  Tunneling processes through black hole horizons have recently been
investigated in the framework of WKB theory discovering interesting interplay
with the Hawking radiation. In this paper we instead adopt the point of view
proper of QFT in curved spacetime, namely, we use a suitable scaling limit
technique to obtain the leading order of the correlation function related with
some tunneling process through a Killing horizon. The computation is done for
certain large class of reference quantum states for scalar fields. In the limit
of sharp localization either on the external side or on opposite sides of the
horizon, the quantum correlation functions appear to have thermal nature, where
in both cases the characteristic temperature is the Hawking one. Our approach
is valid for every stationary charged rotating non extremal black hole,
however, since the computation is completely local, it covers the case of a
Killing horizon which just temporarily exists in some finite region too. These
results give a strong support to the idea that the Hawking radiation, which is
detected at future infinity and needs some global structures to be defined, is
actually related to a local phenomenon taking place even for local geometric
structures (local Killing horizons) existing just for a while.
