We consider sequential testing procedures for detection of criticality of Galton-Watson branching process with or without immigration. We develop a t-test from fixed accuracy estimation theory and a sequential probability ratio test (SPRT). We provide local asymptotic normality (LAN) of the t-test and some asymptotic optimality of the SPRT. We consider a general framework of diffusion approximations from discrete-time processes and develop sequential tests for one-dimensional diffusion processes to investigate the operating characteristics of sequential tests of the discrete-time processes. Especially the Bessel process with constant drift plays a important role for the sequential test of criticality of branching process with immigration.

（Joint work with K. Hitomi (Kyoto Institute of Technology) and Y. Nishiyama (Kyoto Univ.))