Kemp, Gordon (2020) 'Uniform Convergence in Extended Probability of Sub-Gradients of Convex Functions.' Economics Letters, 188. p. 108809. ISSN 0165-1765
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Abstract
It is well known that if a sequence of stochastic convex functions on $\mathbb{R}^{d}$ 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.
Item Type: | Article |
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Uncontrolled Keywords: | Convex functions, sub-gradients, convergence in probability, extended probability measure, uniform convergence |
Divisions: | Faculty of Social Sciences Faculty of Social Sciences > Economics, Department of |
SWORD Depositor: | Elements |
Depositing User: | Elements |
Date Deposited: | 18 Nov 2019 11:56 |
Last Modified: | 18 Aug 2022 12:20 |
URI: | http://repository.essex.ac.uk/id/eprint/25893 |
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