Abstract: For a fairly wide range of boundary function and quasi weight function, the author obtained a general form of precise asymptotics in complete moment convergence for the largest eigenvalue by using the weak convergence theorem and the small deviation conclusion for largest eigenvalue of β-Hermite random matrix and tail probability inequality of the general β-Tracy-Wisdom distribution.

Key words: general β-Tracy-Wisdom distribution, precise asymptotics,  largest eigenvalue, complete moment convergence, general form