Recursive moment formulas for regenerative simulation

Abstract

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 → ∞.