= Solution
A <pure birth process> $N=(N(t):t\ge0)$ is a <continuous-time Markov chain> on $\mathbb N_0$ whose only possible transition from state $n$ is
$$
n\longrightarrow n+1
\quad\text{at rate }\lambda_n>0.
$$
Equivalently, after entering state $n$ it waits an independent exponential time $X_n\sim\operatorname{Exp}(\lambda_n)$ and then moves to $n+1$.
The process is nonexplosive when it makes only finitely many jumps during every bounded time interval. If it starts from zero, this is equivalent to its explosion time
$$
T_\infty=\sum_{n=0}^{\infty}X_n
$$
being infinite almost surely.
Back to article page
