Past exam of the probability course of the University of Oxford / 2021 / 3 / c / i / Solution

DefineAn interior jump cannot go below $-c-b$ or above $c+a$, and every exterior reset lands in $I$. Thus $C$ is closed.

To prove that it is one communicating class, start anywhere in $C$. From $L$ or $U$, a reset can reach $I$ with positive probability. From any state of $I$, repeated $+a$ steps reach some state of $U$, after staying in $I$ up to the last step. A reset from this state can reach every point of the consecutive blockEvery integer $j\in \{-c-b,\dots ,c\}$ has the form $t-kb$ for some $t\in B$ and an integer $k\ge 0$. Repeated $-b$ steps from $t$ reach $j$, with all departure states in $I$ until a possible last step into $L$. This gives access to all of $I\cup L$. Finally, each $j\in U$ can be reached from $j-a\in I$. Every path described has strictly positive probability. Therefore every state in $C$ reaches every other state in $C$.

Each state outside $C$ can only stay put or reset into $I$, and no transition from $C$ returns to it. Different states outside $C$ cannot reach one another. Hence the complete list of communicating classes isThe finite class $C$ is the unique closed communicating class. The exterior singleton classes are transient communicating classes, since their self-transition probabilities are strictly less than one. Notice that this reachability proof does not require $a,b$ to be coprime.