  Prophet inequalities are a central object of study in optimal stopping
theory. In the iid model, a gambler sees values in an online fashion, sampled
independently from a given distribution. Upon observing each value, the gambler
either accepts it as a reward or irrevocably rejects it and proceeds to observe
the next value. The goal of the gambler, who cannot see the future, is
maximising the expected value of the reward while competing against the
expectation of a prophet (the offline maximum). In other words, one seeks to
maximise the gambler-to-prophet ratio of the expectations.
  This model has been studied with infinite, finite and unknown number of
values. When the gambler faces a random number of values, the model is said to
have random horizon. We consider the model in which the gambler is given a
priori knowledge of the horizon's distribution. Alijani et al. (2020) designed
a single-threshold algorithms achieving a ratio of $1/2$ when the random
horizon has an increasing hazard rate and is independent of the values. We
prove that with a single-threshold, a ratio of $1/2$ is actually achievable for
several larger classes of horizon distributions, with the largest being known
as the $\mathcal{G}$ class in reliability theory. Moreover, we extend this
result to its dual, the $\overline{\mathcal{G}}$ class (which includes the
decreasing hazard rate class), and to low-variance horizons. Finally, we
construct the first example of a family of horizons, for which multiple
thresholds are necessary to achieve a nonzero ratio. We establish that the
Secretary Problem optimal stopping rule provides one such algorithm, paving the
way towards the study of the model beyond single-threshold algorithms.
