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