Estimation of integrals with respect to infinite measures using regenerative sequences

Abstract

Letfbe an integrable function on an infinite measure space (S,, π). We show that if aregenerative sequence{Xn}n≥0with canonical measureπcould 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 ℝdusing a simple symmetric random walk on ℤ.