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Comparing weak and strong convergence of density functions

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  • Walker, Stephen G.

Abstract

Weak convergence of a distribution does not imply the density converges with respect to the L1 metric. We prove that strong convergence can be established by showing that a smoothed and non-smoothed sequence of the densities converge to each other.

Suggested Citation

  • Walker, Stephen G., 2023. "Comparing weak and strong convergence of density functions," Statistics & Probability Letters, Elsevier, vol. 200(C).
    • Handle: RePEc:eee:stapro:v:200:y:2023:i:c:s0167715223001025
    DOI: 10.1016/j.spl.2023.109878
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    References listed on IDEAS

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    1. Sylvia. Richardson & Peter J. Green, 1997. "On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(4), pages 731-792.
    2. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265, November.
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    Keywords

    Kernel density; Modes; Smoothing;
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