability

Noncommutative probability I: motivations and examples

In many areas of mathematics, one can usually assign a commutative algebraic structure to the “space” that one is studying. For example, instead of studying an (affine) algebraic variety, one can study algebraic functions on that variety, which gives rise … Continue reading

Posted in Functional analysis, Probability | Leave a comment

Zeros of random polynomials

Given a polynomial , where the coefficients are random, what can we say about the distribution of the roots (on )? Of course, it would depend on what “random” means. Here, “random” means that the sequence is an i.i.d. sequence … Continue reading

Posted in Algebra, Complex analysis, Potential theory, Probability | Leave a comment

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading

Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability | 1 Comment

Two little remarks on sphere

1. “Take a sphere of radius in dimensions, large; then most points inside the sphere are in fact very close to the surface.” David Ruelle Let and . Let The fraction of the volume of to the volume of is … Continue reading

Posted in Geometry, Miscellaneous, Probability | Tagged , , | Leave a comment

From combinatorics to entropy

Let and . Then by Stirling’s formula. This is probably well-known to people who have studied statistical physics, but some of us including myself may not be very aware of this kind of relation. I wonder if this was the … Continue reading

Posted in Analysis, Probability | Tagged | Leave a comment

Martingale Theory III: Optional stopping theorem

This is a sequel to Martingale Theory II: Conditional Expectation. Our aim here is to prove the main theorems about discrete time martingales. More advanced probability texts (e.g. those on stochastic calculus) assume that these theorems are well known to … Continue reading

Posted in Probability | Leave a comment

Potential theory on finite sets

Motivation Consider the following Dirichlet problem on the unit square : on where is a given function on the boundary. To get an approximate solution, we may replace the unit square by a grid. Let be large and set . … Continue reading

Posted in Potential theory, Probability | Leave a comment