Recursive moment formulas for regenerative simulation

Let f be a real-valued function defined on the state space of a regenerative process \( \mathop X\limits_ \eqsim = \left\{ {X\left( t \right):t \geqslant 0} \right\}\) with regeneration times 0 = T0 < T1 <..., and suppose that $$ {r_t} = \frac{1}{t}\int_0^t {f\left( {X\left( s \right)} \right)ds \to r\quad a.s.} $$ (1.1) as t → ∞.