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Uniform convergence in extended probability of sub-gradients of convex functions

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  • Kemp, Gordon C.R.

Abstract

It is well known that if a sequence of stochastic convex functions on Rd converges in probability point-wise to some non-stochastic function then the limit function is convex and the convergence is uniform on compact sets; see Andersen and Gill (1982) and Pollard (1991). In the present paper, I establish that if the limiting function is differentiable then any sequence of measurable sub-gradients of the stochastic convex functions converges in extended probability to the gradient of the limit function uniformly on compact sets.

Suggested Citation

  • Kemp, Gordon C.R., 2020. "Uniform convergence in extended probability of sub-gradients of convex functions," Economics Letters, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:ecolet:v:188:y:2020:i:c:s0165176519304100
    DOI: 10.1016/j.econlet.2019.108809
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    References listed on IDEAS

    as
    1. Maxwell B. Stinchcombe & Halbert White, 1992. "Some Measurability Results for Extrema of Random Functions Over Random Sets," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 59(3), pages 495-514.
    2. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, June.
    3. Pollard, David, 1991. "Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory, Cambridge University Press, vol. 7(2), pages 186-199, June.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Convex functions; Sub-gradients; Convergence in probability; Extended probability measure; Uniform convergence;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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