Catoni-style Confidence Sequences under Infinite Variance

  title={Catoni-style Confidence Sequences under Infinite Variance},
  author={Sujay Bhatt and Guanhua Fang and P. Li and Gennady Samorodnitsky},
In this paper, we provide an extension of confidence sequences for settings where the variance of the data-generating distribution does not exist or is infinite. Confidence sequences furnish confidence intervals that are valid at arbitrary data-dependent stopping times, naturally having a wide range of applications. We first establish a lower bound for the width of the Catoni-style confidence sequences for the finite variance case to highlight the looseness of the existing results. Next, we derive… 
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  • D. DarlingH. Robbins
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1967
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  • L. DubinsL. J. Savage
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1965
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