Surveillance can be based, in some change-point detection problems,
on a sequence of invariant statistics. Gordon and Pollak prove that, under
certain conditions, the average run length (ARL) to false alarm of invariance-
based Shiryayev–Roberts detection schemes is asymptotically the same as
that of the dual classical scheme that is based on the original sequence of
observations. In this paper we give alternative conditions under which the two
ARLs coincide and demonstrate that these conditions are satisfied in cases
where Gordon and Pollak’s conditions are difficult to check.