This question was asked a long time ago but never answered. I'm a professor in the USA who teaches math and stats courses, has trained many actuaries, and has sent students to stats PhD programs. An actuarial science master's student probably has sufficient stats to start stats grad school. On the "minimum requirements" end, I have sent students to stats PhD programs with coursework in statistical modeling (e.g., the GLM), time series analysis, and probability theory. Because there is so much variability in stats programs, there's not widespread agreement on what coursework, specifically, successful PhD applicants should have.
I tell students interested in stats grad school that the most useful courses are Statistical Theory, Advanced Regression, and Real Analysis. Measure theory would help, but there's not an expectation from PhD admissions committees in the USA that students will have had measure theory. They'll certainly learn probability theory in a deep way in their qualifying exam courses. European PhD programs likely expect more, because the culture is that one should have done a master's degree in stats before starting a PhD. An actuary who could pass the old S exam (as well as, of course, the P exam) is probably mostly ready to start a PhD program but might need to beef up a bit on statistical theory.
Most PhD programs have websites where they describe what successful applicants are supposed to have learned prior to starting. It might be possible to audit any missing courses at a local university (while still working as an actuary) before applying. This has the added benefit of potentially giving you another strong letter of recommendation.
