A new class of strongly consistent variance estimators for steady-state simulations (Q1110224)

Consider a stochastic process \(\{\) X(t): \(t\geq 0\}\) representing a steady-state simulation, i.e. \[ r(t)=t^{- 1}\int^{t}_{0}X_{(s)}ds\Rightarrow r\quad (as\quad t\to \infty), \] where \(\Rightarrow\) denotes weak convergence and r is a finite constant. If the process is even such that \(t^{1/2}(r(t)-r)\Rightarrow \sigma N(0,1)\) (as \(t\to \infty)\) for a positive constant \(\sigma\), then an asymptotic confidence interval for r can be easily established, provided there exists a consistent estimator \(s=s(t)\) of \(\sigma\). Assuming a strong approximation (with rates) for the process \(\{\) tr(t):t\(\geq 0\}\), the authors suggest some consistent estimators for \(\sigma\) as needed above. The latter estimators take advantage of the strong limit behaviour of the increments of Wiener processes. Conditions under which the assumed strong appoximation holds are briefly discussed. Finally, convergence rates of the new estimators are compared with those obtained when \(\sigma\) is estimated via the regenerative method of simulation.