On a central limit theorem in renewal theory (Q6067032)

Let \(T_{0}=0\) and \(T_{n}=\sum_{i=1}^{n}X_{i}\), where \((X_{i})\) are i.i.d. positive random variables. Let \(N(t)=\max (n:T_{n}\leq t)\) and \(\tau (t)=\min (n:T_{n}>t)=N(t)+1\). Clearly we have \(T_{N(t)}\leq t<T_{\tau (t)}\). Let \(Z(.)\) denote another stochastic process that ``starts again'' at times \( T_{n}\), and assume that \(Z(t)\) has regenerative increments in the sense that \(\left\{ X_{n},Z(t=t+T_{n-1})-Z(T_{n-1}),0\leq t\leq X_{n}\right\} \) are i.i.d. Define \(\eta _{n}=Z(T_{n})-Z(T_{n-1})\),\(V_{n}=Z(T_{n})\) and assume \( M_{1}=\sup_{0\leq t\leq T_{1}}\left\vert Z(t)\right\vert <\infty \) \(a.s.\). In this paper the author proves a variety of theorems related to \(Z(t)\). Under suitable assumptions we have a central limit theorem (Th.1.4), convergence of moments (Th. 3.1), law of large numbers (Th.4.1), asymptotics of \(E(Z(t))\) (Th. 5.1, Th. 5.4). The paper is well written with clear proofs.