Owen (1980) A table of normal integrals

November 8, 2020 paper

This is an artcle of my recent acquaintance. It is about various integrals involving the standard normal distribution \(\Phi(x)\) and its derivative. Besides it is a handy reference, it is interesting to see how the author organize the hundreds of integrals in a manner that is easy to lookup. [more]

A rough description of Radon-Nikodym derivative

November 8, 2020 blog math

Radon-Nikodym theorem suggests, in simplified terms, that if we have two measures \(\mu,\lambda\) of the same space with \(\mu\) absolutely continuous with respect to \(\lambda\) and a function \(\phi\) is \(\mu\)-integrable, then \(\int_A \phi d\mu = \int_A \phi \frac{d\mu}{d\lambda} d\lambda\) which the term \(d\mu/d\lambda\) is called the Radon-Nikodym derivative. [more]

Martingale and local martingale

October 28, 2020 blog math

Martingale is a stochastic process with the martingale property. If we have \(X_t\) as the stochastic process, the martingale property says that \(\mathbf{E}[X_t\mid\mathcal{F}_s] = X_s\), for \(s\lt t\). Closely related to this is the local martingale. However, the Wikipedia page does not have it clearly explained. Here is my narrative.... [more]

Akra & Bazzi (1998) On the solution of linear recurrence equations

September 7, 2020 paper

This is a paper that extends the Master Theorem 1 for more general use. To recap, the Master Theorem is about the complexity of a recurrence algorithm. It assumed the recurrence relation has the form Jon L. Bentley, Dorothea Haken, and James B. Saxe. A general method for solving divide-and-conquer... [more]