# Uniform Convergence in Extended Probability of Sub-Gradients of Convex Functions

Kemp, Gordon (2020) 'Uniform Convergence in Extended Probability of Sub-Gradients of Convex Functions.' Economics Letters, 188. 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 Convex functions, sub-gradients, convergence in probability, extended probability measure, uniform convergence Faculty of Social Sciences > Economics, Department of Elements 18 Nov 2019 11:56 06 Feb 2020 13:15 http://repository.essex.ac.uk/id/eprint/25893