While working on GLM and related methods I encountered a strange formulation of the central limit theorem: For the (quasi) score functions $U_\beta$, Peter McCullagh (1983) states for example: $$ N^{-1/2} U_\beta \sim N(0,\sigma^2i_\beta/N) + O_p(N^{-\frac{1}{2}}) $$ He claims that this result should follow from the central limit theorem.
Does anyone know which central limit theorem they used to obtain the rates of convergences as the above statement is clearly more informative than just saying
$$N^{-1/2} U_\beta \stackrel{d}{\to} N(0,\sigma^2i_\beta/N).$$