If the survival function S(t) tends to zero as t grows without bound, which statement is true under almost sure death assumption?

• A. S(t) tends to 0
• B. S(t) tends to 1
• C. S(t) oscillates
• D. S(t) tends to infinity

Answer: (A)

The survival function S(t) is defined as P(T > t), where T is the lifetime. If death occurs almost surely, then T is finite with probability 1, so as time grows large the chance of still being alive past t must shrink to zero. Therefore S(t) tends to 0. This also means the probability of having died by time t, F(t) = P(T ≤ t), tends to 1, which is consistent with almost sure death. The alternative that S(t) tends to 1 would imply a nonzero chance of living forever, which contradicts almost sure death. S(t) cannot oscillate, since survival probabilities are nonincreasing in t, and it cannot tend to infinity since S(t) is always between 0 and 1.