Estimation of integrals with respect to infinite measures using regenerative sequences

Abstract

Let f be an integrable function on an infinite measure space (S, S, π). We show that if a regenerative sequence {X_n}_n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫_Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on R^d using a simple symmetric random walk on Z.