Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 26I / f

Here $\vartheta <1$ is equivalent to $\mu \alpha <\lambda \beta$, which gives $b<1$ from its explicit formula. Any nonnegative solution is of the formsince its second component at level zero must be one. Letting $i\to \infty$ forces $a\ge 0$ for nonnegativity. The choice $a=0$ is nonnegative, and subtracting it from any other such solution gives $a[(1,1{)}^T-{\vartheta }^i(b,1{)}^T]\ge 0$. Therefore minimality selectsIn particular the return probability after leaving $W_0$ is $b<1$, so the irreducible chain is transient. These formulas are the hitting probabilities for a velocity-switching Markov ladder.