Delayed Versus Non-delayed Regenerations. Let X˜(t) be a realvalued delayed regenerative process over Tn. Then X(t) = X˜(T1+t), t ≥ 0 is a regenerative process. Assuming X˜ (t) is bounded, show that...

Delayed Versus Non-delayed Regenerations. Let X˜(t) be a realvalued delayed regenerative process over Tn. Then X(t) = X˜(T1+t), t ≥ 0 is a regenerative process. Assuming X˜ (t) is bounded, show that if limt→∞ E[X(t)] exists (such as by Theorem 45), then E[X˜ (t)] has the same limit. Hint: Take the limit as t → ∞ of