Category Archives: 2017 Math

Construction of Bergman kernel, evaluation of Bergman kernel

In this note, first we construct a Bergman kernel of simply connected smooth bounded domain in . Next, we evaluate the Bergman kernel when the domain is a unit disc. Finally, we introduce Bergman metirc on a complex manifold. 1. Construction of Bergman kernel Let be a bounded, simply connected and smooth domain in .… Read More »

Marcinkiewicz-Zygmund’s SLLN

First, we present well-known Khintchine-Kolmogorov Theorem. Theorem (Khintchine-Kolmogorov). Let be independent random variables with finite expectation. If , then converges a.e. Now we present the Marcinkiewicz-Zygmund theorem which is our object of this article. Theorem (Marcinkiewicz-Zygmund Theorem). Suppose that are i.i.d with for some . Then     where     Proof. Set    … Read More »