  Recent observations of a zero bias conductance peak in tunneling transport
measurements in superconductor--semiconductor nanowire devices provide evidence
for the predicted zero--energy Majorana modes, but not the conclusive proof for
their existence. We establish that direct observation of a splitting of the
zero bias conductance peak can serve as the smoking gun evidence for the
existence of the Majorana mode. We show that the splitting has an oscillatory
dependence on the Zeeman field (chemical potential) at fixed chemical potential
(Zeeman field). By contrast, when the density is constant rather than the
chemical potential -- the likely situation in the current experimental set-ups
-- the splitting oscillations are generically suppressed. Our theory predicts
the conditions under which the splitting oscillations can serve as the smoking
gun for the experimental confirmation of the elusive Majorana mode.
