Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 27K (Applied Probability)

The total jump rate is ${\lambda }_0$ at zero and ${\lambda }_i(1+{\rho }_i)$ at $i>0$. Hence the jump chain has transition probabilitiesand, for $i>0$,All other entries vanish.

A reversible Markov chain must satisfy detailed balance. For every $i\ge 1$, the rate from $i$ to $i+1$ is positive, while the reverse rate from $i+1$ to $i$ is zero. No positive measure can satisfy ${\pi }_i{g}_{i,i+1}={\pi }_{i+1}{g}_{i+1,i}$, so the irreducible chain is not reversible.

After leaving zero, the jump chain avoids every future reset with probabilityThis probability is positive exactly when ${\prod }_{i\ge 1}(1+{\rho }_i)<\infty$. If $q>0$, the probability of ever returning to zero is less than one, so the irreducible chain is transient. If $q=0$, a reset to zero occurs almost surely after each departure; by the Strong Markov property this happens repeatedly, so the chain is recurrent. Therefore

Under the transience assumption, the number of returns to zero is geometrically distributed and hence finite almost surely. The final excursion is a pure ascent through all sufficiently large states. Its holding times are independent exponential random variables with meansThe nonexplosion criterion for a pure birth process says that their sum is finite almost surely exactly when the sum of their means is finite. Thus

Let $m$ be a nonzero invariant measure. Balance at state $i\ge 1$ givesand hence, after choosing $m_0>0$,Balance at zero requiresThe sum telescopes:Consequently a nonzero invariant measure exists exactly whenIt can be normalized to an invariant probability distribution exactly when, in addition,where the empty product for $i=0$ is one and an overall constant has been suppressed.