I have two very simple continuous-time Markov chains and defined on states by the following transition rates: For
and for
where 0<z<1 and the omitted transition rates are zero.
Assume that both chains start in state (0,1). Now, consider the random variable , i.e., the sum of the state indices i and j over time, for , and analogously.
Is it true that has the same distribution of for all t?
Is it true that ? (or ?)
Why?
Thanks