“…, k} such that θ 0,i = θ, the process Y (i ) satisfies Assumption (A) with the objects θ 0,i , j i , α i ,n , W i , ν i and η i such that j i ≡ 0, I i = {1} and η i > 0 on E i (note that we omit the second index for η i ,1 and ν i ,1 ). Many references provide practical criteria to check Assumption (A) with I S = {1}, j S ≡ 0, η S > 0 and with α S,n converging exponentially fast to 0, which corresponds to the classical irreducible situation, see for example [16,9,10,11,15,20,1,18,19,21,4]. The process Y (i ) is called the process X restricted to E i .…”