This study collects recorded ground motions from the near-source region of large earthquakes and considers to what extent this historic record can inform expectations of future ground motions at similar sites. The distribution of observed peak ground acceleration (PGA) is well approximated by the lognormal distribution, and we expect the observed distribution to remain unchanged with the addition of data from future earthquakes. However, the distribution of peak ground displacements (PGD) will likely change after a well-recorded large earthquake. Specifically we expect future observations of PGD greater than those previously recorded. We use seismic scaling relations to motivate the expected distribution of PGD as uniform on the logarithmic scale, or at least fat-tailed. Because PGA does not scale with fault rupture area or slip on the fault, there are no such scaling relations to predict the observed distribution of PGA. The observed records show that there is essentially no correlation between PGD and PGA for near-source ground motions from large events. The large uncertainty in a future value of PGD in the near-source region of a large earthquake exists despite the ability of Earth scientists to accurately model long-period ground motions. In contrast, the relative certainty in a future value of PGA exists despite the inability to model short-period ground motions reliably. The stability of the observed distribution of PGA with respect to new ground-motion records enables us to predict the distribution of future PGA and to calculate the probability of exceeding the largest recorded PGA.