The lasing transition of "class-A" lasers, where the photon lifetime is much longer than the carrier lifetime, is analogous to the second-order phase transition and, in addition, there is the Ginzburg-Landau (GL) type theory. We theoretically examine the validity of the GL theory for lasers in terms of various parameters, particularly, the ratio between photon and carrier lifetimes. For this purpose, we propose a higher-order photon correlation (g (q) ) measurement to check whether or not the photon statistics are described by the GL theory. The measurement of g (q) has an important experimental advantage because of its quantum efficiency independence. We found that the applicability of the GL theory of lasers is broader than conventionally imagined. Namely, for low-β lasers, GL theory is applicable even when the photon lifetime is comparable to the carrier lifetime. We show that our finding can be understand in the framework of center manifold reduction, which is an example of the "slaving principle" proposed by Haken.