A three-dimensional Monte Carlo model of the uniform relativistic runaway electron breakdown in air in the presence of static electric and magnetic fields is used to calculate electron distribution functions, avalanche rates, and the direction and velocity of avalanche propagation. We also derive the conditions required for an electron with a given momentum to start an avalanche in the absence of a magnetic field. The results are compared to previously developed kinetic and analytical models and our own analytical estimates, and it is concluded that the rates used in many early models [e.g., Lehtinen et al., 1997; Taranenko and Roussel-Dupr¿, 1996; Yukhimuk et al., 1998; Roussel-Dupr¿ et al., 1998> are overestimated by a factor of ~10. The Monte Carlo simulation results are applied to a fluid model of runaway electron beams in the middle atmosphere accelerated by quasi-electrostatic fields following a positive lightning stroke. In particular, we consider the case of lightning discharges which drain positive charge from remote regions of a laterally extensive (>100 km) thundercloud, using a Cartesian two-dimensional model. The resulting optical emission intensities in red sprites associated with the runaway electrons are found to be negligible compared to the emissions from thermal electrons heated in the conventional type of breakdown. The calculated gamma ray flux is of the same order as the terrestrial gamma ray flashes observed by the Burst and Transient Source Experiment detector on the Compton Gamma Ray Observatory. ¿ 1999 American Geophysical Union