A pure birth process is nonexplosive exactly when
(1)
Its explosion time is the sum of independent exponential holding times. If the displayed series converges, that sum has finite expectation. If it diverges, the Laplace transform of the sum is
(2)
forcing the sum to be infinite almost surely.

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  1. Pure birth process
  2. Birth-death process
  3. Continuous-time Markov chain
  4. Markov chain
  5. Markov process
  6. Probability theory
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