In this talk, we will introduce a notion of a noncommutative probability space and useful properties.

Then we will discuss various convergence results of weighted sums in a noncommutative probability space, e.g., weak law of large numbers, convergence rates and precise asymptotics, etc.

Also, we will discuss some noncommutative inequalities, e.g., Fuk-Nagaev inequalities, Bennett inequality and Rosenthal inequality, etc