Estimation of integrals with respect to infinite measures using regenerative sequences

Let f be an integrable function on an infinite measure space (S, , π). We show that if a regenerative sequence {Xn } n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫ S f dπ can be produced. We further show that under appropriate second moment conditions, a confide...

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Bibliographic Details
Published in:Journal of applied probability 2015-12, Vol.52 (4), p.1133-1145
Main Authors: Athreya, Krishna B., Roy, Vivekananda
Format: Article
Language:English
Subjects:
60F05
65C05
Confidence intervals
Estimating techniques
Estimators
improper target
Integrals
Markov chain
Mathematical functions
Monte Carlo
Random walk
Regenerative
regenerative sequence
Research Papers
Studies
Sums
Symbols
Symmetry
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Summary:Let f be an integrable function on an infinite measure space (S, , π). We show that if a regenerative sequence {Xn } n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫ S f dπ 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.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1450802757