Context : Earlier cross-correlation studies for AM Her were performed in various energy range from optical to X-ray and suggested that it mostly shows a high level of correlation but on occasion it shows a low level of correlation or uncorrelation .

Aims : To investigate the degree of correlation between soft ( 2-4 keV ) and hard ( 9-20 keV ) X-rays , we perform the cross-correlation study of the X-ray data sets of AM Her obtained with RXTE .

Methods : We cross-correlate the background-subtracted soft and hard X-ray light curves using the XRONOS program crosscor and fit a model to the obtained cross-correlation functions .

Results : We detect a hard X-ray lag of 192 \pm 33 s in a specific section of energy-dependent light curve , where the soft X-ray ( 2-4 keV ) intensity decreases but the hard X-ray ( 9-20 keV ) intensity increases .
From a spectral analysis , we find that the X-ray emission temperature increases during the anti-correlated intensity variation .
In two other observations , the cross-correlation functions show a low level of correlation , which is consistent with the earlier results performed in a different energy range .

Conclusions : We report a detection of an anti-correlated hard X-ray lag of \sim 190 s from the proto-type polar AM Her .
The hard X-ray lag is detected for the first time in the given energy range , and it is the longest lag among those reported in magnetic cataclysmic variables .
We discuss the implications of our findings regarding the origin of the hard X-ray lag and the anti-correlated intensity variation .