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Consider a pure-birth process $X(t)$ with rates $\lambda_i$ that satisfies $\sum_{i=0}^\infty \frac{1}{\lambda_i} = \infty.$ By Reuter's criterion this is sufficient for $X(t)$ to be regular, ie $X(t) < \infty$ for all $t \ge 0$ holds a.s.

For $\lambda > 0$ let $\hat{X}(\lambda) := \int_0^\infty \lambda e^{-\lambda t} X(t) dt$ be the formal Laplace-Transform of $X(t)$.

Suppose there is a $\lambda^* > 0$ so that \begin{align} E\hat{X}(\lambda^*) &= 1 \\ E\int_0^\infty t e^{-\lambda^* t} X(t) dt & < \infty \end{align}

holds. Is the expected number of jumps $EX(a) \le 1$ for some $a$?

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If $\mathrm E(X(t))\gt c$ for every $t\gt0$, then for every $\lambda\gt0$, $ \mathrm E(\hat X(\lambda))\gt\int_0^{+\infty}\lambda\,\mathrm e^{-\lambda t}\,c\,\mathrm dt=c. $ Apply this to $c=1$.

(Hence, the second integrability hypothesis is not needed.)

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    Hint: if $\lambda\gt\lambda^*$, there exists some finite $C$ such that $t\mathrm e^{-\lambda t}\leqslant C\mathrm e^{-\lambda^* t}$ for every nonnegative $t$.2012-08-03