  We analyze the noise properties of both electric charge and heat currents as
well as their correlations in a quantum-dot based thermoelectric engine. The
engine is a three-terminal conductor with crossed heat and charge flows where
heat fluctuations can be monitored by a charge detector. We investigate the
mutual influence of charge and heat dynamics and how it is manifested in the
current and noise properties. In the presence of energy-dependent tunneling,
operating conditions are discussed where a charge current can be generated by
heat conversion. In addition, heat can be pumped into the hot source by driving
a charge current in the coupled conductor. An optimal configuration is found
for structures in which the energy dependence of tunneling maximizes asymmetric
transmission with maximal charge-heat cross-correlations. Remarkably, at a
voltage that stalls the heat engine we find that in the optimal case the
non-equilibrium state is maintained by fluctuations in the heat and charge
current only.
