Applying the CUSUM algorithm to a correlated random process
If you look at the derivation of CUSUM from first principles, the idea is to construct a likelihood ratio test statistic and wait till it exceeds a threshold. In case of CUSUM, the assumption is that samples are uncorrelated and so the test statistic simplifies to a sum. Since you are dealing with a different statistical change model, you can try writing a likelihood ratio statistic and see if you can simplify it to anything useful. There are also other algorithms out there for testing "shifts in variance" or "shifts in spectral content" for ARMA processes which you may want to read about.

See also: Ch 6 and 7 Detection of abrupt changes: theory and application
M Basseville, IV Nikiforov - 1993.