  The cumulative distribution function (CDF) of a non-central
$\chi^2$-distributed random variable (RV) is often used when measuring the
outage probability of communication systems. For ultra-reliable low-latency
communication (URLLC), it is important but mathematically challenging to
determine the outage threshold for an extremely small outage target. This
motivates us to investigate lower bounds of the outage threshold, and it is
found that the one derived from the Chernoff inequality (named Cher-LB) is the
most effective lower bound. This finding is associated with three rigorously
established properties of the Cher-LB with respect to the mean, variance,
reliability requirement, and degrees of freedom of the non-central
$\chi^2$-distributed RV. The Cher-LB is then employed to predict the
beamforming gain in URLLC for both conventional multi-antenna systems (i.e.,
MIMO) under first-order Markov time-varying channel and reconfigurable
intellgent surface (RIS) systems. It is exhibited that, with the proposed
Cher-LB, the pessimistic prediction of the beamforming gain is made
sufficiently accurate for guaranteed reliability as well as the transmit-energy
efficiency.
